Hydrodynamic and phoretic interactions of active particles in Python

This talk is about the Python library PyStokes and its research applications. PyStokes is freely available on GitHub: https://github.com/rajeshrinet/pystokes

Abstract: In this talk, I present a modular Python library for computing many-body hydrodynamic and phoretic interactions between spherical active particles in suspension when these are given by solutions of the Stokes and Laplace equations. Underpinning the library is a grid-free methodology that combines dimensionality reduction, spectral expansion, and Ritz-Galerkin discretization, thereby reducing the computation to the solution of a linear system. The system can be solved analytically as a series expansion or numerically at a cost quadratic in the number of particles. The library is agnostic to boundary conditions and includes, amongst others, confinement by plane walls or liquid-liquid interfaces. The use of the library is demonstrated with examples in experimentally relevant settings.

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Talk presented in Edwards Meeting, 10th January 2020

Transcript

1. m m m https://github.com/rajeshrinet/pystokes Hydrodynamic and phoretic interactions of active particles in Python no-shear interface crystallization at a no-slip wall Convective rolls of active particles in a harmonic trap experiment & theory electrohydrodynamic flow in a rectangular geometry arrested clusters at a wall In-silico experiments with active matter emergent optofluidic potential bound states of two active particles
2. m m m https://github.com/rajeshrinet/pystokes Hydrodynamic and phoretic interactions of active particles in Python no-shear interface crystallization at a no-slip wall Convective rolls of active particles in a harmonic trap experiment & theory electrohydrodynamic flow in a rectangular geometry arrested clusters at a wall In-silico experiments with active matter emergent optofluidic potential bound states of two active particles Rajesh Singh Postdoctoral researcher Department of applied mathematics and theoretical physics (DAMTP) University of Cambridge Collaborators R. Adhikari, M. E. Cates, A. Caciagli and E. Eiser Cavendish Laboratory DAMTP
3. Active colloids • Microorganisms Gollub lab, PRL 2010; Goldstein lab, PRL 2009 spontaneous symmetry breaking • Self-propelling droplets Thutupalli et al NJP 2011 self-propulsion by chemical asymmetry • Synthetic microswimmers Palacci et al Science 2013; Ebbens and Howse SM 2010
4. Active colloids • Microorganisms Gollub lab, PRL 2010; Goldstein lab, PRL 2009 spontaneous symmetry breaking • Self-propelling droplets Thutupalli et al NJP 2011 self-propulsion by chemical asymmetry • Synthetic microswimmers Palacci et al Science 2013; Ebbens and Howse SM 2010 non-equilibrium processes on the surface drive exterior fluid flow, even when the colloid is stationary the fluid stress may react back and cause self-propulsion in absence of external forces and torques fluid flow mediates long-range hydrodynamic interactions (HI) universal mechanisms due to the scale-separation for propulsion
5. Ideal active colloid A spherical colloid with slip boundary condition boundary velocity = rigid body motion + active slip v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i )
6. Electrophoretic slip E Charged colloid in a dielectric fluid V = qE 6⇡⌘b Size-dependent velocity: Long-ranged flow v ⇠ 1 r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989)
7. Electrophoretic slip E Charged colloid in a dielectric fluid V = qE 6⇡⌘b Size-dependent velocity: Long-ranged flow v ⇠ 1 r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) v ⇠ 1 r3 Charged colloid in an electrolytic fluid Colloidal charge is balanced by a diffused cloud of counter ions
8. Electrophoretic slip E Charged colloid in a dielectric fluid V = qE 6⇡⌘b Size-dependent velocity: Long-ranged flow v ⇠ 1 r Slip flow Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) v ⇠ 1 r3 Charged colloid in an electrolytic fluid Colloidal charge is balanced by a diffused cloud of counter ions
9. Electrophoretic slip E Charged colloid in a dielectric fluid V = qE 6⇡⌘b Size-dependent velocity: Long-ranged flow v ⇠ 1 r Slip flow Electrophoretic slip vA = ✏⇣ 4⇡⌘ Es <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">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</latexit> <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">AAACJXicbVDLSgMxFM34rPVVdekmWAQ3lhkp6EKhKoLLCvYBnbZk0jttaCYzJJlCHeZn3PgrblxYRHDlr5g+Ftp6IHA451xy7/EizpS27S9raXlldW09s5Hd3Nre2c3t7VdVGEsKFRryUNY9ooAzARXNNId6JIEEHoea178d+7UBSMVC8aiHETQD0hXMZ5RoI7Vzl4NW4gZE9yjh+DrFV/jU9SWhiQuRYjwU7hNogtOk6EbMNTSdpD0f37VUO5e3C/YEeJE4M5JHM5TbuZHbCWkcgNCUE6Uajh3pZkKkZpRDmnVjBRGhfdKFhqGCBKCayeTKFB8bpYP9UJonNJ6ovycSEig1DDyTHK+o5r2x+J/XiLV/0UyYiGINgk4/8mOOdYjHleEOk0A1HxpCqGRmV0x7xJSkTbFZU4Izf/IiqZ4VHLvgPBTzpZtZHRl0iI7QCXLQOSqhe1RGFUTRM3pF72hkvVhv1of1OY0uWbOZA/QH1vcP54qlgQ==</latexit> <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">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</latexit> <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">AAACJXicbVDLSgMxFM34rPVVdekmWAQ3lhkp6EKhKoLLCvYBnbZk0jttaCYzJJlCHeZn3PgrblxYRHDlr5g+Ftp6IHA451xy7/EizpS27S9raXlldW09s5Hd3Nre2c3t7VdVGEsKFRryUNY9ooAzARXNNId6JIEEHoea178d+7UBSMVC8aiHETQD0hXMZ5RoI7Vzl4NW4gZE9yjh+DrFV/jU9SWhiQuRYjwU7hNogtOk6EbMNTSdpD0f37VUO5e3C/YEeJE4M5JHM5TbuZHbCWkcgNCUE6Uajh3pZkKkZpRDmnVjBRGhfdKFhqGCBKCayeTKFB8bpYP9UJonNJ6ovycSEig1DDyTHK+o5r2x+J/XiLV/0UyYiGINgk4/8mOOdYjHleEOk0A1HxpCqGRmV0x7xJSkTbFZU4Izf/IiqZ4VHLvgPBTzpZtZHRl0iI7QCXLQOSqhe1RGFUTRM3pF72hkvVhv1of1OY0uWbOZA/QH1vcP54qlgQ==</latexit> Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) v ⇠ 1 r3 Charged colloid in an electrolytic fluid Colloidal charge is balanced by a diffused cloud of counter ions
10. Thermophoretic slip Response of a colloid to an inhomogeneous temperature field. Anderson, Ann Rev Fluid Mech (1989) v✓(⇢) = µ✓(⇢)rs ✓(⇢). <latexit sha1_base64="j/Ci2ekGBiqz0BmFBVZAZZoofkM=">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</latexit> <latexit sha1_base64="j/Ci2ekGBiqz0BmFBVZAZZoofkM=">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</latexit> <latexit sha1_base64="j/Ci2ekGBiqz0BmFBVZAZZoofkM=">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</latexit> <latexit sha1_base64="j/Ci2ekGBiqz0BmFBVZAZZoofkM=">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</latexit>
11. Ebbens and Howse, Soft matter 2010 vA(⇢i ) = µc(⇢i )rs c(⇢i ), Autophoretic colloids: chemical slip Response of a colloid to an inhomogeneous distribution of solutes
12. Summary: Phoretic motion ‣ Local field gradients leads to interfacial slip flows ‣ These gradients can be of different origin: chemical, electrical, thermal, etc. It can also be due to squirming motion ‣ The gradients can be external or self-generated ‣ The slip velocity can also be of biological origin (microorganisms) ‣ How to derive universal mechanisms for a given slip? ‣ What is the role of boundaries in the flow? ‣ How is the slip modified by phoretic interactions?
13. Galerkin expansion of the slip boundary velocity = rigid body motion + active slip v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i )
14. Galerkin expansion of the slip boundary velocity = rigid body motion + active slip Expanding the slip in the basis of tensorial spherical harmonics Y(l) vA = 1 X l=1 1 (l 1)!(2l 3)!! V(l) · Y(l 1)(ˆ ⇢) v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i )
15. Galerkin expansion of the slip boundary velocity = rigid body motion + active slip Expanding the slip in the basis of tensorial spherical harmonics Y(l) vA = 1 X l=1 1 (l 1)!(2l 3)!! V(l) · Y(l 1)(ˆ ⇢) Y(l) are dimensionless, symmetric, irreducible Cartesian tensors of rank l that form a complete, orthogonal basis on the sphere Y(l)(ˆ ⇢) = ( 1)l⇢l+1r(l)⇢ 1 v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i )
16. Galerkin expansion of the slip boundary velocity = rigid body motion + active slip Expanding the slip in the basis of tensorial spherical harmonics Y(l) vA = 1 X l=1 1 (l 1)!(2l 3)!! V(l) · Y(l 1)(ˆ ⇢) Y(l) are dimensionless, symmetric, irreducible Cartesian tensors of rank l that form a complete, orthogonal basis on the sphere Y(l)(ˆ ⇢) = ( 1)l⇢l+1r(l)⇢ 1 v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) Y (0) = 1, Y (1) ↵ = ˆ ⇢↵, Y (2) ↵ = ⇣ ˆ ⇢↵ ˆ ⇢ ↵ 3 ⌘ , Y (3) ↵ = ˆ ⇢↵ ˆ ⇢ ˆ ⇢ 1 5 [ˆ ⇢↵ + ˆ ⇢ ↵ + ˆ ⇢ ↵ ] .
17. vA ⇠ X V(l) · Y(l 1)
18. vA ⇠ X V(l) · Y(l 1)
19. vA ⇠ X V(l) · Y(l 1) V(l) = V(ls) + V(la) + V(lt) symmetric antisymmetric trace
20. l v(l )⇠ ( r(l 1)G ) 1 2 3 vA ⇠ X V(l) · Y(l 1) V(l) = V(ls) + V(la) + V(lt) symmetric antisymmetric trace
21. l v(l )⇠ ( r(l 1)G ) 1 2 3 Symmetric v(l )⇠ ( r(l 1)G )r(l 2)(r ⇥ G) vA ⇠ X V(l) · Y(l 1) V(l) = V(ls) + V(la) + V(lt) symmetric antisymmetric trace
22. l v(l )⇠ ( r(l 1)G ) 1 2 3 Symmetric v(l )⇠ ( r(l 1)G )r(l 2)(r ⇥ G) Antisymmetric ⇠ ( r(l 1)G )r(l 2)(r ⇥ G)( r(l 3)(r2G) ) vA ⇠ X V(l) · Y(l 1) V(l) = V(ls) + V(la) + V(lt) symmetric antisymmetric trace
23. l v(l )⇠ ( r(l 1)G ) 1 2 3 Symmetric v(l )⇠ ( r(l 1)G )r(l 2)(r ⇥ G) Antisymmetric ⇠ ( r(l 1)G )r(l 2)(r ⇥ G)( r(l 3)(r2G) ) Trace r(l 2)(r ⇥ G)( r(l 3)(r2G) ) vA ⇠ X V(l) · Y(l 1) V(l) = V(ls) + V(la) + V(lt) symmetric antisymmetric trace
24. l v(l )⇠ ( r(l 1)G ) 1 2 3 Symmetric v(l )⇠ ( r(l 1)G )r(l 2)(r ⇥ G) Antisymmetric ⇠ ( r(l 1)G )r(l 2)(r ⇥ G)( r(l 3)(r2G) ) Trace r(l 2)(r ⇥ G)( r(l 3)(r2G) ) vA ⇠ X V(l) · Y(l 1) in an unbounded domain of fluid flow v / r l, for slip V(l ) flow from the l-th mode has three independent terms: (a) symmetric irreducible gradients of G (b) its curl and, (c) its Laplacian V(l) = V(ls) + V(la) + V(lt) symmetric antisymmetric trace
26. Drescher et al PNAS 2011
27. Drescher et al PNAS 2011 Herminghaus et al Soft Matter
28. vA i (⇢i ) = VA i + 1 15 V(3t) i · Y(2)(⇢i ) + V(2s) i · Y(1)(⇢i ) 1 75 V(4t) i · Y(3)(⇢i ). S. Thutupalli, D. Geyer, RS, R. Adhikari, and H. A. Stone, PNAS 2018 Experiments: Fluid flow of 2s + 3t + 4t mode
29. vA i (⇢i ) = VA i + 1 15 V(3t) i · Y(2)(⇢i ) + V(2s) i · Y(1)(⇢i ) 1 75 V(4t) i · Y(3)(⇢i ). Experimental flow Theoretical flow S. Thutupalli, D. Geyer, RS, R. Adhikari, and H. A. Stone, PNAS 2018 Experiments: Fluid flow of 2s + 3t + 4t mode
30. friction tensors for slip slip coefficients FH i = T T ij ·V i T R ij ·⌦ i 1 X l =1s (T, l ) ij · V(l ) j , TH i = RT ij ·V i RR ij ·⌦ i 1 X l =1s (R, l ) ij · V(l ) j . Generalized Stokes laws The forces depend on the positions of colloids through the friction tensors obtained in terms of a Green’s function of Stoke’s equation RS and Adhikari, PRL 2016
31. friction tensors for slip slip coefficients FH i = T T ij ·V i T R ij ·⌦ i 1 X l =1s (T, l ) ij · V(l ) j , TH i = RT ij ·V i RR ij ·⌦ i 1 X l =1s (R, l ) ij · V(l ) j . Generalized Stokes laws The forces depend on the positions of colloids through the friction tensors obtained in terms of a Green’s function of Stoke’s equation FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic We use the above in Newton’s laws to obtain the rigid body motion RS and Adhikari, PRL 2016
32. Rigid body motion of active colloids T T ij ·Vj T R ij · ⌦j + FP i + ˆ Fi 1 X l =1s (T, l ) ij · V(l ) j = 0, RT ij ·Vj RR ij · ⌦j + TP i + ˆ Ti 1 X l =1s (R, l ) ij · V(l ) j = 0. RS and Adhikari, EJCM 2017, JPC 2018
33. Rigid body motion of active colloids T T ij ·Vj T R ij · ⌦j + FP i + ˆ Fi 1 X l =1s (T, l ) ij · V(l ) j = 0, RT ij ·Vj RR ij · ⌦j + TP i + ˆ Ti 1 X l =1s (R, l ) ij · V(l ) j = 0. Invert for rigid body motion RS and Adhikari, EJCM 2017, JPC 2018
34. Rigid body motion of active colloids T T ij ·Vj T R ij · ⌦j + FP i + ˆ Fi 1 X l =1s (T, l ) ij · V(l ) j = 0, RT ij ·Vj RR ij · ⌦j + TP i + ˆ Ti 1 X l =1s (R, l ) ij · V(l ) j = 0. Propulsion tensors relate modes of slip to rigid body motion White noises Mobility matrices connectors for forces and torques Vi = µT T ij · FP j + µT R ij · TP j + q 2kBTµT T ij · ⌘T j + q 2kBTµT R ij · ⇣R j + 1 X l =2s ⇡(T, l ) ij · V(l ) j + VA i ⌦i = µRT ij · FP j + µRR ij · TP j | {z } Passive + q 2kBTµRT ij · ⇣T j + q 2kBTµRR ij · ⌘R j | {z } Brownian + 1 X l =2s ⇡(R, l ) ij · V(l ) j + ⌦A i | {z } Active Invert for rigid body motion RS and Adhikari, EJCM 2017, JPC 2018
35. Rigid body motion of active colloids T T ij ·Vj T R ij · ⌦j + FP i + ˆ Fi 1 X l =1s (T, l ) ij · V(l ) j = 0, RT ij ·Vj RR ij · ⌦j + TP i + ˆ Ti 1 X l =1s (R, l ) ij · V(l ) j = 0. Propulsion tensors relate modes of slip to rigid body motion White noises Mobility matrices connectors for forces and torques Vi = µT T ij · FP j + µT R ij · TP j + q 2kBTµT T ij · ⌘T j + q 2kBTµT R ij · ⇣R j + 1 X l =2s ⇡(T, l ) ij · V(l ) j + VA i ⌦i = µRT ij · FP j + µRR ij · TP j | {z } Passive + q 2kBTµRT ij · ⇣T j + q 2kBTµRR ij · ⌘R j | {z } Brownian + 1 X l =2s ⇡(R, l ) ij · V(l ) j + ⌦A i | {z } Active ⇡(T, l ) ij = µT T ik · (T, l ) kj + µT R ik · (R, l ) kj , ⇡(R, l ) ij = µRT ik · (T, l ) kj + µRR ik · (R, l ) kj . Invert for rigid body motion RS and Adhikari, EJCM 2017, JPC 2018
37. active particles which swim into a wall, induced a monopole, when stalled by it. This leads to an attractive flow causing crystallization RS and Adhikari PRL 2016.
38. Faster time-scales at a no-shear surface when compared to a no-slip wall (Thutupalli et al PNAS 2018). viscosity ratio
42. RS and Adhikari PRL 2016, Squires JFM 2001
43. RS and Adhikari PRL 2016, Squires JFM 2001
44. Colloidal tethered to an interface - free to move in the plane Caciagli, RS, Joshi, Adhikari and Eiser, arXiv:2003.04284
45. Caciagli, RS, Joshi, Adhikari and Eiser, under review Optically trap one of the colloids and study the optofluidic interactions
46. Water Oil t=0.00s t=1.05s t=1.10s t=1.15s t=1.20s t=0.00s t=0.45s t=0.55s t=0.75s t=0.85s Caciagli, RS, Joshi, Adhikari and Eiser, arXiv:2003.04284; RS and Adhikari PRL 2016; Squires and Brenner PRL 2000
47. Water Oil t=0.00s t=1.05s t=1.10s t=1.15s t=1.20s t=0.00s t=0.45s t=0.55s t=0.75s t=0.85s F✓ 1 <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> Thermophoresis: quadrupolar flow Caciagli, RS, Joshi, Adhikari and Eiser, arXiv:2003.04284; RS and Adhikari PRL 2016; Squires and Brenner PRL 2000
48. Water Oil t=0.00s t=1.05s t=1.10s t=1.15s t=1.20s t=0.00s t=0.45s t=0.55s t=0.75s t=0.85s F✓ 1 <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> Thermophoresis: quadrupolar flow FP 1 <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> Monopolar flow once the colloid is stalled Caciagli, RS, Joshi, Adhikari and Eiser, arXiv:2003.04284; RS and Adhikari PRL 2016; Squires and Brenner PRL 2000
49. Water Oil Flow-induced attraction FH FH 0 1 t=0.00s t=1.05s t=1.10s t=1.15s t=1.20s t=0.00s t=0.45s t=0.55s t=0.75s t=0.85s F✓ 1 <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> Thermophoresis: quadrupolar flow FP 1 <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> Monopolar flow once the colloid is stalled Caciagli, RS, Joshi, Adhikari and Eiser, arXiv:2003.04284; RS and Adhikari PRL 2016; Squires and Brenner PRL 2000
50. Water Oil Flow-induced attraction FH FH 0 1 t=0.00s t=1.05s t=1.10s t=1.15s t=1.20s t=0.00s t=0.45s t=0.55s t=0.75s t=0.85s F✓ 1 <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> Thermophoresis: quadrupolar flow FP 1 <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> Monopolar flow once the colloid is stalled Caciagli, RS, Joshi, Adhikari and Eiser, arXiv:2003.04284; RS and Adhikari PRL 2016; Squires and Brenner PRL 2000
51. Water Oil Flow-induced attraction FH FH 0 1 t=0.00s t=1.05s t=1.10s t=1.15s t=1.20s t=0.00s t=0.45s t=0.55s t=0.75s t=0.85s F✓ 1 <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> <latexit sha1_base64="/OAjlDRJNddmkS90Od74rveYZZc=">AAAB/nicbVDLSsNAFJ34rPVVFVduBovgqiQi6LIoiMsK9gFNDJPppB06mYSZG6GEgL/ixoUibv0Od/6NkzYLbT0wcDjnXu6ZEySCa7Dtb2tpeWV1bb2yUd3c2t7Zre3td3ScKsraNBax6gVEM8ElawMHwXqJYiQKBOsG4+vC7z4ypXks72GSMC8iQ8lDTgkYya8duhGBURBmN7nvPGQujBiQ3K/V7YY9BV4kTknqqETLr325g5imEZNABdG679gJeBlRwKlgedVNNUsIHZMh6xsqScS0l03j5/jEKAMcxso8CXiq/t7ISKT1JArMZBFWz3uF+J/XTyG89DIukxSYpLNDYSowxLjoAg+4YhTExBBCFTdZMR0RRSiYxqqmBGf+y4ukc9Zw7IZzd15vXpV1VNAROkanyEEXqIluUQu1EUUZekav6M16sl6sd+tjNrpklTsH6A+szx+XWZXh</latexit> Thermophoresis: quadrupolar flow FP 1 <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> Monopolar flow once the colloid is stalled Caciagli, RS, Joshi, Adhikari and Eiser, arXiv:2003.04284; RS and Adhikari PRL 2016; Squires and Brenner PRL 2000
52. Petroff et al: Fast-Moving Bacteria Self-Organize into Active Two-Dimensional Crystals of Rotating Cells PRL 2015
53. Petroff et al: Fast-Moving Bacteria Self-Organize into Active Two-Dimensional Crystals of Rotating Cells PRL 2015
54. What about phoretic interactions?
55. Known chemical surface flux jA = D⇢ · rc <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> RS, Adhikari, and Cates JCP 2019
56. Known chemical surface flux Coordinate system Water Plane wall: no-slip of the fluid flow and no-flux of chemical x <latexit sha1_base64="MPCBDNMNYRvSq3NGxeROFM5i9Uc=">AAAB9HicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoMuiG5cV7APaoWQymTY0k4xJpliGfocbF4q49WPc+Tdm2llo64GQwzn3kpMTJJxp47rfTmltfWNzq7xd2dnd2z+oHh61tUwVoS0iuVTdAGvKmaAtwwyn3URRHAecdoLxbe53JlRpJsWDmSbUj/FQsIgRbKzk9wPJQz2N7YWeBtWaW3fnQKvEK0gNCjQH1a9+KEkaU2EIx1r3PDcxfoaVYYTTWaWfappgMsZD2rNU4JhqP5uHnqEzq4QoksoeYdBc/b2R4VjnyexkjM1IL3u5+J/XS0107WdMJKmhgiweilKOjER5AyhkihLDp5ZgopjNisgIK0yM7aliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFBB7hGV7hzZk4L86787EYLTnFzjH8gfP5A+Dxkic=</latexit> <latexit 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sha1_base64="22K8pTQwJQiQxeeQ5U5CBHP9N0M=">AAAB9HicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoMuiG5cV7APaoWQymTY0k0yTTKEM/Q43LhRx68e482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT5pa5kqQltEcqm6AdaUM0FbhhlOu4miOA447QTj+9zvTKnSTIonM0uoH+OhYBEj2FjJ7weSh3oW2wuNBtWaW3cXQOvEK0gNCjQH1a9+KEkaU2EIx1r3PDcxfoaVYYTTeaWfappgMsZD2rNU4JhqP1uEnqMLq4QoksoeYdBC/b2R4VjnyexkjM1Ir3q5+J/XS01062dMJKmhgiwfilKOjER5AyhkihLDZ5ZgopjNisgIK0yM7aliS/BWv7xO2ld1z617j9e1xl1RRxnO4BwuwYMbaMADNKEFBCbwDK/w5kydF+fd+ViOlpxi5xT+wPn8Acixkhc=</latexit> <latexit sha1_base64="22K8pTQwJQiQxeeQ5U5CBHP9N0M=">AAAB9HicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoMuiG5cV7APaoWQymTY0k0yTTKEM/Q43LhRx68e482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT5pa5kqQltEcqm6AdaUM0FbhhlOu4miOA447QTj+9zvTKnSTIonM0uoH+OhYBEj2FjJ7weSh3oW2wuNBtWaW3cXQOvEK0gNCjQH1a9+KEkaU2EIx1r3PDcxfoaVYYTTeaWfappgMsZD2rNU4JhqP1uEnqMLq4QoksoeYdBC/b2R4VjnyexkjM1Ir3q5+J/XS01062dMJKmhgiwfilKOjER5AyhkihLDZ5ZgopjNisgIK0yM7aliS/BWv7xO2ld1z617j9e1xl1RRxnO4BwuwYMbaMADNKEFBCbwDK/w5kydF+fd+ViOlpxi5xT+wPn8Acixkhc=</latexit> jA = D⇢ · rc <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> RS, Adhikari, and Cates JCP 2019
57. Known chemical surface flux jA = D⇢ · rc <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> Boundary condition Differential laws Chemical problem r · j = 0 j = Drc <latexit 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sha1_base64="22K8pTQwJQiQxeeQ5U5CBHP9N0M=">AAAB9HicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoMuiG5cV7APaoWQymTY0k0yTTKEM/Q43LhRx68e482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT5pa5kqQltEcqm6AdaUM0FbhhlOu4miOA447QTj+9zvTKnSTIonM0uoH+OhYBEj2FjJ7weSh3oW2wuNBtWaW3cXQOvEK0gNCjQH1a9+KEkaU2EIx1r3PDcxfoaVYYTTeaWfappgMsZD2rNU4JhqP1uEnqMLq4QoksoeYdBC/b2R4VjnyexkjM1Ir3q5+J/XS01062dMJKmhgiwfilKOjER5AyhkihLDZ5ZgopjNisgIK0yM7aliS/BWv7xO2ld1z617j9e1xl1RRxnO4BwuwYMbaMADNKEFBCbwDK/w5kydF+fd+ViOlpxi5xT+wPn8Acixkhc=</latexit> jA = D⇢ · rc <latexit 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sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> RS, Adhikari, and Cates JCP 2019
58. Known chemical surface flux Chemo-hydrodynamic coupling vA(⇢) = µc(⇢)rs c(⇢), <latexit sha1_base64="YfuLFv7dfPJ2/X7xzzU02PY+5Ck=">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</latexit> <latexit 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sha1_base64="YfuLFv7dfPJ2/X7xzzU02PY+5Ck=">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</latexit> jA = D⇢ · rc <latexit 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sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> Boundary condition Differential laws Chemical problem r · j = 0 j = Drc <latexit 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sha1_base64="UgI5vWDsWiyvN4uaIL50dpnqIgU=">AAACO3icbVDLSgMxFM3UVx1foy7dBIvgxjIjgm4KRV24rGIf0Cklk0nb2EwyJBmhDP0vN/6EOzduXCji1r2Ztou+LoQczrmHe+8JYkaVdt13K7eyura+kd+0t7Z3dvec/YOaEonEpIoFE7IRIEUY5aSqqWakEUuCooCRetC/yfT6M5GKCv6oBzFpRajLaYdipA3Vdh78QLBQDSLzpT5HAUNDH4dCT/NPQ7vk2r5vz5Kls9slbtx2Cm7RHRVcBN4EFMCkKm3nzQ8FTiLCNWZIqabnxrqVIqkpZmRo+4kiMcJ91CVNAzmKiGqlo9uH8MQwIewIaR7XcMROO1IUqWxB0xkh3VPzWkYu05qJ7ly1UsrjRBOOx4M6CYNawCxIGFJJsGYDAxCW1OwKcQ9JhLWJ2zYhePMnL4LaedFzi979RaF8PYkjD47AMTgFHrgEZXAHKqAKMHgBH+ALfFuv1qf1Y/2OW3PWxHMIZsr6+wcot6/t</latexit> <latexit sha1_base64="UgI5vWDsWiyvN4uaIL50dpnqIgU=">AAACO3icbVDLSgMxFM3UVx1foy7dBIvgxjIjgm4KRV24rGIf0Cklk0nb2EwyJBmhDP0vN/6EOzduXCji1r2Ztou+LoQczrmHe+8JYkaVdt13K7eyura+kd+0t7Z3dvec/YOaEonEpIoFE7IRIEUY5aSqqWakEUuCooCRetC/yfT6M5GKCv6oBzFpRajLaYdipA3Vdh78QLBQDSLzpT5HAUNDH4dCT/NPQ7vk2r5vz5Kls9slbtx2Cm7RHRVcBN4EFMCkKm3nzQ8FTiLCNWZIqabnxrqVIqkpZmRo+4kiMcJ91CVNAzmKiGqlo9uH8MQwIewIaR7XcMROO1IUqWxB0xkh3VPzWkYu05qJ7ly1UsrjRBOOx4M6CYNawCxIGFJJsGYDAxCW1OwKcQ9JhLWJ2zYhePMnL4LaedFzi979RaF8PYkjD47AMTgFHrgEZXAHKqAKMHgBH+ALfFuv1qf1Y/2OW3PWxHMIZsr6+wcot6/t</latexit> c ! 0, as r ! 1 <latexit 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sha1_base64="crCRNfo1kaSoj+xYqg7P2tC7uEw=">AAACGnicbVBNS8NAEN34WetX1KOXxSJ4kJKIoMeiF48V7Ac0oWy2m3bpZhN2J2oI/R1e/CtePCjiTbz4b9y2OdjWBwOP92aYmRckgmtwnB9raXlldW29tFHe3Nre2bX39ps6ThVlDRqLWLUDopngkjWAg2DtRDESBYK1guH12G/dM6V5LO8gS5gfkb7kIacEjNS1XYo9xfsDIErFD9g5xR6wR8gx0XiE1YzpcRlC1rUrTtWZAC8StyAVVKDetb+8XkzTiEmggmjdcZ0E/Jwo4FSwUdlLNUsIHZI+6xgqScS0n09eG+Fjo/RwGCtTEvBE/TuRk0jrLApMZ0RgoOe9sfif10khvPRzLpMUmKTTRWEqMMR4nBPuccUoiMwQQhU3t2I6IIpQMGmWTQju/MuLpHlWdZ2qe3teqV0VcZTQITpCJ8hFF6iGblAdNRBFT+gFvaF369l6tT6sz2nrklXMHKAZWN+/cdCggw==</latexit> <latexit sha1_base64="crCRNfo1kaSoj+xYqg7P2tC7uEw=">AAACGnicbVBNS8NAEN34WetX1KOXxSJ4kJKIoMeiF48V7Ac0oWy2m3bpZhN2J2oI/R1e/CtePCjiTbz4b9y2OdjWBwOP92aYmRckgmtwnB9raXlldW29tFHe3Nre2bX39ps6ThVlDRqLWLUDopngkjWAg2DtRDESBYK1guH12G/dM6V5LO8gS5gfkb7kIacEjNS1XYo9xfsDIErFD9g5xR6wR8gx0XiE1YzpcRlC1rUrTtWZAC8StyAVVKDetb+8XkzTiEmggmjdcZ0E/Jwo4FSwUdlLNUsIHZI+6xgqScS0n09eG+Fjo/RwGCtTEvBE/TuRk0jrLApMZ0RgoOe9sfif10khvPRzLpMUmKTTRWEqMMR4nBPuccUoiMwQQhU3t2I6IIpQMGmWTQju/MuLpHlWdZ2qe3teqV0VcZTQITpCJ8hFF6iGblAdNRBFT+gFvaF369l6tT6sz2nrklXMHKAZWN+/cdCggw==</latexit> No-flux wall: j · n = 0 <latexit sha1_base64="OHh8jO6/nf28rAeGL/fzHleBMdo=">AAACInicbVDLSgMxFM34tr6qLt0Ei+DGMiOCDxBEN66kgtVCZyiZTEZjM8mQ3NGWod/ixl9x40JRV4IfY9rOwqoHQg7n3Mu994Sp4AZc99MZG5+YnJqemS3NzS8sLpWXVy6NyjRldaqE0o2QGCa4ZHXgIFgj1YwkoWBXYfuk71/dMW24khfQTVmQkGvJY04JWKlV3veBdSA/U1uxyDr4nghxgHvYD5WITDexH8a3Po0UjGjy0G2VK27VHQD/JV5BKqhArVV+9yNFs4RJoIIY0/TcFIKcaOBUsF7JzwxLCW2Ta9a0VJKEmSAfnNjDG1aJcKy0fRLwQP3ZkZPE9HezlQmBG/Pb64v/ec0M4r0g5zLNgEk6HBRnAoPC/bxwxDWjILqWEKq53RXTG6IJBZtqyYbg/T75L7ncrnpu1TvfqRwdF3HMoDW0jjaRh3bRETpFNVRHFD2gJ/SCXp1H59l5cz6GpWNO0bOKRuB8fQP0L6Pi</latexit> <latexit 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sha1_base64="AiumVr1ajMzj/1KjNdVsTTbEkdE=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoMuiG5dV7APasWQymTY0kwxJRilD/8ONC0Xc+i/u/Bsz7Sy09UDI4Zx7yckJEs60cd1vp7Syura+Ud6sbG3v7O5V9w/aWqaK0BaRXKpugDXlTNCWYYbTbqIojgNOO8H4Ovc7j1RpJsW9mSTUj/FQsIgRbKz00A8kD/Uktld2Nx1Ua27dnQEtE68gNSjQHFS/+qEkaUyFIRxr3fPcxPgZVoYRTqeVfqppgskYD2nPUoFjqv1slnqKTqwSokgqe4RBM/X3RoZjnUezkzE2I73o5eJ/Xi810aWfMZGkhgoyfyhKOTIS5RWgkClKDJ9YgoliNisiI6wwMbaoii3BW/zyMmmf1T237t2e1xpXRR1lOIJjOAUPLqABN9CEFhBQ8Ayv8OY8OS/Ou/MxHy05xc4h/IHz+QMZjpLj</latexit> <latexit sha1_base64="AiumVr1ajMzj/1KjNdVsTTbEkdE=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoMuiG5dV7APasWQymTY0kwxJRilD/8ONC0Xc+i/u/Bsz7Sy09UDI4Zx7yckJEs60cd1vp7Syura+Ud6sbG3v7O5V9w/aWqaK0BaRXKpugDXlTNCWYYbTbqIojgNOO8H4Ovc7j1RpJsW9mSTUj/FQsIgRbKz00A8kD/Uktld2Nx1Ua27dnQEtE68gNSjQHFS/+qEkaUyFIRxr3fPcxPgZVoYRTqeVfqppgskYD2nPUoFjqv1slnqKTqwSokgqe4RBM/X3RoZjnUezkzE2I73o5eJ/Xi810aWfMZGkhgoyfyhKOTIS5RWgkClKDJ9YgoliNisiI6wwMbaoii3BW/zyMmmf1T237t2e1xpXRR1lOIJjOAUPLqABN9CEFhBQ8Ayv8OY8OS/Ou/MxHy05xc4h/IHz+QMZjpLj</latexit> h <latexit sha1_base64="22K8pTQwJQiQxeeQ5U5CBHP9N0M=">AAAB9HicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoMuiG5cV7APaoWQymTY0k0yTTKEM/Q43LhRx68e482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT5pa5kqQltEcqm6AdaUM0FbhhlOu4miOA447QTj+9zvTKnSTIonM0uoH+OhYBEj2FjJ7weSh3oW2wuNBtWaW3cXQOvEK0gNCjQH1a9+KEkaU2EIx1r3PDcxfoaVYYTTeaWfappgMsZD2rNU4JhqP1uEnqMLq4QoksoeYdBC/b2R4VjnyexkjM1Ir3q5+J/XS01062dMJKmhgiwfilKOjER5AyhkihLDZ5ZgopjNisgIK0yM7aliS/BWv7xO2ld1z617j9e1xl1RRxnO4BwuwYMbaMADNKEFBCbwDK/w5kydF+fd+ViOlpxi5xT+wPn8Acixkhc=</latexit> <latexit sha1_base64="22K8pTQwJQiQxeeQ5U5CBHP9N0M=">AAAB9HicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoMuiG5cV7APaoWQymTY0k0yTTKEM/Q43LhRx68e482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT5pa5kqQltEcqm6AdaUM0FbhhlOu4miOA447QTj+9zvTKnSTIonM0uoH+OhYBEj2FjJ7weSh3oW2wuNBtWaW3cXQOvEK0gNCjQH1a9+KEkaU2EIx1r3PDcxfoaVYYTTeaWfappgMsZD2rNU4JhqP1uEnqMLq4QoksoeYdBC/b2R4VjnyexkjM1Ir3q5+J/XS01062dMJKmhgiwfilKOjER5AyhkihLDZ5ZgopjNisgIK0yM7aliS/BWv7xO2ld1z617j9e1xl1RRxnO4BwuwYMbaMADNKEFBCbwDK/w5kydF+fd+ViOlpxi5xT+wPn8Acixkhc=</latexit> <latexit sha1_base64="22K8pTQwJQiQxeeQ5U5CBHP9N0M=">AAAB9HicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoMuiG5cV7APaoWQymTY0k0yTTKEM/Q43LhRx68e482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT5pa5kqQltEcqm6AdaUM0FbhhlOu4miOA447QTj+9zvTKnSTIonM0uoH+OhYBEj2FjJ7weSh3oW2wuNBtWaW3cXQOvEK0gNCjQH1a9+KEkaU2EIx1r3PDcxfoaVYYTTeaWfappgMsZD2rNU4JhqP1uEnqMLq4QoksoeYdBC/b2R4VjnyexkjM1Ir3q5+J/XS01062dMJKmhgiwfilKOjER5AyhkihLDZ5ZgopjNisgIK0yM7aliS/BWv7xO2ld1z617j9e1xl1RRxnO4BwuwYMbaMADNKEFBCbwDK/w5kydF+fd+ViOlpxi5xT+wPn8Acixkhc=</latexit> <latexit sha1_base64="22K8pTQwJQiQxeeQ5U5CBHP9N0M=">AAAB9HicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoMuiG5cV7APaoWQymTY0k0yTTKEM/Q43LhRx68e482/MtLPQ1gMhh3PuJScnSDjTxnW/ndLG5tb2Tnm3srd/cHhUPT5pa5kqQltEcqm6AdaUM0FbhhlOu4miOA447QTj+9zvTKnSTIonM0uoH+OhYBEj2FjJ7weSh3oW2wuNBtWaW3cXQOvEK0gNCjQH1a9+KEkaU2EIx1r3PDcxfoaVYYTTeaWfappgMsZD2rNU4JhqP1uEnqMLq4QoksoeYdBC/b2R4VjnyexkjM1Ir3q5+J/XS01062dMJKmhgiwfilKOjER5AyhkihLDZ5ZgopjNisgIK0yM7aliS/BWv7xO2ld1z617j9e1xl1RRxnO4BwuwYMbaMADNKEFBCbwDK/w5kydF+fd+ViOlpxi5xT+wPn8Acixkhc=</latexit> jA = D⇢ · rc <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> RS, Adhikari, and Cates JCP 2019
59. Known chemical surface flux Chemo-hydrodynamic coupling vA(⇢) = µc(⇢)rs c(⇢), <latexit sha1_base64="YfuLFv7dfPJ2/X7xzzU02PY+5Ck=">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</latexit> <latexit sha1_base64="YfuLFv7dfPJ2/X7xzzU02PY+5Ck=">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</latexit> <latexit sha1_base64="YfuLFv7dfPJ2/X7xzzU02PY+5Ck=">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</latexit> <latexit sha1_base64="YfuLFv7dfPJ2/X7xzzU02PY+5Ck=">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</latexit> jA = D⇢ · rc <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> Boundary condition Differential laws Chemical problem r · j = 0 j = Drc <latexit 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sha1_base64="crCRNfo1kaSoj+xYqg7P2tC7uEw=">AAACGnicbVBNS8NAEN34WetX1KOXxSJ4kJKIoMeiF48V7Ac0oWy2m3bpZhN2J2oI/R1e/CtePCjiTbz4b9y2OdjWBwOP92aYmRckgmtwnB9raXlldW29tFHe3Nre2bX39ps6ThVlDRqLWLUDopngkjWAg2DtRDESBYK1guH12G/dM6V5LO8gS5gfkb7kIacEjNS1XYo9xfsDIErFD9g5xR6wR8gx0XiE1YzpcRlC1rUrTtWZAC8StyAVVKDetb+8XkzTiEmggmjdcZ0E/Jwo4FSwUdlLNUsIHZI+6xgqScS0n09eG+Fjo/RwGCtTEvBE/TuRk0jrLApMZ0RgoOe9sfif10khvPRzLpMUmKTTRWEqMMR4nBPuccUoiMwQQhU3t2I6IIpQMGmWTQju/MuLpHlWdZ2qe3teqV0VcZTQITpCJ8hFF6iGblAdNRBFT+gFvaF369l6tT6sz2nrklXMHKAZWN+/cdCggw==</latexit> No-flux wall: j · n = 0 <latexit 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sha1_base64="OHh8jO6/nf28rAeGL/fzHleBMdo=">AAACInicbVDLSgMxFM34tr6qLt0Ei+DGMiOCDxBEN66kgtVCZyiZTEZjM8mQ3NGWod/ixl9x40JRV4IfY9rOwqoHQg7n3Mu994Sp4AZc99MZG5+YnJqemS3NzS8sLpWXVy6NyjRldaqE0o2QGCa4ZHXgIFgj1YwkoWBXYfuk71/dMW24khfQTVmQkGvJY04JWKlV3veBdSA/U1uxyDr4nghxgHvYD5WITDexH8a3Po0UjGjy0G2VK27VHQD/JV5BKqhArVV+9yNFs4RJoIIY0/TcFIKcaOBUsF7JzwxLCW2Ta9a0VJKEmSAfnNjDG1aJcKy0fRLwQP3ZkZPE9HezlQmBG/Pb64v/ec0M4r0g5zLNgEk6HBRnAoPC/bxwxDWjILqWEKq53RXTG6IJBZtqyYbg/T75L7ncrnpu1TvfqRwdF3HMoDW0jjaRh3bRETpFNVRHFD2gJ/SCXp1H59l5cz6GpWNO0bOKRuB8fQP0L6Pi</latexit> Coordinate system Water Plane wall: no-slip of the fluid flow and no-flux of chemical x <latexit 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sha1_base64="jDDQIw6+b+1mUfEzicwgj66P8ok=">AAACD3icbVDLSgMxFM34rPU16tJNsChuLDMiKIJQdONKKtgHdErJpJk2NDMZkjvVMvQP3Pgrblwo4tatO//GTNuFth4IOZxzL/fe48eCa3Ccb2tufmFxaTm3kl9dW9/YtLe2q1omirIKlUKquk80EzxiFeAgWD1WjIS+YDW/d5X5tT5TmsvoDgYxa4akE/GAUwJGatkHHrAHSG/kkRY8xvdEiHM8xJ4vRVsPQvNh3L9wWnbBKToj4FniTkgBTVBu2V9eW9IkZBFQQbRuuE4MzZQo4FSwYd5LNIsJ7ZEOaxgakZDpZjq6Z4j3jdLGgVTmRYBH6u+OlIQ6W85UhgS6etrLxP+8RgLBWTPlUZwAi+h4UJAIDBJn4eA2V4yCGBhCqOJmV0y7RBEKJsK8CcGdPnmWVI+LrlN0b08KpctJHDm0i/bQIXLRKSqha1RGFUTRI3pGr+jNerJerHfrY1w6Z016dtAfWJ8/4E+b4g==</latexit> <latexit sha1_base64="jDDQIw6+b+1mUfEzicwgj66P8ok=">AAACD3icbVDLSgMxFM34rPU16tJNsChuLDMiKIJQdONKKtgHdErJpJk2NDMZkjvVMvQP3Pgrblwo4tatO//GTNuFth4IOZxzL/fe48eCa3Ccb2tufmFxaTm3kl9dW9/YtLe2q1omirIKlUKquk80EzxiFeAgWD1WjIS+YDW/d5X5tT5TmsvoDgYxa4akE/GAUwJGatkHHrAHSG/kkRY8xvdEiHM8xJ4vRVsPQvNh3L9wWnbBKToj4FniTkgBTVBu2V9eW9IkZBFQQbRuuE4MzZQo4FSwYd5LNIsJ7ZEOaxgakZDpZjq6Z4j3jdLGgVTmRYBH6u+OlIQ6W85UhgS6etrLxP+8RgLBWTPlUZwAi+h4UJAIDBJn4eA2V4yCGBhCqOJmV0y7RBEKJsK8CcGdPnmWVI+LrlN0b08KpctJHDm0i/bQIXLRKSqha1RGFUTRI3pGr+jNerJerHfrY1w6Z016dtAfWJ8/4E+b4g==</latexit> v ! 0 as r ! 1 <latexit sha1_base64="wcClWvz2lIlOWZJ/ReM2WJdMAEk=">AAACGXicbVBNS8NAEN34WetX1KOXxSJ4KokIeix68VjBfkATyma7aZduNmF3Ug2hf8OLf8WLB0U86sl/47bNwbY+GHi8N8PMvCARXIPj/Fgrq2vrG5ulrfL2zu7evn1w2NRxqihr0FjEqh0QzQSXrAEcBGsnipEoEKwVDG8mfmvElOaxvIcsYX5E+pKHnBIwUtd2RthTvD8AolT8gB3sAXuEHBONx1jNeR6XIWRdu+JUnSnwMnELUkEF6l37y+vFNI2YBCqI1h3XScDPiQJOBRuXvVSzhNAh6bOOoZJETPv59LMxPjVKD4exMiUBT9W/EzmJtM6iwHRGBAZ60ZuI/3mdFMIrP+cySYFJOlsUpgJDjCcx4R5XjILIDCFUcXMrpgOiCAUTZtmE4C6+vEya51XXqbp3F5XadRFHCR2jE3SGXHSJaugW1VEDUfSEXtAbereerVfrw/qcta5YxcwRmoP1/Qse5KBg</latexit> <latexit sha1_base64="wcClWvz2lIlOWZJ/ReM2WJdMAEk=">AAACGXicbVBNS8NAEN34WetX1KOXxSJ4KokIeix68VjBfkATyma7aZduNmF3Ug2hf8OLf8WLB0U86sl/47bNwbY+GHi8N8PMvCARXIPj/Fgrq2vrG5ulrfL2zu7evn1w2NRxqihr0FjEqh0QzQSXrAEcBGsnipEoEKwVDG8mfmvElOaxvIcsYX5E+pKHnBIwUtd2RthTvD8AolT8gB3sAXuEHBONx1jNeR6XIWRdu+JUnSnwMnELUkEF6l37y+vFNI2YBCqI1h3XScDPiQJOBRuXvVSzhNAh6bOOoZJETPv59LMxPjVKD4exMiUBT9W/EzmJtM6iwHRGBAZ60ZuI/3mdFMIrP+cySYFJOlsUpgJDjCcx4R5XjILIDCFUcXMrpgOiCAUTZtmE4C6+vEya51XXqbp3F5XadRFHCR2jE3SGXHSJaugW1VEDUfSEXtAbereerVfrw/qcta5YxcwRmoP1/Qse5KBg</latexit> <latexit sha1_base64="wcClWvz2lIlOWZJ/ReM2WJdMAEk=">AAACGXicbVBNS8NAEN34WetX1KOXxSJ4KokIeix68VjBfkATyma7aZduNmF3Ug2hf8OLf8WLB0U86sl/47bNwbY+GHi8N8PMvCARXIPj/Fgrq2vrG5ulrfL2zu7evn1w2NRxqihr0FjEqh0QzQSXrAEcBGsnipEoEKwVDG8mfmvElOaxvIcsYX5E+pKHnBIwUtd2RthTvD8AolT8gB3sAXuEHBONx1jNeR6XIWRdu+JUnSnwMnELUkEF6l37y+vFNI2YBCqI1h3XScDPiQJOBRuXvVSzhNAh6bOOoZJETPv59LMxPjVKD4exMiUBT9W/EzmJtM6iwHRGBAZ60ZuI/3mdFMIrP+cySYFJOlsUpgJDjCcx4R5XjILIDCFUcXMrpgOiCAUTZtmE4C6+vEya51XXqbp3F5XadRFHCR2jE3SGXHSJaugW1VEDUfSEXtAbereerVfrw/qcta5YxcwRmoP1/Qse5KBg</latexit> <latexit sha1_base64="wcClWvz2lIlOWZJ/ReM2WJdMAEk=">AAACGXicbVBNS8NAEN34WetX1KOXxSJ4KokIeix68VjBfkATyma7aZduNmF3Ug2hf8OLf8WLB0U86sl/47bNwbY+GHi8N8PMvCARXIPj/Fgrq2vrG5ulrfL2zu7evn1w2NRxqihr0FjEqh0QzQSXrAEcBGsnipEoEKwVDG8mfmvElOaxvIcsYX5E+pKHnBIwUtd2RthTvD8AolT8gB3sAXuEHBONx1jNeR6XIWRdu+JUnSnwMnELUkEF6l37y+vFNI2YBCqI1h3XScDPiQJOBRuXvVSzhNAh6bOOoZJETPv59LMxPjVKD4exMiUBT9W/EzmJtM6iwHRGBAZ60ZuI/3mdFMIrP+cySYFJOlsUpgJDjCcx4R5XjILIDCFUcXMrpgOiCAUTZtmE4C6+vEya51XXqbp3F5XadRFHCR2jE3SGXHSJaugW1VEDUfSEXtAbereerVfrw/qcta5YxcwRmoP1/Qse5KBg</latexit> v = V + ⌦ ⇥ ⇢ + vA <latexit sha1_base64="hJPWoh7HbPHKy0ZmWJ9JUfY9JB0=">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</latexit> <latexit sha1_base64="hJPWoh7HbPHKy0ZmWJ9JUfY9JB0=">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</latexit> <latexit sha1_base64="hJPWoh7HbPHKy0ZmWJ9JUfY9JB0=">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</latexit> <latexit sha1_base64="hJPWoh7HbPHKy0ZmWJ9JUfY9JB0=">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</latexit> r · = 0 <latexit sha1_base64="KriQaJdZXNt75v6Bv+MDHOhzgAo=">AAACE3icbVDLSgMxFM3UV62vUZdugkUQF2VGBN0IRTcuK9gHdEq5k8m0oZlkSDJCKf0HN/6KGxeKuHXjzr8x03ZRWw+EHM65l3vvCVPOtPG8H6ewsrq2vlHcLG1t7+zuufsHDS0zRWidSC5VKwRNORO0bpjhtJUqCknIaTMc3OZ+85EqzaR4MMOUdhLoCRYzAsZKXfcsCCWP9DCxHw4EhBwCEkkzJwea9RK49rpu2at4E+Bl4s9IGc1Q67rfQSRJllBhCAet276Xms4IlGGE03EpyDRNgQygR9uWCkio7owmN43xiVUiHEtlnzB4os53jCDR+X62MgHT14teLv7ntTMTX3VGTKSZoYJMB8UZx0biPCAcMUWJ4UNLgChmd8WkDwqIsTGWbAj+4snLpHFe8b2Kf39Rrt7M4iiiI3SMTpGPLlEV3aEaqiOCntALekPvzrPz6nw4n9PSgjPrOUR/4Hz9AvTIns8=</latexit> <latexit sha1_base64="KriQaJdZXNt75v6Bv+MDHOhzgAo=">AAACE3icbVDLSgMxFM3UV62vUZdugkUQF2VGBN0IRTcuK9gHdEq5k8m0oZlkSDJCKf0HN/6KGxeKuHXjzr8x03ZRWw+EHM65l3vvCVPOtPG8H6ewsrq2vlHcLG1t7+zuufsHDS0zRWidSC5VKwRNORO0bpjhtJUqCknIaTMc3OZ+85EqzaR4MMOUdhLoCRYzAsZKXfcsCCWP9DCxHw4EhBwCEkkzJwea9RK49rpu2at4E+Bl4s9IGc1Q67rfQSRJllBhCAet276Xms4IlGGE03EpyDRNgQygR9uWCkio7owmN43xiVUiHEtlnzB4os53jCDR+X62MgHT14teLv7ntTMTX3VGTKSZoYJMB8UZx0biPCAcMUWJ4UNLgChmd8WkDwqIsTGWbAj+4snLpHFe8b2Kf39Rrt7M4iiiI3SMTpGPLlEV3aEaqiOCntALekPvzrPz6nw4n9PSgjPrOUR/4Hz9AvTIns8=</latexit> <latexit sha1_base64="KriQaJdZXNt75v6Bv+MDHOhzgAo=">AAACE3icbVDLSgMxFM3UV62vUZdugkUQF2VGBN0IRTcuK9gHdEq5k8m0oZlkSDJCKf0HN/6KGxeKuHXjzr8x03ZRWw+EHM65l3vvCVPOtPG8H6ewsrq2vlHcLG1t7+zuufsHDS0zRWidSC5VKwRNORO0bpjhtJUqCknIaTMc3OZ+85EqzaR4MMOUdhLoCRYzAsZKXfcsCCWP9DCxHw4EhBwCEkkzJwea9RK49rpu2at4E+Bl4s9IGc1Q67rfQSRJllBhCAet276Xms4IlGGE03EpyDRNgQygR9uWCkio7owmN43xiVUiHEtlnzB4os53jCDR+X62MgHT14teLv7ntTMTX3VGTKSZoYJMB8UZx0biPCAcMUWJ4UNLgChmd8WkDwqIsTGWbAj+4snLpHFe8b2Kf39Rrt7M4iiiI3SMTpGPLlEV3aEaqiOCntALekPvzrPz6nw4n9PSgjPrOUR/4Hz9AvTIns8=</latexit> <latexit sha1_base64="KriQaJdZXNt75v6Bv+MDHOhzgAo=">AAACE3icbVDLSgMxFM3UV62vUZdugkUQF2VGBN0IRTcuK9gHdEq5k8m0oZlkSDJCKf0HN/6KGxeKuHXjzr8x03ZRWw+EHM65l3vvCVPOtPG8H6ewsrq2vlHcLG1t7+zuufsHDS0zRWidSC5VKwRNORO0bpjhtJUqCknIaTMc3OZ+85EqzaR4MMOUdhLoCRYzAsZKXfcsCCWP9DCxHw4EhBwCEkkzJwea9RK49rpu2at4E+Bl4s9IGc1Q67rfQSRJllBhCAet276Xms4IlGGE03EpyDRNgQygR9uWCkio7owmN43xiVUiHEtlnzB4os53jCDR+X62MgHT14teLv7ntTMTX3VGTKSZoYJMB8UZx0biPCAcMUWJ4UNLgChmd8WkDwqIsTGWbAj+4snLpHFe8b2Kf39Rrt7M4iiiI3SMTpGPLlEV3aEaqiOCntALekPvzrPz6nw4n9PSgjPrOUR/4Hz9AvTIns8=</latexit> = pI + ⌘a(rva + (rva)T ) <latexit sha1_base64="MxHV9PVnYKd7yqcVmEx3rfdBoDU=">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</latexit> <latexit sha1_base64="MxHV9PVnYKd7yqcVmEx3rfdBoDU=">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</latexit> <latexit sha1_base64="MxHV9PVnYKd7yqcVmEx3rfdBoDU=">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</latexit> <latexit sha1_base64="MxHV9PVnYKd7yqcVmEx3rfdBoDU=">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</latexit> jA = D⇢ · rc <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> RS, Adhikari, and Cates JCP 2019
60. Chemo-hydrodynamics of active colloids RS, Adhikari, and Cates JCP 2019 Using integral representation of Stokes and Laplace equations, spectral expansion, and Galerkin discretization, to compute hydrodynamic and phoretic interactions between spheres with slip boundary conditions on their surfaces, we obtain
61. Chemo-hydrodynamics of active colloids RS, Adhikari, and Cates JCP 2019 Using integral representation of Stokes and Laplace equations, spectral expansion, and Galerkin discretization, to compute hydrodynamic and phoretic interactions between spheres with slip boundary conditions on their surfaces, we obtain Many-body Slip V(l ) i = X m (l ,m) ik · J(m) k , <latexit sha1_base64="NZWxusQptmurYHb9N3L3cEWezP0=">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</latexit> <latexit sha1_base64="NZWxusQptmurYHb9N3L3cEWezP0=">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</latexit> <latexit sha1_base64="NZWxusQptmurYHb9N3L3cEWezP0=">AAACTHicbVBNS8MwGE7n15xfVY9eikPYYIxWBL0IQy/iaYL7gHWWNE23sKQpSSqM0h/oxYM3f4UXD4oIptuEuflCyMPzPO+bN48fUyKVbb8ahZXVtfWN4mZpa3tnd8/cP2hLngiEW4hTLro+lJiSCLcUURR3Y4Eh8ynu+KPrXO88YiEJj+7VOMZ9BgcRCQmCSlOeiVwG1dAP03bmpSR7SCvUlWTAYDW7dGXCPOb6nAZyzPSVulRPDmBuHc15a6yauSjg6nfYrXZMDFqolTyzbNftSVnLwJmBMphV0zNf3ICjhOFIIQql7Dl2rPopFIogirOSm0gcQzSCA9zTMIIMy346CSOzTjQTWCEX+kTKmrDzHSlkMv+OdubbykUtJ//TeokKL/opieJE4QhNHwoTailu5claAREYKTrWACJB9K4WGkIBkdL55yE4i19eBu3TumPXnbuzcuNqFkcRHIFjUAEOOAcNcAOaoAUQeAJv4AN8Gs/Gu/FlfE+tBWPWcwj+VGH9B1A+tk4=</latexit> <latexit sha1_base64="NZWxusQptmurYHb9N3L3cEWezP0=">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</latexit>
62. Generalized Stokes laws FA(l ) i = 1 X l0 0=1s (l ,l0 0) ik · V(l0 0) k . <latexit sha1_base64="SIkyPEuN4ZOz8nejVCyXmQ3+vfw=">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</latexit> <latexit sha1_base64="SIkyPEuN4ZOz8nejVCyXmQ3+vfw=">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</latexit> <latexit sha1_base64="SIkyPEuN4ZOz8nejVCyXmQ3+vfw=">AAACfXicbVFbb9MwFHbCbZTLCjzyYlFNW9GIEoQ0XiYNkBCPQ6LdpKZEJ47TWfUlsk+QKiv/gl/GG3+FF3DajsvGkSx//s53ri4bKRym6fcovnHz1u07O3cH9+4/eLg7fPR46kxrGZ8wI409L8FxKTSfoEDJzxvLQZWSn5XLd73/7Au3Thj9CVcNnytYaFELBhioYvg1V4AXZe3fd4UX3We/fjOQ/k13IHMnFgrG3fGL3LWq8HJ/w+wfZ67XCl3jqstLIyu3UuHy+QKUgj7XMgguMxz+Dhx3OasMXladBuVG+EeQDIrhKE3StdHrINuCEdnaaTH8lleGtYprZBKcm2Vpg3MPFgWTvBvkreMNsCUs+CxADYq7uV9vr6N7galobWw4Guma/TvCg3L9dEHZd+2u+nryf75Zi/XruRe6aZFrtilUt5Kiof1X0EpYzlCuAgBmReiVsguwwDB8WL+E7OrI18H0ZZKlSfbx1ejk7XYdO+QpeUYOSEaOyAn5QE7JhDDyI6LROHoe/Yz34sM42UjjaBvzhPxj8dEvOE/EXQ==</latexit> <latexit sha1_base64="SIkyPEuN4ZOz8nejVCyXmQ3+vfw=">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</latexit> Chemo-hydrodynamics of active colloids RS, Adhikari, and Cates JCP 2019 Using integral representation of Stokes and Laplace equations, spectral expansion, and Galerkin discretization, to compute hydrodynamic and phoretic interactions between spheres with slip boundary conditions on their surfaces, we obtain Many-body Slip V(l ) i = X m (l ,m) ik · J(m) k , <latexit sha1_base64="NZWxusQptmurYHb9N3L3cEWezP0=">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</latexit> <latexit sha1_base64="NZWxusQptmurYHb9N3L3cEWezP0=">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</latexit> <latexit sha1_base64="NZWxusQptmurYHb9N3L3cEWezP0=">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</latexit> <latexit sha1_base64="NZWxusQptmurYHb9N3L3cEWezP0=">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</latexit>
63. Generalized Stokes laws FA(l ) i = 1 X l0 0=1s (l ,l0 0) ik · V(l0 0) k . <latexit sha1_base64="SIkyPEuN4ZOz8nejVCyXmQ3+vfw=">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</latexit> <latexit sha1_base64="SIkyPEuN4ZOz8nejVCyXmQ3+vfw=">AAACfXicbVFbb9MwFHbCbZTLCjzyYlFNW9GIEoQ0XiYNkBCPQ6LdpKZEJ47TWfUlsk+QKiv/gl/GG3+FF3DajsvGkSx//s53ri4bKRym6fcovnHz1u07O3cH9+4/eLg7fPR46kxrGZ8wI409L8FxKTSfoEDJzxvLQZWSn5XLd73/7Au3Thj9CVcNnytYaFELBhioYvg1V4AXZe3fd4UX3We/fjOQ/k13IHMnFgrG3fGL3LWq8HJ/w+wfZ67XCl3jqstLIyu3UuHy+QKUgj7XMgguMxz+Dhx3OasMXladBuVG+EeQDIrhKE3StdHrINuCEdnaaTH8lleGtYprZBKcm2Vpg3MPFgWTvBvkreMNsCUs+CxADYq7uV9vr6N7galobWw4Guma/TvCg3L9dEHZd+2u+nryf75Zi/XruRe6aZFrtilUt5Kiof1X0EpYzlCuAgBmReiVsguwwDB8WL+E7OrI18H0ZZKlSfbx1ejk7XYdO+QpeUYOSEaOyAn5QE7JhDDyI6LROHoe/Yz34sM42UjjaBvzhPxj8dEvOE/EXQ==</latexit> <latexit sha1_base64="SIkyPEuN4ZOz8nejVCyXmQ3+vfw=">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</latexit> <latexit sha1_base64="SIkyPEuN4ZOz8nejVCyXmQ3+vfw=">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</latexit> Chemo-hydrodynamics of active colloids RS, Adhikari, and Cates JCP 2019 Using integral representation of Stokes and Laplace equations, spectral expansion, and Galerkin discretization, to compute hydrodynamic and phoretic interactions between spheres with slip boundary conditions on their surfaces, we obtain Many-body Slip V(l ) i = X m (l ,m) ik · J(m) k , <latexit sha1_base64="NZWxusQptmurYHb9N3L3cEWezP0=">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</latexit> <latexit sha1_base64="NZWxusQptmurYHb9N3L3cEWezP0=">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</latexit> <latexit sha1_base64="NZWxusQptmurYHb9N3L3cEWezP0=">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</latexit> <latexit sha1_base64="NZWxusQptmurYHb9N3L3cEWezP0=">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</latexit>
64. Generalized Stokes laws FA(l ) i = 1 X l0 0=1s (l ,l0 0) ik · V(l0 0) k . <latexit sha1_base64="SIkyPEuN4ZOz8nejVCyXmQ3+vfw=">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</latexit> <latexit sha1_base64="SIkyPEuN4ZOz8nejVCyXmQ3+vfw=">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</latexit> <latexit sha1_base64="SIkyPEuN4ZOz8nejVCyXmQ3+vfw=">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</latexit> <latexit sha1_base64="SIkyPEuN4ZOz8nejVCyXmQ3+vfw=">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</latexit> Chemical Surface Flux Many-Body Slip Exterior Fluid Flow Chemical interactions Hydrodynamic interactions Chemo-hydrodynamics of active colloids RS, Adhikari, and Cates JCP 2019 Using integral representation of Stokes and Laplace equations, spectral expansion, and Galerkin discretization, to compute hydrodynamic and phoretic interactions between spheres with slip boundary conditions on their surfaces, we obtain Many-body Slip V(l ) i = X m (l ,m) ik · J(m) k , <latexit sha1_base64="NZWxusQptmurYHb9N3L3cEWezP0=">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</latexit> <latexit 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65. RS, Adhikari, and Cates JCP 2019
66. RS, Adhikari, and Cates JCP 2019
67. Experiments: Active colloids at a plane wall Light on Light off Palacci et al Science 2013: chemical reactions in presence of light make colloids move (active) Crystallisation at a wall in presence of light Crystals melt as light is turned off
68. Experiments: Active colloids at a plane wall Theurkauff et al PRL 2012, Buttinoni et al PRL 2013: Dynamic clusters of autophoretic particles Cluster size increases with speed of isolated colloids Light on Light off Palacci et al Science 2013: chemical reactions in presence of light make colloids move (active) Crystallisation at a wall in presence of light Crystals melt as light is turned off
69. Experiments: Active colloids at a plane wall Petroff et al PRL 2015: Crystallization has also been reported for microorganisms , T. Majus, at a plane wall. But there is no arrest of clustering here. The activity here is biological and these colloids interact by hydrodynamic interactions alone Theurkauff et al PRL 2012, Buttinoni et al PRL 2013: Dynamic clusters of autophoretic particles Cluster size increases with speed of isolated colloids Light on Light off Palacci et al Science 2013: chemical reactions in presence of light make colloids move (active) Crystallisation at a wall in presence of light Crystals melt as light is turned off
70. Formalism to study the hydrodynamic and phoretic interactions between colloids that does not need to resolve explicit fluid degrees of freedom. Solution obtained in terms of irreducible modes Boundary conditions in the flow modify the active forces and torques, and thus, determine the collective behaviour. Summary
71. ‣ Grid-free method to compute hydrodynamic and phoretic interactions between spheres with slip boundary conditions ‣ Identify mechanisms using interactions computed from each irreducible mode ‣ Deepen theoretical understanding and suggest newer experiments Formalism to study the hydrodynamic and phoretic interactions between colloids that does not need to resolve explicit fluid degrees of freedom. Solution obtained in terms of irreducible modes Boundary conditions in the flow modify the active forces and torques, and thus, determine the collective behaviour. Summary
72. ‣ Grid-free method to compute hydrodynamic and phoretic interactions between spheres with slip boundary conditions ‣ Identify mechanisms using interactions computed from each irreducible mode ‣ Deepen theoretical understanding and suggest newer experiments Formalism to study the hydrodynamic and phoretic interactions between colloids that does not need to resolve explicit fluid degrees of freedom. Solution obtained in terms of irreducible modes Boundary conditions in the flow modify the active forces and torques, and thus, determine the collective behaviour. Summary Thank You !