Active Suspensions Confined by Planar Walls

Chapter II: Active Matter with Spatially Varying Transport Properties

2.3 Active Suspensions Confined by Planar Walls

D

, D

, U

D

<latexit sha1_base64="/uUUGO6/1vWXPO86iJLxgG/JF+E=">AAACAHicbZC7SgNBFIbPxluMt1ULC5vBIFhI2A2ClgEtLCyiZJNAsiyzk0kyZPbCzKwQlm18FRsLRWx9DDvfxslmC038YeDjP+dw5vx+zJlUlvVtlFZW19Y3ypuVre2d3T1z/6Ato0QQ6pCIR6LrY0k5C6mjmOK0GwuKA5/Tjj+5ntU7j1RIFoUtNY2pG+BRyIaMYKUtzzy68dJWPTvvIw0POTheWs88s2rVrFxoGewCqlCo6Zlf/UFEkoCGinAsZc+2YuWmWChGOM0q/UTSGJMJHtGexhAHVLppfkCGTrUzQMNI6BcqlLu/J1IcSDkNfN0ZYDWWi7WZ+V+tl6jhlZuyME4UDcl80TDhSEVolgYaMEGJ4lMNmAim/4rIGAtMlM6sokOwF09ehna9Zls1+/6i2rgr4ijDMZzAGdhwCQ24hSY4QCCDZ3iFN+PJeDHejY95a8koZg7hj4zPH23blQk=</latexit><latexit sha1_base64="/uUUGO6/1vWXPO86iJLxgG/JF+E=">AAACAHicbZC7SgNBFIbPxluMt1ULC5vBIFhI2A2ClgEtLCyiZJNAsiyzk0kyZPbCzKwQlm18FRsLRWx9DDvfxslmC038YeDjP+dw5vx+zJlUlvVtlFZW19Y3ypuVre2d3T1z/6Ato0QQ6pCIR6LrY0k5C6mjmOK0GwuKA5/Tjj+5ntU7j1RIFoUtNY2pG+BRyIaMYKUtzzy68dJWPTvvIw0POTheWs88s2rVrFxoGewCqlCo6Zlf/UFEkoCGinAsZc+2YuWmWChGOM0q/UTSGJMJHtGexhAHVLppfkCGTrUzQMNI6BcqlLu/J1IcSDkNfN0ZYDWWi7WZ+V+tl6jhlZuyME4UDcl80TDhSEVolgYaMEGJ4lMNmAim/4rIGAtMlM6sokOwF09ehna9Zls1+/6i2rgr4ijDMZzAGdhwCQ24hSY4QCCDZ3iFN+PJeDHejY95a8koZg7hj4zPH23blQk=</latexit><latexit sha1_base64="/uUUGO6/1vWXPO86iJLxgG/JF+E=">AAACAHicbZC7SgNBFIbPxluMt1ULC5vBIFhI2A2ClgEtLCyiZJNAsiyzk0kyZPbCzKwQlm18FRsLRWx9DDvfxslmC038YeDjP+dw5vx+zJlUlvVtlFZW19Y3ypuVre2d3T1z/6Ato0QQ6pCIR6LrY0k5C6mjmOK0GwuKA5/Tjj+5ntU7j1RIFoUtNY2pG+BRyIaMYKUtzzy68dJWPTvvIw0POTheWs88s2rVrFxoGewCqlCo6Zlf/UFEkoCGinAsZc+2YuWmWChGOM0q/UTSGJMJHtGexhAHVLppfkCGTrUzQMNI6BcqlLu/J1IcSDkNfN0ZYDWWi7WZ+V+tl6jhlZuyME4UDcl80TDhSEVolgYaMEGJ4lMNmAim/4rIGAtMlM6sokOwF09ehna9Zls1+/6i2rgr4ijDMZzAGdhwCQ24hSY4QCCDZ3iFN+PJeDHejY95a8koZg7hj4zPH23blQk=</latexit><latexit sha1_base64="/uUUGO6/1vWXPO86iJLxgG/JF+E=">AAACAHicbZC7SgNBFIbPxluMt1ULC5vBIFhI2A2ClgEtLCyiZJNAsiyzk0kyZPbCzKwQlm18FRsLRWx9DDvfxslmC038YeDjP+dw5vx+zJlUlvVtlFZW19Y3ypuVre2d3T1z/6Ato0QQ6pCIR6LrY0k5C6mjmOK0GwuKA5/Tjj+5ntU7j1RIFoUtNY2pG+BRyIaMYKUtzzy68dJWPTvvIw0POTheWs88s2rVrFxoGewCqlCo6Zlf/UFEkoCGinAsZc+2YuWmWChGOM0q/UTSGJMJHtGexhAHVLppfkCGTrUzQMNI6BcqlLu/J1IcSDkNfN0ZYDWWi7WZ+V+tl6jhlZuyME4UDcl80TDhSEVolgYaMEGJ4lMNmAim/4rIGAtMlM6sokOwF09ehna9Zls1+/6i2rgr4ijDMZzAGdhwCQ24hSY4QCCDZ3iFN+PJeDHejY95a8koZg7hj4zPH23blQk=</latexit> T2

, D

, U

Region 1 Region 2

<latexit sha1_base64="m+7QFSLs2FAlwS8scE7LYBqtjqk=">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</latexit>

x

<latexit sha1_base64="nq2cX3hvR2q8JliFPSwxwAOsbrQ=">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</latexit>

0

<latexit sha1_base64="NiEcK2CVn+eAOek10JEzttir118=">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</latexit>

L

₁ <latexit sha1_base64="QDgFjpmvEaQGwRuMyYHWscxpoKA=">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</latexit>

L

₂

Figure 2.7: A schematic of a suspension of ABPs with an abrupt change in activity bounded between two parallel planar walls. In regions 1 (−𝐿₁ ≤ 𝑥 < 0) and 2 (0 ≤ 𝑥 ≤ 𝐿₂) the ABPs have swim speeds𝑈_𝑖, translational diffusivities 𝐷_{𝑇 𝑖}, and rotational diffusivities𝐷_𝑅𝑖, where the subscript𝑖 (=1 or 2)represents the index of a region.

Now finite suspensions of ABPs in the presence of an abrupt variation in activity are considered with two examples. We first explore a suspension confined between two

parallel walls at𝑥 = −𝐿₁ and𝑥 = 𝐿₂ with a step change in transport properties at 𝑥 =0 as described in Fig. 2.7. The governing Smoluchowski equations (2.2)-(2.4) and associated moment equations (2.5)-(2.8) remain the same but the spatial domain for the two regions are now finite in𝑥direction.

The walls are assumed to be hard and act as no-flux boundaries, i.e. 𝒏· 𝒋_𝑇 = 0, at𝑥 = −𝐿₁ and𝑥 = 𝐿₂, where 𝒏 is the unit normal vector to the surface of walls.

Continuity of field variables and fluxes at the boundary of the two regions (𝑥 =0) still applies and the overall number density is (𝐿₁ + 𝐿₂) ⟨𝑛⟩ = ∫ ^𝐿₂

−𝐿₁

𝑛 𝑑𝑥. The steady state analytical solution for bounded suspensions with the 𝑸 =0-closure is straightforward to obtain:

𝑛_𝑖 𝑛₀

=𝛾_𝑖𝑎_𝑖(cosh(𝜆_𝑖𝑥) −1) + 𝛾_𝑖 𝑚_𝑥₀

𝑛₀

sinh(𝜆_𝑖𝑥) + 1, (2.24) 𝑚_𝑖,𝑥

𝑛₀

= 𝑚_𝑥₀

𝑛₀ cosh(𝜆_𝑖𝑥) + 𝑎_𝑖sinh(𝜆_𝑖𝑥) , (2.25) where

𝑚_𝑥₀ 𝑛₀

= 1 𝑏

Λ₁Λ₂(𝑈₁−𝑈₂) (cosh(𝜆₁𝐿₁) −1) (cosh(𝜆₂𝐿₂) −1) + Λ₁𝑈₁(cosh(𝜆₁𝐿₁) −1) − Λ₂𝑈₂(cosh(𝜆₂𝐿₂) −1)

(2.26)

𝑎₁ = 1 𝑏

Λ₁Λ₂(𝑈₁−𝑈₂)sinh(𝜆₁𝐿₁) (cosh(𝜆₂𝐿₂) −1)) + Λ₁𝑈₁sinh(𝜆₁𝐿₁) + 𝛾₁

𝛾₂

Λ₂𝑈₂sinh(𝜆₂𝐿₂)

(2.27)

𝑎₂ = 1 𝑏

Λ₁Λ₂(𝑈₂−𝑈₁)sinh(𝜆₂𝐿₂) (cosh(𝜆₁𝐿₁) −1)) + 𝛾₂

𝛾₁

Λ₁𝑈₁sinh(𝜆₁𝐿₁) + Λ₂𝑈₂sinh(𝜆₂𝐿₂)

(2.28)

𝑏 = 𝑑 𝛾₁

Λ₁𝑈₁sinh(𝜆₁𝐿₁) (1+Λ₂(cosh(𝜆₂𝐿₂) −1)) + 𝑑

𝛾₂

Λ₂𝑈₂sinh(𝜆₂𝐿₂) (1+Λ₁(cosh(𝜆₁𝐿₁) −1)),

(2.29)

𝜆_𝑖 =

√ 𝑑−1

𝛿_𝑖 vt

1+ 𝑃𝑒_𝑖

𝑑(𝑑−1) , (2.30)

𝛾_𝑖 = vu uu uu ut

𝑑

𝑑(𝑑−1) 𝑃𝑒_𝑖

, (2.31)

Λ𝑖 = 1 1+

𝑃𝑒_𝑖 𝑑(𝑑−1)

. (2.32)

Here, 𝑛₀ and 𝑚_𝑥₀ are the number density and polar order at the step change in transport properties (𝑥 = 0). In order to determine the value of 𝑛₀, the particle conservation equation ⟨𝑛⟩ = ∫𝐿₂

−𝐿₁𝑛 𝑑𝑥/(𝐿₁+𝐿₂) is used for given overall number density⟨𝑛⟩.

D<latexit sha1_base64="1cL2yT0DI3+xFQCILVbdP4GdFE0=">AAACAHicbZBPS8MwGMZT/875r+rBg5fgEDzIaETQ40APHjxMWbfBVkqapVtYmpYkFUbpxa/ixYMiXv0Y3vw2Zl0PuvlA4Mfzvi9v3idIOFPacb6tpeWV1bX1ykZ1c2t7Z9fe22+rOJWEuiTmsewGWFHOBHU105x2E0lxFHDaCcbX03rnkUrFYtHSk4R6ER4KFjKCtbF8+/DGz1ooP+tDAw8FuH6Gct+uOXWnEFwEVEINlGr69ld/EJM0okITjpXqISfRXoalZoTTvNpPFU0wGeMh7RkUOKLKy4oDcnhinAEMY2me0LBwf09kOFJqEgWmM8J6pOZrU/O/Wi/V4ZWXMZGkmgoyWxSmHOoYTtOAAyYp0XxiABPJzF8hGWGJiTaZVU0IaP7kRWif15FTR/cXtcZdGUcFHIFjcAoQuAQNcAuawAUE5OAZvII368l6sd6tj1nrklXOHIA/sj5/AGkzlQY=</latexit><latexit sha1_base64="1cL2yT0DI3+xFQCILVbdP4GdFE0=">AAACAHicbZBPS8MwGMZT/875r+rBg5fgEDzIaETQ40APHjxMWbfBVkqapVtYmpYkFUbpxa/ixYMiXv0Y3vw2Zl0PuvlA4Mfzvi9v3idIOFPacb6tpeWV1bX1ykZ1c2t7Z9fe22+rOJWEuiTmsewGWFHOBHU105x2E0lxFHDaCcbX03rnkUrFYtHSk4R6ER4KFjKCtbF8+/DGz1ooP+tDAw8FuH6Gct+uOXWnEFwEVEINlGr69ld/EJM0okITjpXqISfRXoalZoTTvNpPFU0wGeMh7RkUOKLKy4oDcnhinAEMY2me0LBwf09kOFJqEgWmM8J6pOZrU/O/Wi/V4ZWXMZGkmgoyWxSmHOoYTtOAAyYp0XxiABPJzF8hGWGJiTaZVU0IaP7kRWif15FTR/cXtcZdGUcFHIFjcAoQuAQNcAuawAUE5OAZvII368l6sd6tj1nrklXOHIA/sj5/AGkzlQY=</latexit><latexit sha1_base64="1cL2yT0DI3+xFQCILVbdP4GdFE0=">AAACAHicbZBPS8MwGMZT/875r+rBg5fgEDzIaETQ40APHjxMWbfBVkqapVtYmpYkFUbpxa/ixYMiXv0Y3vw2Zl0PuvlA4Mfzvi9v3idIOFPacb6tpeWV1bX1ykZ1c2t7Z9fe22+rOJWEuiTmsewGWFHOBHU105x2E0lxFHDaCcbX03rnkUrFYtHSk4R6ER4KFjKCtbF8+/DGz1ooP+tDAw8FuH6Gct+uOXWnEFwEVEINlGr69ld/EJM0okITjpXqISfRXoalZoTTvNpPFU0wGeMh7RkUOKLKy4oDcnhinAEMY2me0LBwf09kOFJqEgWmM8J6pOZrU/O/Wi/V4ZWXMZGkmgoyWxSmHOoYTtOAAyYp0XxiABPJzF8hGWGJiTaZVU0IaP7kRWif15FTR/cXtcZdGUcFHIFjcAoQuAQNcAuawAUE5OAZvII368l6sd6tj1nrklXOHIA/sj5/AGkzlQY=</latexit><latexit sha1_base64="1cL2yT0DI3+xFQCILVbdP4GdFE0=">AAACAHicbZBPS8MwGMZT/875r+rBg5fgEDzIaETQ40APHjxMWbfBVkqapVtYmpYkFUbpxa/ixYMiXv0Y3vw2Zl0PuvlA4Mfzvi9v3idIOFPacb6tpeWV1bX1ykZ1c2t7Z9fe22+rOJWEuiTmsewGWFHOBHU105x2E0lxFHDaCcbX03rnkUrFYtHSk4R6ER4KFjKCtbF8+/DGz1ooP+tDAw8FuH6Gct+uOXWnEFwEVEINlGr69ld/EJM0okITjpXqISfRXoalZoTTvNpPFU0wGeMh7RkUOKLKy4oDcnhinAEMY2me0LBwf09kOFJqEgWmM8J6pOZrU/O/Wi/V4ZWXMZGkmgoyWxSmHOoYTtOAAyYp0XxiABPJzF8hGWGJiTaZVU0IaP7kRWif15FTR/cXtcZdGUcFHIFjcAoQuAQNcAuawAUE5OAZvII368l6sd6tj1nrklXOHIA/sj5/AGkzlQY=</latexit> _T1, D_R1, U₁ D<latexit sha1_base64="/uUUGO6/1vWXPO86iJLxgG/JF+E=">AAACAHicbZC7SgNBFIbPxluMt1ULC5vBIFhI2A2ClgEtLCyiZJNAsiyzk0kyZPbCzKwQlm18FRsLRWx9DDvfxslmC038YeDjP+dw5vx+zJlUlvVtlFZW19Y3ypuVre2d3T1z/6Ato0QQ6pCIR6LrY0k5C6mjmOK0GwuKA5/Tjj+5ntU7j1RIFoUtNY2pG+BRyIaMYKUtzzy68dJWPTvvIw0POTheWs88s2rVrFxoGewCqlCo6Zlf/UFEkoCGinAsZc+2YuWmWChGOM0q/UTSGJMJHtGexhAHVLppfkCGTrUzQMNI6BcqlLu/J1IcSDkNfN0ZYDWWi7WZ+V+tl6jhlZuyME4UDcl80TDhSEVolgYaMEGJ4lMNmAim/4rIGAtMlM6sokOwF09ehna9Zls1+/6i2rgr4ijDMZzAGdhwCQ24hSY4QCCDZ3iFN+PJeDHejY95a8koZg7hj4zPH23blQk=</latexit><latexit sha1_base64="/uUUGO6/1vWXPO86iJLxgG/JF+E=">AAACAHicbZC7SgNBFIbPxluMt1ULC5vBIFhI2A2ClgEtLCyiZJNAsiyzk0kyZPbCzKwQlm18FRsLRWx9DDvfxslmC038YeDjP+dw5vx+zJlUlvVtlFZW19Y3ypuVre2d3T1z/6Ato0QQ6pCIR6LrY0k5C6mjmOK0GwuKA5/Tjj+5ntU7j1RIFoUtNY2pG+BRyIaMYKUtzzy68dJWPTvvIw0POTheWs88s2rVrFxoGewCqlCo6Zlf/UFEkoCGinAsZc+2YuWmWChGOM0q/UTSGJMJHtGexhAHVLppfkCGTrUzQMNI6BcqlLu/J1IcSDkNfN0ZYDWWi7WZ+V+tl6jhlZuyME4UDcl80TDhSEVolgYaMEGJ4lMNmAim/4rIGAtMlM6sokOwF09ehna9Zls1+/6i2rgr4ijDMZzAGdhwCQ24hSY4QCCDZ3iFN+PJeDHejY95a8koZg7hj4zPH23blQk=</latexit><latexit sha1_base64="/uUUGO6/1vWXPO86iJLxgG/JF+E=">AAACAHicbZC7SgNBFIbPxluMt1ULC5vBIFhI2A2ClgEtLCyiZJNAsiyzk0kyZPbCzKwQlm18FRsLRWx9DDvfxslmC038YeDjP+dw5vx+zJlUlvVtlFZW19Y3ypuVre2d3T1z/6Ato0QQ6pCIR6LrY0k5C6mjmOK0GwuKA5/Tjj+5ntU7j1RIFoUtNY2pG+BRyIaMYKUtzzy68dJWPTvvIw0POTheWs88s2rVrFxoGewCqlCo6Zlf/UFEkoCGinAsZc+2YuWmWChGOM0q/UTSGJMJHtGexhAHVLppfkCGTrUzQMNI6BcqlLu/J1IcSDkNfN0ZYDWWi7WZ+V+tl6jhlZuyME4UDcl80TDhSEVolgYaMEGJ4lMNmAim/4rIGAtMlM6sokOwF09ehna9Zls1+/6i2rgr4ijDMZzAGdhwCQ24hSY4QCCDZ3iFN+PJeDHejY95a8koZg7hj4zPH23blQk=</latexit><latexit sha1_base64="/uUUGO6/1vWXPO86iJLxgG/JF+E=">AAACAHicbZC7SgNBFIbPxluMt1ULC5vBIFhI2A2ClgEtLCyiZJNAsiyzk0kyZPbCzKwQlm18FRsLRWx9DDvfxslmC038YeDjP+dw5vx+zJlUlvVtlFZW19Y3ypuVre2d3T1z/6Ato0QQ6pCIR6LrY0k5C6mjmOK0GwuKA5/Tjj+5ntU7j1RIFoUtNY2pG+BRyIaMYKUtzzy68dJWPTvvIw0POTheWs88s2rVrFxoGewCqlCo6Zlf/UFEkoCGinAsZc+2YuWmWChGOM0q/UTSGJMJHtGexhAHVLppfkCGTrUzQMNI6BcqlLu/J1IcSDkNfN0ZYDWWi7WZ+V+tl6jhlZuyME4UDcl80TDhSEVolgYaMEGJ4lMNmAim/4rIGAtMlM6sokOwF09ehna9Zls1+/6i2rgr4ijDMZzAGdhwCQ24hSY4QCCDZ3iFN+PJeDHejY95a8koZg7hj4zPH23blQk=</latexit> _T₂, D_R2, U₂

Region 1 Region 2

x

L₁ <latexit sha1_base64="QDgFjpmvEaQGwRuMyYHWscxpoKA=">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</latexit>

L₂

Boundary Layers Walls

An active suspension with one wall Yan and Brady (2015)

An infinite suspension of ABPs with a step change in transport properties A homogeneous active suspension

DT1, DR1, U1

<latexit sha1_base64="1cL2yT0DI3+xFQCILVbdP4GdFE0=">AAACAHicbZBPS8MwGMZT/875r+rBg5fgEDzIaETQ40APHjxMWbfBVkqapVtYmpYkFUbpxa/ixYMiXv0Y3vw2Zl0PuvlA4Mfzvi9v3idIOFPacb6tpeWV1bX1ykZ1c2t7Z9fe22+rOJWEuiTmsewGWFHOBHU105x2E0lxFHDaCcbX03rnkUrFYtHSk4R6ER4KFjKCtbF8+/DGz1ooP+tDAw8FuH6Gct+uOXWnEFwEVEINlGr69ld/EJM0okITjpXqISfRXoalZoTTvNpPFU0wGeMh7RkUOKLKy4oDcnhinAEMY2me0LBwf09kOFJqEgWmM8J6pOZrU/O/Wi/V4ZWXMZGkmgoyWxSmHOoYTtOAAyYp0XxiABPJzF8hGWGJiTaZVU0IaP7kRWif15FTR/cXtcZdGUcFHIFjcAoQuAQNcAuawAUE5OAZvII368l6sd6tj1nrklXOHIA/sj5/AGkzlQY=</latexit><latexit sha1_base64="1cL2yT0DI3+xFQCILVbdP4GdFE0=">AAACAHicbZBPS8MwGMZT/875r+rBg5fgEDzIaETQ40APHjxMWbfBVkqapVtYmpYkFUbpxa/ixYMiXv0Y3vw2Zl0PuvlA4Mfzvi9v3idIOFPacb6tpeWV1bX1ykZ1c2t7Z9fe22+rOJWEuiTmsewGWFHOBHU105x2E0lxFHDaCcbX03rnkUrFYtHSk4R6ER4KFjKCtbF8+/DGz1ooP+tDAw8FuH6Gct+uOXWnEFwEVEINlGr69ld/EJM0okITjpXqISfRXoalZoTTvNpPFU0wGeMh7RkUOKLKy4oDcnhinAEMY2me0LBwf09kOFJqEgWmM8J6pOZrU/O/Wi/V4ZWXMZGkmgoyWxSmHOoYTtOAAyYp0XxiABPJzF8hGWGJiTaZVU0IaP7kRWif15FTR/cXtcZdGUcFHIFjcAoQuAQNcAuawAUE5OAZvII368l6sd6tj1nrklXOHIA/sj5/AGkzlQY=</latexit><latexit sha1_base64="1cL2yT0DI3+xFQCILVbdP4GdFE0=">AAACAHicbZBPS8MwGMZT/875r+rBg5fgEDzIaETQ40APHjxMWbfBVkqapVtYmpYkFUbpxa/ixYMiXv0Y3vw2Zl0PuvlA4Mfzvi9v3idIOFPacb6tpeWV1bX1ykZ1c2t7Z9fe22+rOJWEuiTmsewGWFHOBHU105x2E0lxFHDaCcbX03rnkUrFYtHSk4R6ER4KFjKCtbF8+/DGz1ooP+tDAw8FuH6Gct+uOXWnEFwEVEINlGr69ld/EJM0okITjpXqISfRXoalZoTTvNpPFU0wGeMh7RkUOKLKy4oDcnhinAEMY2me0LBwf09kOFJqEgWmM8J6pOZrU/O/Wi/V4ZWXMZGkmgoyWxSmHOoYTtOAAyYp0XxiABPJzF8hGWGJiTaZVU0IaP7kRWif15FTR/cXtcZdGUcFHIFjcAoQuAQNcAuawAUE5OAZvII368l6sd6tj1nrklXOHIA/sj5/AGkzlQY=</latexit><latexit sha1_base64="1cL2yT0DI3+xFQCILVbdP4GdFE0=">AAACAHicbZBPS8MwGMZT/875r+rBg5fgEDzIaETQ40APHjxMWbfBVkqapVtYmpYkFUbpxa/ixYMiXv0Y3vw2Zl0PuvlA4Mfzvi9v3idIOFPacb6tpeWV1bX1ykZ1c2t7Z9fe22+rOJWEuiTmsewGWFHOBHU105x2E0lxFHDaCcbX03rnkUrFYtHSk4R6ER4KFjKCtbF8+/DGz1ooP+tDAw8FuH6Gct+uOXWnEFwEVEINlGr69ld/EJM0okITjpXqISfRXoalZoTTvNpPFU0wGeMh7RkUOKLKy4oDcnhinAEMY2me0LBwf09kOFJqEgWmM8J6pOZrU/O/Wi/V4ZWXMZGkmgoyWxSmHOoYTtOAAyYp0XxiABPJzF8hGWGJiTaZVU0IaP7kRWif15FTR/cXtcZdGUcFHIFjcAoQuAQNcAuawAUE5OAZvII368l6sd6tj1nrklXOHIA/sj5/AGkzlQY=</latexit> DT2, DR2, U2<latexit sha1_base64="/uUUGO6/1vWXPO86iJLxgG/JF+E=">AAACAHicbZC7SgNBFIbPxluMt1ULC5vBIFhI2A2ClgEtLCyiZJNAsiyzk0kyZPbCzKwQlm18FRsLRWx9DDvfxslmC038YeDjP+dw5vx+zJlUlvVtlFZW19Y3ypuVre2d3T1z/6Ato0QQ6pCIR6LrY0k5C6mjmOK0GwuKA5/Tjj+5ntU7j1RIFoUtNY2pG+BRyIaMYKUtzzy68dJWPTvvIw0POTheWs88s2rVrFxoGewCqlCo6Zlf/UFEkoCGinAsZc+2YuWmWChGOM0q/UTSGJMJHtGexhAHVLppfkCGTrUzQMNI6BcqlLu/J1IcSDkNfN0ZYDWWi7WZ+V+tl6jhlZuyME4UDcl80TDhSEVolgYaMEGJ4lMNmAim/4rIGAtMlM6sokOwF09ehna9Zls1+/6i2rgr4ijDMZzAGdhwCQ24hSY4QCCDZ3iFN+PJeDHejY95a8koZg7hj4zPH23blQk=</latexit><latexit sha1_base64="/uUUGO6/1vWXPO86iJLxgG/JF+E=">AAACAHicbZC7SgNBFIbPxluMt1ULC5vBIFhI2A2ClgEtLCyiZJNAsiyzk0kyZPbCzKwQlm18FRsLRWx9DDvfxslmC038YeDjP+dw5vx+zJlUlvVtlFZW19Y3ypuVre2d3T1z/6Ato0QQ6pCIR6LrY0k5C6mjmOK0GwuKA5/Tjj+5ntU7j1RIFoUtNY2pG+BRyIaMYKUtzzy68dJWPTvvIw0POTheWs88s2rVrFxoGewCqlCo6Zlf/UFEkoCGinAsZc+2YuWmWChGOM0q/UTSGJMJHtGexhAHVLppfkCGTrUzQMNI6BcqlLu/J1IcSDkNfN0ZYDWWi7WZ+V+tl6jhlZuyME4UDcl80TDhSEVolgYaMEGJ4lMNmAim/4rIGAtMlM6sokOwF09ehna9Zls1+/6i2rgr4ijDMZzAGdhwCQ24hSY4QCCDZ3iFN+PJeDHejY95a8koZg7hj4zPH23blQk=</latexit><latexit sha1_base64="/uUUGO6/1vWXPO86iJLxgG/JF+E=">AAACAHicbZC7SgNBFIbPxluMt1ULC5vBIFhI2A2ClgEtLCyiZJNAsiyzk0kyZPbCzKwQlm18FRsLRWx9DDvfxslmC038YeDjP+dw5vx+zJlUlvVtlFZW19Y3ypuVre2d3T1z/6Ato0QQ6pCIR6LrY0k5C6mjmOK0GwuKA5/Tjj+5ntU7j1RIFoUtNY2pG+BRyIaMYKUtzzy68dJWPTvvIw0POTheWs88s2rVrFxoGewCqlCo6Zlf/UFEkoCGinAsZc+2YuWmWChGOM0q/UTSGJMJHtGexhAHVLppfkCGTrUzQMNI6BcqlLu/J1IcSDkNfN0ZYDWWi7WZ+V+tl6jhlZuyME4UDcl80TDhSEVolgYaMEGJ4lMNmAim/4rIGAtMlM6sokOwF09ehna9Zls1+/6i2rgr4ijDMZzAGdhwCQ24hSY4QCCDZ3iFN+PJeDHejY95a8koZg7hj4zPH23blQk=</latexit><latexit sha1_base64="/uUUGO6/1vWXPO86iJLxgG/JF+E=">AAACAHicbZC7SgNBFIbPxluMt1ULC5vBIFhI2A2ClgEtLCyiZJNAsiyzk0kyZPbCzKwQlm18FRsLRWx9DDvfxslmC038YeDjP+dw5vx+zJlUlvVtlFZW19Y3ypuVre2d3T1z/6Ato0QQ6pCIR6LrY0k5C6mjmOK0GwuKA5/Tjj+5ntU7j1RIFoUtNY2pG+BRyIaMYKUtzzy68dJWPTvvIw0POTheWs88s2rVrFxoGewCqlCo6Zlf/UFEkoCGinAsZc+2YuWmWChGOM0q/UTSGJMJHtGexhAHVLppfkCGTrUzQMNI6BcqlLu/J1IcSDkNfN0ZYDWWi7WZ+V+tl6jhlZuyME4UDcl80TDhSEVolgYaMEGJ4lMNmAim/4rIGAtMlM6sokOwF09ehna9Zls1+/6i2rgr4ijDMZzAGdhwCQ24hSY4QCCDZ3iFN+PJeDHejY95a8koZg7hj4zPH23blQk=</latexit>

<latexit sha1_base64="GYgYvgqWQArVHhp/YU3rNKui/dE=">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</latexit>

n= const.

<latexit sha1_base64="AgbDc0H5Qem367oSrbNp35L+dAc=">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</latexit>

m=0

Figure 2.8: A schematic of the singular perturbation analysis with matched asymp- totic expansions when the length of a region is much larger than the boundary-layer thickness 𝐿_𝑖 ≫ 𝐷_{𝑇 𝑖}/𝑈_𝑖. Red lines represent the number density in boundary layers near the walls. The leading order solution insided the boundary layers has been ob- tained by Yan and Brady [15] for the number density of ABPs near a wall. The blue line represents the number density in the boundary layer where transport properties change. To leading order, the suspension can be treated as an infinite suspension with a step change in transport properties inside the boundary layer at𝑥 = 0. The number densities in the boundary layers are matched with number densities in the bulk, or outer regions, where the number density is constant to leading order.

The resulting formula, however, is not particularly illuminating and difficult to evaluate because of the sharp boundary layers at both walls and at the point of discontinuity in properties. Instead, a singular perturbation analysis with three

Dalam dokumen Effect and Motility-Induced Phase Separation (Halaman 38-43)