Electroosmotic Flow Around a Particle Assume a spherical particle of diameter a λD is suspended and held stationary inside a microchannel with diameter d a and that the particle and channel are made from the same material and have the same surface potential. An electric field E∞ is applied to

Question

Electroosmotic Flow Around a Particle

Assume a spherical particle of diameter [latex]a \gg \lambda_{D}[/latex] is suspended and held stationary inside a microchannel with diameter [latex]d \gg a[/latex] and that the particle and channel are made from the same material and have the same surface potential. An electric field [latex]E_{\infty}[/latex] is applied to the system. Calculate the flow solution (in terms of velocity potential [latex]\phi[/latex] ) around this particle.

Accepted Answer

Recall from Eqns. (7.71)-(7.74) and Fig. [latex]7.17[/latex] that potential flow around a sphere can be modeled as the superposition of a uniform stream [latex](\phi=U r \cos heta)[/latex] with a doublet of strength [latex]s=-U a^{3} / 2[/latex], for which [latex]\phi=U a^{3} \cos heta / 2 r^{2}[/latex]. This solution identically solves the electroosmotic flow around a particle, with [latex]U[/latex] given by [latex]\varepsilon \zeta E_{\infty} / \mu[/latex]. Since [latex]d \gg a[/latex], it is correct to model the electroosmotic flow as a uniform stream.

[latex]\phi=U z=U r \cos heta, \quad \psi=-\frac{1}{2} U r^{2} \sin ^{2} heta.[/latex] (7.71)

[latex]\psi=-\frac{1}{2} U r^{2}\left(1-\frac{a^{3}}{r^{3}} ight) \sin ^{2} heta.[/latex] (7.74)

Question 12.3: Electroosmotic Flow Around a Particle Assume a spherical par...

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