Helmholtz-Smoluchowski slip-flow

A slip boundary condition is assumed to exist at the walls due to electroosmotic flow given by the Helmholtz-Smoluchowski equation... 

This formula for the electroosmotic velocity past a plane charged surface is known as the Helmholtz-Smoluchowski equation. Note that within this picture, where the double layer thickness is very small compared with the characteristic length, say alX t> 100, the fluid moves as in plug flow. Thus the velocity slips at the wall that is, it goes from U to zero discontinuously. For a finite-thickness diffuse layer the actual velocity profile has a behavior similar to that shown in Fig. 6.5.1, where the velocity drops continuously across the layer to zero at the wall. The constant electroosmotic velocity therefore represents the velocity at the edge of the diffuse layer. A typical zeta potential is about 0.1 V. Thus for = 10 V m" with viscosity that of water, the electroosmotic velocity U 10 " ms, a very small value. 

Extension to 2D and 3D Systems In the majority of microfluidic cases where 1/k is much smaller than the channel height, the Helmholtz-Smoluchowski equation provides a reasonable estimate of the flow velocity at the edge of the double layer field. As such when modeling two- and three-dimensional flow systems, it is common to apply this equation as a slip boundary condition on the bulk flow field. Since beyond the double layer by definition... 

Particle trajectories are determined by a combination of fluid-flow, electrophoresis and DEP. The fluid-flow is driven by electroosmosis for the case of interest here. For the thin Debye layer approximation, electroosmotic flow may be simply modeled with a slip velocity adjacent to the channel walls that is proportional to the tangential component of the local electric field, as shown by the Helmholtz-Smoluchowski equation 12). Here, the proportionality constant between the velocity and field is called the electroosmotic mobility, tigo- Fluid-flow in microchannels becomes even simpler for ideal flow conditions where the zeta potential, and hence /ieo, is uniform over all walls and where there are no pressure gradients. For these conditions, it can be shown that the fluid velocity at all points in the fluid domain is given by the product of the local electric field and// o(/i). 

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