Navier-Stokes equation with electric force

The Basic Equations of Electrokinetics In the continuum model, electrokinetic flows are described by the incompressible Navier-Stokes equations with a volume density of electric forces —peVcj) where pe is the electric charge density and 4> is the electric potential ... 

Outside of the Debye layer, the fluid is electrically neutral. Inside the Debye layer, the fluid obeys the Navier-Stokes equation with an electrical body force... 

A careful account of the problem can be found in Ref. [95]. Ohshima et al. [96] first found a numerical solution of the problem, valid for arbitrary values of the zeta potential or the product Ka. In the same paper, they dealt with the problem of finding the sedimentation potential and the DC conductivity of a suspension of mercury drops. The problems are solved following the lines of the electrophoresis theory of rigid particles previously derived by O Brien and White [18]. The liquid drop is assumed to behave as an ideal conductor, so that electric fields and currents inside the drop are zero, and its surface is equipotential. The main difference between the treatment of the electrophoresis of rigid particles and that of drops is that there is a velocity distribution of the fluid inside the drop, Vj, governed by the Navier-Stokes equation with zero body force (in the case of electrophoresis), and related to the velocity outside the drop, v, by the boundary conditions ... 

Except for the theories of DERJAGUIN and co-workers [2.121], the theory in this regime has been developed for a stationary particle in a temperature gradient where the thermal force on a particle is exactly balanced by some external force such as an electric field for a charged particle, as in Millikan-cell experiments. The first approximate theory in this case was developed by EPSTEIN [2.131] who used the Navier-Stokes equation with a "thermal creep" boundary condition to obtain ... 

Henry [ 157] solved the steady-flow continuity and Navier-Stokes equations in spherical geometry, neglecting inertial terms but including pressure and electrical force terms, coupled with Poisson s equation. The electrical force term in Henry s analysis consisted of the sum of the externally applied electric field and the field due to the double layers. His major assumptions are low surface potential (i.e., potentials less than approximately 25 mV) and undistorted double layers. The additional parameter ku appearing in the Henry... 

Assume that the resistance to the cylinder motion is due to the shear stress associated with the electroosmotic flow that is generated, so that the Navier-Stokes equation reduces to a balance between viscous and electrical forces. Show that the solution for the electrophoretic velocity of the cylinder is the same as that for a sphere of the same zero potential with the Debye length small. 

By measuring velocity of a spherical particle sinking in a liquid under gravity force the viscosity of the liquid can be found (the buoyancy effect should be taken into account). Note that in Section 7.3.3, using an electric field as an action force, the same Stokes law has been applied (with some precautions) to evaluation of velocity and mobility of spherical ions in isotropic liquids or nematic liquid crystals For large Reynolds numbers, Re = pv//ri>l the flow in no longer laminar and even becomes turbulent. Then, the convective term (vV)v should be added to the left part of the Navier-Stokes equation... 

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