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Nonlinear Electrokinetic Phenomena in Nanosized Dispersions
Published in Victor M. Starov, Nanoscience, 2010
Equation 5.21 with boundary conditions 5.22 and 5.29 expresses the traditional formulation of the Stokes problem for particle motion under the action of force F→ and torque M→ in a liquid, free from volume forces. The solution of this problem in the case of a spherical particle results in traditional formulae: Equation 5.4 for dielectrophoresis velocity, 5.6 for the angular velocity of electrorotation, and 5.7 for the energy of the particle’s electric orientation.
One-Phase and Two-Phase Flow in Highly Permeable Porous Media
Published in Heat Transfer Engineering, 2019
This macro-scale law is supported by many experimental results, in particular by Darcy’s own experiments. It is also confirmed by theoretical developments. Upscaling Stokes equations at the pore-scale (Eqs. (7)–(9)) via homogenization theory [25] or volume averaging [20], [26] yields a sound physical basis for the use of Darcy’s law. Key steps of the mathematical developments are as follows. Linearization of the Navier–Stokes equations in the limit Re → 0 to obtain Stokes’ problem,Introduction of a coupled problem for macro-scale values and pore-scale deviations defined as Simplifications of the problem based on the separation of spatial scales,Introduction of an approximated solution of the coupled problem through a closure, which emerges from the mathematical structure of the Stokes problem. At first-order, this closure reads where the vector b and tensor B are called mapping variables. These are solutions of a local problem that is usually solved over a representative unit cell (see the Appendix),Derivation of the macro-scale equation from the averaged equations and the closure: one obtains Darcy’s law, Eq. (13) and the intrinsic permeability, given by which is calculated from the solution of the closure problem over the representative unit cell. Nowadays, this is almost a routine operation to calculate such effective properties from reconstructed 3D images, such as those obtained using X-ray microtomography.
Numerical study of the wall effect on particle sedimentation
Published in International Journal of Computational Fluid Dynamics, 2018
Liang-Hsia Tsai, Chien-Cheng Chang, Tsorng-Whay Pan, Roland Glowinski
In Stage 1, a degenerated generalised Stokes problem (i.e. with zero viscosity) is solved by a Neumann preconditioned Uzawa/conjugate gradient algorithm as discussed in Hao et al. (2009). The elliptic problems in the preconditioned steps are solved using the fast solver FISHPAK developed by Adams, Swarztrauber, and Sweet (1980).
Viscous fluid-thin cylindrical elastic body interaction: asymptotic analysis on contrasting properties
Published in Applicable Analysis, 2019
and for the limit velocity pressure, , the following Stokes problem with no-slip condition on the coupling boundary