Nonlinear Darcy fluid flow with deposition

Abstract

In this article we consider a model of a filtration process. The process is modeled using the nonlinear Darcy fluid flow equations with a varying permeability, coupled with a deposition equation. Existence and uniqueness of the solution to the modeling equations is established. A numerical approximation scheme is proposed and analyzed, with an a priori error estimate derived. Numerical experiments are presented which support the obtained theoretical results.The lack of regularity of the fluid velocity, u ∈ H div (Ω), leads to an open question of the existence of a solution to (1.1)-(1.3). In order to obtain a modeling system of equations for which a solution can be shown to exist, we replace η in (1.1) by an smoothed porosity, η s . The approach of regularizing the model with the introduction of η s is, in part, motivated by the Darcy fluid flow equations, which can be derived by averaging, e.g. volume averaging [20], homogenization [1], mixture theory [17]. Recently in [9] we considered the case of (steady-state) generalized Newtonian fluid flow through a porous medium, modeled by equations (1.1), (1.2), with µ ef f κ(η) −→ β(|u|). With the general assumptions that β(·) was a positive, bounded, Lipschitz continuous function, bounded away from zero, and with β(|u|) replaced with β(|u s |), existence of a solution was established. Two smoothing operators for u were presented. One was a local averaging operator, whereby u s (x) is obtained by averaging u in a neighborhood of x. The second smoothing operator, which is nonlocal, computes u s (x) using a differential filter applied to u. That is, u s is given by the solution to an elliptic differential equation whose right hand side is u. For establishing the existence of a solution the key property that the smoothing operator needs to satisfy is that it transform a weakly convergent sequence in L 2 (Ω) into a sequence which converges strongly in L ∞ (Ω).

Heat Transfer in a Fractional Nanofluid Flow through a Permeable Medium

“…Darcy flow is analyzed by various researchers. Nonlinear Darcy flow is examined by Ervin et al [31]. e Lorentz force shows a significant role in viscoelastic problems such as in MHD generators and control theory.…”