L. E. Payne

Analysis of the boundary condition at the interface between a viscous fluid and a porous medium and related modelling questions

Abstract This paper deals with three fundamental modelling questions for the Darcy and Brinkman equations for flow in a porous medium. It is first shown that in the Dirichlet initial — boundary value problem for the Brinkman equations the solution depends continuously on the viscous coefficient. Then L 2 convergence of the solution of this problem to the solution of an analogous problem for the Darcy equations is established. Finally, it is proved that for flow in a domain occupied by a viscous fluid in contact with a porous solid, the solution depends continuously on a coefficient in the interface boundary condition. The continuous dependence holds for Stokes flow in the fluid, and the analogous Navier-Stokes situation is discussed.

Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity

The equations for convective fluid motion in a porous medium of Brinkman or Forchheimer type are analysed when the viscosity varies with either temperature or a salt concentration. Mundane situations such as salinization require models which incorporate strong viscosity variation. Therefore, we establish rigorous a priori bounds with coefficients which depend only on boundary data, initial data and the geometry of the problem, which demonstrate continuous dependence of the solution on changes in the viscosity. A convergence result is established for the Darcy equations when the variable viscosity is allowed to tend to a constant viscosity.

Structural stability for the Darcy equations of flow in porous media

A priori bounds are derived for the Darcy equations of flow in porous media when the porous body is subject to boundary conditions of Newton cooling type. With the aid of these a priori bounds we are able to demonstrate continuous dependence on the cooling coefficient when the boundary condition of Newton cooling type is employed. We further show that the solution depends continuously on a change in the equation of state employed in the body force in the Darcy equation. The model is allowed to change from one of Boussinesq convection type to one more general, and structural stability is established.

Unconditional Nonlinear Stability in Temperature-Dependent Viscosity Flow in a Porous Medium

The equations of flow in porous media attributable to Forchheimer are considered. In particular, the problem of thermal convection in such a medium is addressed when the viscosity varies with temperature. It is shown that nonlinear stability may be achieved naturally for all initial data by working with L 3 or L 4 norms. It is also shown that L 2 theory is not sufficient for such unconditional stability. Previous work has established nonlinear stability for vanishingly small initial data thresholds, but we believe this is the first analysis that addresses the important physical problem of unconditional stability. It is shown how to extend the nonlinear analysis for a viscosity linear in temperature to the cases when the viscosity may be quadratic or when penetrative convection is allowed in the layer.

Convection in thawing subsea permafrost

Detailed quantitative values are obtained for the critical values of the salt Rayleigh number for both linear and nonlinear stability, for a simplified model appropriate to the onset of buoyant, relatively fresh water motion in a layer of salty subsea sediments. The geophysical problem that motivates this work arises because of the formation of substantial permafrost around the Earth’s shores some 18000 years ago. With the rise of sea levels the permafrost has responded to the relatively warm and salty sea, which has created a thawing front and a layer of salty sediments beneath the sea bed. This phenomenon has been studied extensively off the coast of Alaska by W. Harrison and coworkers and our analysis is based on a model developed by W. Harrison and D. Swift. From the mathematical viewpoint the analysis reduces to studying convection in a porous medium with a nonlinear boundary condition. We find the critical Rayleigh number for convection according to linear theory, but our main thrust is directed toward the nonlinear problem. Here we use an energy method to determine a critical Rayleigh number below which convection cannot develop. We first show there is a critical Rayleigh number close to that of linear theory, which guarantees unconditional nonlinear stability. Then we demonstrate conditional nonlinear stability (i. e. conditional upon the existence of some finite threshold amplitude, which we calculate) provided the critical Rayleigh number of linear theory is not exceeded. The latter analysis requires two approaches according to whether the two-dimensional or three-dimensional problem is considered. In particular, a novel energy has to be introduced to make the three-dimensional problem tractable.

Convergence of the equations for a Maxwell fluid

The equations for the flow of a viscoelastic fluid of the Maxwell type are analyzed in a linear approximation. First, we establish that the solution depends continuously on changes in the relaxation time. Next, we investigate how the solution to the linearized Maxwell system converges to the solution to Stokes flow as the relaxation time tends to zero. Convergence in different measures is examined and specific a priori bounds are derived.

Effect of errors in the spatial geometry for temperature dependent stokes flow

Abstract A priori bounds are established for the solution to the problem of Stokes flow in a bounded domain, for a viscous, heat conducting, incompressible fluid, when changes in the spatial geometry are admitted. These bounds demonstrate how the velocity field and the temperature field depend on changes in the spatial geometry and also yield a convergence theorem in terms of boundary perturbations. The results have a direct bearing on an error analysis for a numerical approximation to non-isothermal Stokes flow when the boundary of a complicated domain is approximated by a simpler one, e.g., in the procedure of triangulation combined with finite elements.

A naturally efficient numerical technique for porous convection stability with non-trivial boundary conditions

A highly efficient numerical technique is presented for solving eigenvalue problems which arise in complicated convection — instability studies in porous media. The differential equations are written as a system of ‘natural’ variables which are suggested by the way the boundary conditions arise. The method easily gives high resolution in boundary layers, yields all the eigenvalues and eigenfunctions, deals with complex coefficients, and can handle spatially dependent coefficients in a very efficient manner. The numerical technique is motivated by the practical problem of salinization in porous sands in arid zones as is beautifully modelled by Gilman and Bear. Since the salinization study of Gilman and Bear is a prototype for the field of convective motion in unsaturated porous soils and this field is one which is increasingly occupying geotechnical attention, we believe the numerical method presented here has much potential. Copyright