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 and 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.
This paper investigates the time-dependent Stokes flow of a viscous fluid in a channel with nonzero net entry flow. Assuming the fluid to be initially at rest with entry flow at the finite end of a semiinfinite channel, energy bounds for the flow are derived. It is shown that the flow decays exponentially in energy norm to a transient Poiseuille flow as the distance from the finite end tends to infinity. The problem was previously investigated by Lin (SAACM 2 (1992) 249-264) for the case in which the net entry flow was zero. Our methods are patterned after those of Lin, but a somewhat better choice of arbitrary constants yields an improved decay rate for Lin's problem.
a b s t r a c tFor a non-local reaction-diffusion problem with either homogeneous Dirichlet or homogeneous Neumann boundary conditions, the questions of blow-up are investigated. Specifically, if the solutions blow up, lower bounds for the time of blow-up are derived.
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