A variational principle for the equations of nonstationary viscous filtration (Q2719451)

The paper deals with the equations governing the filtration of incompressible fluid in a porous medium NEWLINE\[NEWLINE \dot u = -\nabla p+\mu\Delta u - au+F,\quad\,\,\text{div}\,u=0.\tag{1} NEWLINE\]NEWLINE Here \(u\) is the filtration velocity vector, \(p\) is the pressure, \(F\) is the external force, \(\mu\) is the viscosity, \(a\) is the resistance coefficient which is a given function independent of time, and \(\,a\geq 0\). There are two hypotheses that underline the equations (1). The first is that both steady and unsteady filt ration flows are subjected to the same resistance force [\textit{V.~I.~Aravin} and \textit{S.~N.~Numerov}, ``The Theory of Fluids and Gases in Undeformable Porous Media'', Izda. Tekhn. Lit., Moscow (1953)]. The second is that the Darcy law of homogeneous steady filtration flow should be modified by incorporating viscosity effect when unsteady filtration is observed [\textit{I.~V.~Shirko}, In: ``Chislennoe Issledovanie v Aerogidrodinamike'', Nauka, Moscow (1986)]. The main result of this paper is the desc ription of necessary and sufficient conditions for the numbers \(\gamma_i\), which ensure a variational formulation of \(\Gamma\)-problem.