Anisotropy effect on the convection of a heat-conducting fluid in a porous medium and cosymmetry of the Darcy problem

Darcy Convection Anisotropic Problem with Multi Stability of Stationary Motions for Rectangle

“…Next, a perturbation of the temperature field is introduced T (x, y, t) = T * (y) + θ (x, y, t) and a transition is made to dimensionless quantities [1]. For stream function ψ and temperature deviations θ the following initial-boundary problem is obtained with respect to the linear profile:…”

“…In this case, the system of equations is co-symmetric according to [2], those there is a vector field L, which is orthogonal to the vector field of the problem and does not vanish on a nontrivial stationary solution. IN [1] set the conditions under which the task (1)-( 4) is cosymmetric, these conditions are clarified by the following lemma. (stationary or non-stationary) [3][4][5].…”

“…In the case of anisotropy, the Darcy problem with co-symmetry in the [1] an analysis of the occurrence of stationary convective regimes, branching o from the loss of stability of mechanical equilibrium, is given. The conditions on the coefficients of the system under which the problem has co-symmetry and the branching of the continuous family of stationary modes are analytically determined.…”

“…Table 1 The first row of the table answers the isotropic problem [4], the next three lines are co-symmetries in the anisotropic case. The second line corresponds to the conditions given in [1], and the third and fourth conditions (5). Thus, the conditions for the existence of co-symmetry (5) allow you to expand the set of values of the coefficients for which you can apply the formula for calculating the critical Rayleigh numbers [1].…”

“…The second line corresponds to the conditions given in [1], and the third and fourth conditions (5). Thus, the conditions for the existence of co-symmetry (5) allow you to expand the set of values of the coefficients for which you can apply the formula for calculating the critical Rayleigh numbers [1]. Critical numbers characterize the occurrence of convection as a result of monotonous loss of stability of mechanical equilibrium.…”