Investigation of Nonpotential Flow of Fluid in Porous Medium Taking into Account of Nonlinear Darcy Law and Variable Diffusion Coefficient

We have considered the non-potential flow of the incompressible fluid in the porous medium taking into account nonlinear Darcy law and different types of the diffusion coefficient. The flow is supposed to be cylindrically-symmetric and stationary. The velocity has two components: ⃗ =(,0,). We have considered the flow when = 0 + (,),||≪ 0, ≪ 0,0 = const. The combination of the Euler equations reduces to the equation of second order, and continuity equation reduces to an equation of first order for (,) and (,). These equations are linear differential equations with solutions of the form (,)= ()(), = ()(). For () we have obtained the Bessel equation of zero order with solution √ in the form ()= −0( ), = const. From the relation between () and () we √ √ 1 ′ have obtained (): ()= = 1( ), = const. The system of equations for () and () is reduced to one equation of the third order for (). We have obtained the √︁ Φ 0 exact solution of this equation with fixed diffusion coefficient ()= ℎ +Φ1 where Φ0,Φ1, , , = const. A special case when constants in the equation are connected in the relation 0 =200(1 + 00 2) is fully considered. In this case for function () we have obtained the equation of second order. Exact solutions of this equation are obtained with three types of diffusion: () = 0, ()= 0, ()= 0 − , 0 = const, = const. We have established that for all solutions the component of the velocity (,) decreases exponentially with increasing of .

nonpotential flow, porous medium, Darcy law, diffusion coefficient