Investigation of Potential Flow of Fluid in Porous Medium Taking Account of Darcy Law and Variable Diffusion Coefficient

We have considered the potential flow of the fluid in the porous medium taking into account Darcy low and different types of the diffusion coefficient in a tube with radius a. The flow is supposed to be stationary and cylindrically-symmetric and the Darcy force is a linear function of the velocity. We have established that a result of the potential flow is identity ∂2P∕∂r∂z ≡ ∂2P∕∂z∂r, where ∂P∕∂r and vz = ∂Φ∕∂z are defined from Euler equation for two components of the velocity: vr = ∂Φ∕∂r and vz = ∂Φ∕∂z, where Φ(r,z) is velocity potential. It means that Euler equation system is compatible and integrable, and the solution is reduced to the solution of the continuity equation. Continuity equation is linear differential equation for the potential Φ(r,z) and one assumes solution in divided variable: Φ(r,z) = U(r)W(z). For U(z) we have Bessel equation of zero order. This solution depends on the choice of the diffusion coefficient in the continuity equation. In all the occasions we have exact solution and established that component of the velocity vz descreases like exponent with increase of z.

potential flow, stationary flow, porous medium, diffusion, Darcy low