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Darcyan flow with relaxation effect

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The equation for the flow of a liquid through a porous medium is either elliptic or parabolic which implies that a disturbance in pressure or head is transmitted with infinite velocity. This is unsatisfactory from a physical viewpoint, although not necessarily from a practical one. If Darcy's law is completed with an inertial term the flow equation becomes a strongly damped wave equation. The proposed additional term can be identified from experiments with confined flow and free surface flow when the pressure or head varies harmonically with time.

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c 0 :

characteristic speed

c 1 :

characteristic speed

c w :

characteristic speed

C :

auxiliary parameter

d :

diameter of a bead

E :

modulus of elasticity

F :

auxiliary function

h :

piezometric height

H :

height of groundwater table

I 0(x):

modified Bessel function

I 1(x):

modified Bessel function

k :


K :

hydraulic conductivity

L :

tube length or channel length

L 0 :

reference length

m :

auxiliary parameter

n :


p :


Q :

mass flux

r=(x, y, z):

Cartesian coordinates

\(\hat r = (\xi ,n,\zeta )\) :

dimensionless Cartesian coordinates


tube radius

r 0 :

tube radius

R 0 :

tube radius

r p :

equivalent pore radius

t :


\(\bar \upsilon = (u,\upsilon ,w)\) :

velocity vector

V :


α :

coefficient of compressibility

β :

coefficient of compressibility

κ :

coefficient of compressibility

γ :


ε :

relaxation time

μ :

dynamic viscosity

ν :

Poisson's ratio

ϱ :


τ :

dimensionless time

ω :

angular frequency


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Rehbinder, G. Darcyan flow with relaxation effect. Appl. sci. Res. 46, 45–72 (1989). https://doi.org/10.1007/BF00420002

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  • Porous Medium
  • Free Surface
  • Wave Equation
  • Additional Term
  • Surface Flow