Measurement of the relaxation time in Darcy flow

The inertia of a liquid flowing through a porous medium is normally ignored, but if the acceleration is great, it may be important. The relaxation time, defined so that it alone accounts for the inertia, has been determined experimentally with a simple oscillator. A U-Tube is provided with a porous plug and filled with a liquid. During pendulation of the liquid, the frequency and the damping define the relaxation time. The measured value of the relaxation time is about 10 times the theoretical estimate derived from Navier-Stokes equation.Symbols E modulus of elasticity - E_D dissipated energy - E_k kinetic energy - g acceleration of gravity - G pressure gradient - h height - K₀ permeability - L length of porous plug - n porosity - P dissipated power - pressure - R half the tube length - R_c radius of the tube bend - r radial coordinate - r_o radius of the tube - s coordinate along a streamline in the tube - t time - v flux per unit area - it relaxation time - , auxiliary variables - , v dynamic and kinematic viscosity - , velocity potential for inviscid flow and gravity potential - dissipation function - displacement of the liquid - ,_o frequency of damped and undamped oscillations