Compressible flows generalized expressions

Io(x) and l2(x) are modified Bessel functions of zero and second order, Bb and pe are the bulk elasticity modulus and the density of phase B, / is the capillary length (/ ac). Substituting Eqs. (12) and (14) into (8) one can obtain the general expressions for the bubble (drop) volume variation 6V ,(ico), the pressure variations inside the bubble (drop) 5P, (ici)), and inside the cell 6P (ico) at the given bP, and 6Vp . The effect of both the flow mobility and the compressibility of the medium in the capillary on the oscillating bubble or drop measurements is analysed in details in [26]. Here only the simpler case of quasi-stationary flow of incompressible media in the capillary is described. 

X = linear coordinate in direction opposite to flow In the above notation, the units of Di will be distance /time, and the permeability Pi may be expressed in the same units. If different units are to be adopted for Pi then the form of the gas law or equation of state may be introduced, whereby (Pyi) V = z iii RT, where Pyt is the partial pressure (or absolute activity) of component i, iii is the moles of component i, and z is the compressibility factor (which in general is applicable to singlephase liquids as well as gases). Accordingly,... 

By introducing the appropriate expression for the hydraulic head, i.e. Equation 1.3 or 1.4, into Darcy s equation 1.8, it describes the flow for compressible and incompressible groundwater, respectively. The flow of compressible groundwater can thus be described by a more general form of Darcy s equation, in which the gradient of the groundwater potential is given by Equation 1.5... 

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