please answer all parts with a step by step solution

2-3. Flow in Porous Media: Darcy's Law A relationship used to model flow in isotropic porous media is Darcy's law, v=(Pg)P where is the Darcy permeability (units of m2 ), g is the gravitational acceleration, and P is the dynamic pressure (Chapter 6). The velocity and pressure in Darcy's law are each averaged over a length scale that is large compared with the pore dimensions, but small compared with the dimensions of the system of interest. That is, microstructural details are ignored, but variations in v and P at a more macroscopic level are considered. The pore structure enters only through its effect on , which is usually determined by experiment. (a) For a material of average porosity (volume fraction of pores), use Darcy's law to obtain expressions for conservation of mass in the fluid phase, in integral form and then in differential form. (b) Show that, for an incompressible fluid, the differential equation from part (a) reduces to 2P=0 (c) Consider an idealized porous material consisting of straight, cylindrical pores of diameter d. Assume that all pores are parallel to the x,y, or z axes and that they intersect at points described by a simple cubic lattice of dimension d. The overall dimensions of a sample of this material are of the order of L, where L. The volume flow rate Q in a single pore segment is described by Poiseuille's law, Q=128Pd4 where P is the pressure drop per segment. Evaluate for this material. FUNDAMENTALS OF HEAT AND MASS TRANSFER (d) Suppose now that the pores run only along the x and y axes. What change is required in the form of Darcy's law? Once again evaluate the permeability