Integral-equation solution of potential flow past a porous body of arbitrary shape

Abstract

Potential flow past a porous body of arbitrary shape with constant physical permeability k0, as well as the flow in the corresponding porous medium, are analysed by means of a pair of linear Fredholm integral equations of the second kind. As an example for verification of the proposed general method, the case of a two-dimensional porous circular cylinder is worked out in detail.

Potential Flow Past a Slightly Deformed Porous Circular Cylinder

“…It is observed that, when = 0 in (32), the resultant expression for F x agrees with that of the potential flow past a circular cylinder, given in Power et al (1984).…”

“…In contrast with the case of Power et al (1984) of a porous circular cylinder of constant permeability where the total force is independent of the cylinder size, in this case the total force is a function of the characteristic radius r 1 of the slightly deformed cylinder and the radius r 2 of the circular core. In Fig.…”

“…It may be noted that for both positive and negative values of when m = 2, 4, the contour r = r 1 (1 + cos mθ) tells that the deformation is symmetric with respect to both the axis, while for m = 3 it is not symmetric. It may be noted that Power et al (1984) gave expression for the drag acting on the surface of a porous cylinder r = r 1 with permeability k 0 in case of unform flow along x-axis as…”

“…This problem is reduced to a nonlinear boundary value problem of Poincare type for the normal pressure function inside the unit circle. Power et al (1984) solved a problem of potential flow past a porous body of arbitrary shape with constant permeability k 0 , while the interior flow is represented as a viscous potential flow with the corresponding pressure related to the seepage velocity by Darcy's law. The solution was found by means of a pair of non-linear Fredholm integral equations.…”

“…Therefore, for the best approximation to the natural porous structure, here we will discuss a problem of uniform potential flow past a deformed porous circular cylinder with a porous core of different permeability, and the corresponding solution is calculated using analytical techniques. We also use similar assumption made by Power et al (1984) that the dimensionless parameter K * is small so that second-order terms are neglected. The total force exerted by the exterior flow upon the body is dependent on the thickness of the porous materials and the permeability ratio of the two regions.…”