Low Reynolds Number Shear Flow along an Elliptic Hole in a Wall

Abstract

The influence of an elliptic hole (of semi-axes a₁ and a₂, t=a₂⁄a₁≤1) on the shear flow along a thin solid plane wall is investigated on the basis of the Stokes equation. Exact solution is obtained in terms of ellipsoidal coordinates. If the flow is parallel to the major (or minor) axis, a radial incoming flow induced upstream along ±π⁄4 direction to the wall moves almost parallel to the axis in the neighbourhood of the hole and is reflected into a radial outgoing flow. The total flux of this flow is found to be (Remark: Graphics omitted.), where c=t²K+(1−2t²)E (or (2−t²)E−t²K), K and E being the complete elliptic integrals with the modulus k=(1−t²)^1⁄2.