A novel non-primitive Boundary Integral Equation Method for three-dimensional and axisymmetric Stokes flows

Abstract

A new Boundary Integral Equation (BIE) formulation for Stokes flow is presented for three-dimensional and axisymmetrical problems using non-primitive variables, assuming velocity field is prescribed on the boundary. The formulation involves the vector potential, instead of the classical stream function, and all three components of the vorticity are implied. Furthermore, following the Helmholtz decomposition, a scalar potential is added to represent the solenoidal velocity field. Firstly, the BIEs for three-dimensional flows are formulated for the vector potential and the vorticity by employing the fundamental solutions in free space of vector Laplace and biharmonic equations. The equations for axisymmetric flows are then derived from the three-dimensional formulation in a second step. The outcome is a domain integral free BIE formulation for both three-dimensional and axisymmetric Stokes flows with prescribed velocity boundary condition. Numerical results are included to validate and show the efficiency of the proposed axisymmetric formulation.

Appendices

Appendix A: Vector operations on tensors

The necessary formulas involving the divergence and curl operation of a function, irrespective of the coordinate system are as follows:

(A.1)

(A.2)

(A.3)

(A.4)

Appendix B: Axisymmetric kernel functions in terms of Complete Elliptic Integrals

The complete elliptic integral of the first and second kind, K(m) and E(m), are defined as

(B.1)

(B.2)

where m is the elliptic integral parameter (0≤m≤1). Some functions which can be represented in the form of complete elliptic integrals required later to express the kernel functions are

(B.3)

(B.4)

(B.5)

(B.6)

The axisymmetric integration is carried out in cylindrical coordinates (ρ,ϕ,z), where the distance between the source and evaluation points is

(B.7)

and the elliptic integral parameter is defined as

$$ \mathrm{m} = \frac{4\rho\rho'}{(z - z')^{2} + (\rho+ \rho')^{2}} $$

(B.8)

Taking into account that the plane of computation is the origin plane (ϕ=0), we obtain

$$ \mathrm{r} = \bigl[ a - b\cos\phi' \bigr]^{1/2},\qquad \mathrm{m} = \frac{2b}{a + b} $$

(B.9)

where

$$ a = \bigl(z - z'\bigr)^{2} + \rho^{2} + \rho^{\prime\,2}, \qquad b = 2\rho\rho' $$

(B.10)

For specific values of i and j, the kernel functions appearing in the axisymmetric BIE formulation are

For i=0 and j=1:

$$ G_{01}^{AX} = \frac{4K(\mathrm{m})}{(a + b)^{1/2}} $$

(B.11)

For i=0 and j=3:

(B.12)

For i=1 and j=1:

(B.13)

For i=1 and j=3:

(B.14)

For i=1 and j=−1:

(B.15)