Boundary Integral and Singularity Methods for Linearized Viscous Flow
By C. Pozrikidis
Publisher: Cambridge University Press
Print Publication Year: 1992
Online Publication Date:February 2010
Online ISBN:9780511624124
Hardback ISBN:9780521405027
Paperback ISBN:9780521406932
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511624124.008
Subjects: Numerical analysis and computational science, Differential and integral equations, dynamical systems and control theory
Image View Extract Fullview: Text View | Enlarge Image ‹ Previous Chapter ›Next Chapter
Introduction
In the preceding chapters we developed integral representations of a flow in terms of boundary distributions of Green's functions and their associated stress tensor. Physically, these representations involve boundary distributions of point forces, point sources, and point force dipoles. When we applied the representations at the boundaries of the flow and imposed the required boundary conditions, we obtained Fredholm integral equations of the first or second kind for the unknown boundary functions. Two advantages of solving these equations instead of the primary differential equations are reduction of the dimensionality of the mathematical problem with respect to the physical problem by one unit, and efficient treatment of infinite flows.
Now, in section 6.5 we saw that one important aspect of the numerical treatment of the integral equations resulting from boundary integral representations is the accurate computation of the singular single-layer and double-layer integrals. True, the singularities of the corresponding kernels may be subtracted off or eliminated using a suitable identity or changing the variables of integration, but this places burdens on the computer implementation and increases the cost of the computations. One ad-hoc way to circumvent this difficulty is to move the pole of the Green's function and its associated stress tensor from the boundary D onto a surface exterior to the flow, as illustrated in Figure 7.1.1. Of course, this is an arbitrary remedy that lacks physical foundation, but nevertheless it appears worth consideration. Furthermore, taking this action one step further, we may replace the continuous surface distribution of the singularities with line or point-wise distributions.
pp. i-vi
pp. vii-x
pp. xi-xii
pp. 1-18
2 - The boundary integral equations : Read PDF
pp. 19-75
3 - Green's functions : Read PDF
pp. 76-102
4 - Generalized boundary integral methods : Read PDF
pp. 103-138
5 - Flow due to interfaces : Read PDF
pp. 139-158
6 - The boundary element method : Read PDF
pp. 159-189
7 - The singularity method : Read PDF
pp. 190-214
pp. 215-248
pp. 249-256
pp. 257-259