Complete general solution of Stokes equations for plane boundaries

A general solution of Oseen equations

“…We shall prove that every solution (q, p) of Oseen equations can be expressed in the above form in terms of two scalar functions A and B and hence show that it is a general solution. We state the following result due to Venkatalaxmi et al (2004), which we shall use in our proof: If q ∈ C 2 (R 3 ) and satisfies ∇ • q = 0, then we can find scalar functions A and B such that q is given by ( 7), where A and B are the solutions of…”

“…From the result due to Venkatalaxmi et al (2004), since ∇ • q = 0, the velocity vector can be expressed as (7) in terms of A and B. Now, using (14), we get from (11) that…”

“…Then we show that q represents a possible Oseen flow as follows. Since q is divergence-free, from the result due to Venkatalaxmi et al (2004), we can express q as in ( 7), where A and B satisfy ( 11) and ( 12), respectively. Then using the fact that q and ∇ × q satisfy ( 14) and ( 18), respectively, we can show that the functions A and B satisfy ( 9) and (10), respectively using Lemmas 1-4.…”

Stokes velocity generated by a point force in various geometries

“…Classical Green functions for the Stokes equations in the presence of a plane solid wall, free surface or fluid-fluid interface are still a subject of research [43][44][45], resulting in useful representations and interpretations. In this paper we contribute to this field by demonstrating the graphical addition of velocity fields generated by different multipoles.…”