Potential flow:

6 CHAPTER 1. POTENTIAL FLOW:rRθUFigure 1.1: Sphere of radius R moving with a velocity U in a fluid which is atrest at large distance from the sphere.boundary condition for the fluid velocity u i at the surface of the sphere is,u i n i = U i n i (1.1)The equation for the velocity potential, 1.3, has to be solved subject to theboundary conditions 1.1 in order to determine the velocity field. Since equation1.3 is linear, the potential is a linear function of the velocity of the particle. Inaddition, the potential is a solution of the Laplace equation which decays to zeroat a large distance from the sphere, so it is a linear combination of the sphericalharmonic solutions. It is possible to construct only one scalar function which islinear in the velocity and in one of the spherical harmonics,φ = AU jx jr 3 (1.2)where A is a constant. The velocity field is then given by,u i = ∂φ∂x i(δij= AU jr 3 − 3x )ix jr 5(1.3)The constant A is determined from the normal velocity boundary condition 1.1.The normal fluid velocity at the surface is given by,u i n i = u ix ir= − 2AU jx jr 4= − 2AU jn jr 3 (1.4)