Displacement of inviscid fluid by a sphere moving away from a wall

We develop a theoretical analysis of the displacement of inviscid fluid particles and material surfaces caused by the unsteady flow around a solid body that is moving away from a wall. The body starts at position h(s) from the wall, and the material surface is initially parallel to the wall and at distance h(L) from it. A volume of fluid D-f+ is displaced away from the wall and a volume D-f- towards the wall. D-f+ and D-f- are found to be sensitive to the ratio h(L)/h(s). The results of our specific calculations for a sphere can be extended in general to other shapes of bodies. When the sphere moves perpendicular to the wall the fluid displacement and drift volume D-f+ are calculated numerically by computing the flow around the sphere. These numerical results are compared with analytical expressions calculated by approximating the flow around the sphere as a dipole moving away from the wall. The two methods agree well because displacement is an integrated effect of the fluid flow and the largest contribution to displacement is produced when the sphere is more than two radii away from the wall, i.e. when the dipole approximation adequately describes the flow. Analytic expressions for fluid displacement are used to calculate D-f+ when the sphere moves at an acute angle a away from the wall. In general the presence of the wall reduces the volume displaced forward and this effect is still significant when the sphere starts 100 radii from the wall. A sphere travelling perpendicular to the wall, alpha = 0, displaces forward a volume D-f+(0) = 4 pi a(3)h(L)/3(3/2)h(s) when the marked surface starts downstream, or behind the sphere, and displaces a volume D-f+(0) similar to 2 pi a(3)/3 forward when it is marked upstream or in front of the body. A sphere travelling at an acute angle away from the wall displaces a volume D-f+(alpha) similar to D-f+(0) cos alpha forward when the surface starts downstream of the sphere. When the marked surface is initially upstream of the sphere, there are two separate regions displaced forward and a simple cosine dependence on alpha is not found. These results can all be generalized to calculate material surfaces when the sphere moves at variable speed, displacements no longer being expressed in terms of time, but in relation to the distance travelled by the sphere.