Question

A solid sphere of radius R is rotating with angular velocity ain otherwise still infinite fluid of density ? and viscosity ? (a) For creeping flow assumptions to hold, which condition(s) has to be satisfied? (b) Under creeping flow assumption, solve the velocity field in the flow. Does the solution still satisfy the creeping flow assumption at the far field? fluid p, ?

Answer 1

A solid sphere moving at constant speed through a viscous fluid is the centerpiece of steady creeping flow (Stokes, 1851). The solution may be worked out starting from the field equations (20-1) and the boundary conditions (see problem 20.8) but even if the details of obtaining the solution are a bit complicated, the result is fairly simple. In spherical coordinates with polar axis in the direction of LU the resulting flow is given by ea (20-5) where a is the radius of the sphere. The effective pressure coordinates and r tangent vectors. 3та 2-20 cos θ (20-6)

Figure 20.1 numeric integration of eq. (15-15) starting equidistantly at z =-100 to avoid problems of long range terms (see problem 20.6 for how to obtain the streamlines) flow around the unit sphere. Stream lines have been obtained by is forward-backwards asymmetric, such that the pressure is highest on the part of the sphere that turns towards the incoming fluid (at T). This asymmetry is in marked contrast with the potential flow solution (16-62) for a sphere where the symmetry of the pressure gives rise to d'Alembert's paradox. It is not particularly difficult to demonstrate from the spherical Laplacian (C- 16) that the Stokes flow field is indeed a solution to the creeping flow equations. The vanishing of the azimuthal component vp is a consequence of the symmetry of the problem, and implies that there is no lift. At the surface of the sphere the velocity field vanishes as it should, and at infinity it approaches that of the uniform flow with velocity U in the 2-direction, having Ur-U cos θ and υθ--U sin θ. Notice that the flow pattern is independent of the viscosity η of the fluid, and that the pressure is proportional to the viscosity. This is, as discussed above, common for creeping flow problems where the boundary conditions do not directly involve the pressure but only the velocity field