Stokes Flow: In class we saw that flow around a sphere of radius, a, at very low Reynolds number (Re < < 1) can be computed with a streamfunction, &, in spherical coordinates with v = 0 and no o dependence: Remembering that: a Show that (r, 0) = Ua 4 2 sin(0) [2 (1) - 3 ur (r, 0) = up (r, 0) = 1 r sin(0) 20 1 r sin(0) r 3 a ur(r, 0) = U cos(0) [1- [ - 2 + 1/ (9) ] +#+0] 3 a ue(r, 0) Usin(0) -1 + = b Give a brief explanation why u is independent of viscosity. c Is the flow symmetric? + ()] P(r, 0) = P - (1) 3aU 2r2 Cos (0) (2) (3) d Estimate to how many sphere radius the effect of the sphere is felt. Take the distance at which the velocity deficit is still 10% of free stream velocity U. e Show that the pressure field is: (4) (5) (6) where Poo is the free stream pressure f What can you conclude about the distribution of pressure in the front and back of the sphere? What does this imply for the drag on the sphere? g Show that the total drag, FD, found by integrating the pressure and shear around the sphere, is: Fn = 6Ua The shear stress distribution in the fluid in spherical coordinates with symmetry is: Tre (r, 0) = 1 Jur ????ue ug + 20 r r h Which fraction of the total drag is contributed by pressure and shear stress. (7) (8)