The annular region between two concentric rigid spheres of radii a and ha (with > 1)...

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The annular region between two concentric rigid spheres of radii a and ha (with à> 1) is filled with Newtonian fluid of viscosity u and density p. The outer sphere is held stationary whereas the inner sphere is made to rotate with angular velocity 2. Assume that inertia is negligible so that the fluid is in the Stokes' flow regime. (a) Without any detailed calculation, explain why the pressure field in the fluid must be identically zero. Show that this implies that continuity is trivially satisfied. (b) By solving the Stokes' equations, show that the fluid velocity is [-)]a. 1 u = \r, where r is the position vector and r is the magnitude of r. (c) Calculate the stress field in the fluid. Deduce that the couple L exerted on the inner sphere by the fluid is L-πμα 23 Ω. - 1 Briefly comment on the limit 2 >1. For 1 → 1 , show that the couple is to leading order 8 L = 1 1 and give an physical interpretation of this result. d) What is the condition that inertia is indeed negligible for the case à »1? Explain why this is not the appropriate condition for the case → 1. The annular region between two concentric rigid spheres of radii a and ha (with à> 1) is filled with Newtonian fluid of viscosity u and density p. The outer sphere is held stationary whereas the inner sphere is made to rotate with angular velocity 2. Assume that inertia is negligible so that the fluid is in the Stokes' flow regime. (a) Without any detailed calculation, explain why the pressure field in the fluid must be identically zero. Show that this implies that continuity is trivially satisfied. (b) By solving the Stokes' equations, show that the fluid velocity is [-)]a. 1 u = \r, where r is the position vector and r is the magnitude of r. (c) Calculate the stress field in the fluid. Deduce that the couple L exerted on the inner sphere by the fluid is L-πμα 23 Ω. - 1 Briefly comment on the limit 2 >1. For 1 → 1 , show that the couple is to leading order 8 L = 1 1 and give an physical interpretation of this result. d) What is the condition that inertia is indeed negligible for the case à »1? Explain why this is not the appropriate condition for the case → 1.