In Section 10.2.2 we calculated the velocity field for flow past a stationary sphere. Such a sphere experienced a buoyant force but no lift. To generate lift, we need to rotate the sphere. We can calculate the velocity profile for the rotating sphere by changing the boundary condition on the velocity at the surface of the sphere. For a sphere rotating about the \(\theta\)-direction at a rate of \(B\) revolutions per second (Hint: \(v_{\theta}=2 \pi r_{o} B \sin \theta\) ):

a. What are the boundary conditions on the sphere surface?

b. Solve for the stream function.

c. What are the \(r\) and \(\theta\) velocity components?

d. Substitute the velocity into the momentum equations, determine an expression for the pressure (many pages of algebra).

e. Using the results of part (d) to integrate the normal component of the pressure, and by comparing with the stationary solution, isolate the lift component of the normal force (many more pages of algebra).

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10.2.2 MOMENTUM BALANCE The momentum balance is a statement of Newton's Law of Motion; F=ma. To derive it, we consider the same rectangular differential element we used in deriving the continuity equation (Figure 10.4). The balance equation for the momentum of the fluid element is: In Rate of Out Rate of + Gen Sum of forces = Acc Rate of momentum momentum acting on = momentum in by - out by + the body accumulated convection convection Momentum may flow into and out of the control volume in a variety of ways: by convection, i.e., by bulk fluid motion; by molecular transfer, i.e., by velocity gradients; and by external forces acting on the body such as gravity or pressure. The first mechanism mentioned above arises due to fluid acceleration through the control volume while the second arises from shear stresses acting on the fluid. We consider only the x-component of momentum in detail and include gravitation as the only body force. Other body forces such as surface tension forces or electromagnetic forces may arise depending on the situation. Pressure is included as a separate surface force. The components of our momentum equation include: