Fluid Dynamics | Part IB, 2001

State the form of Bernoulli's theorem appropriate for an unsteady irrotational motion of an inviscid incompressible fluid.

A circular cylinder of radius aa is immersed in unbounded inviscid fluid of uniform density ρ\rho. The cylinder moves in a prescribed direction perpendicular to its axis, with speed UU. Use cylindrical polar coordinates, with the direction θ=0\theta=0 parallel to the direction of the motion, to find the velocity potential in the fluid.

If UU depends on time tt and gravity is negligible, determine the pressure field in the fluid at time tt. Deduce the fluid force per unit length on the cylinder.

[In cylindrical polar coordinates, ϕ=ϕrer+1rϕθeθ\nabla \phi=\frac{\partial \phi}{\partial r} \mathbf{e}_{r}+\frac{1}{r} \frac{\partial \phi}{\partial \theta} \mathbf{e}_{\theta}.]

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