The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a resultant thrust on the whole cylinder is 2mU sin θ/ga, and its thrust is 2πρmU absolute units in the direction Cy, to be counteracted by a support at the centre C; the liquid is streaming past r = a with velocity U reversed, and the cylinder is surrounded by a vortex. Similarly, the streaming velocity V reversed will give rise to a thrust 2πρmV in the direction xC.
Now if the cylinder is released, and the components U and V are reversed so as to become the velocity of the cylinder with respect to space filled with liquid, and at rest at infinity, the cylinder will experience components of force per unit length
(i.) − 2πρmV, 2πρmU, due to the vortex motion;
(ii.) − πρa2 dU/dt, − πρa2 dV/dt, due to the kinetic reaction of the liquid;
(iii.) 0, −π(σ − ρ) a2g, due to gravity,
taking Oy vertically upward, and denoting the density of the cylinder by σ; so that the equations of motion are
| πρa2 | dU | = − πρa2 | dU | − 2πρmV, |
| dt | dt |
(4)
| πρa2 | dV | = − πρa2 | dV | + 2πρmV − π (σ − ρ) a2g, |
| dt | dt |
(5)