The brachistochrone curve for a non-dissipative particle tries to maximizeinertia of the particle but for a fluid filled cylinder, increasing inertiawould amount to increased dissipative losses. Hence the trade-off betweeninertia and dissipation plays a vital role in determining the brachistochronecurve of a fluid filled cylinder. This trade-off manifests itself in the formof an integro-differential equation governing the angular acceleration of thecylinder. Here, we compute the brachistochrone curve of a fluid filled cylinderusing optimal control principles and investigate the effect of theaforementioned trade-off on the deviation of the brachistochrone curve fromthat of a non-dissipative particle. Also, we investigate the effects of thenon-dimensional parameters of the problem on the shape of the brachistochronecurve. We then analyze the stability of the time varying fluid flow in thecylinder and find an admissible region for the terminal point which wouldensure the stability of the fluid flow as the cylinder rolls along thebrachistochrone curve.
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