A large mass M is supported by a piston of diameter D and length L. The...
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A large mass M is supported by a piston of diameter D and length L. The piston sits in a cylinder closed at the bottom, and the gap a between the cylinder wall and piston (a<<D) is filled with oil of viscosity μ. The piston sinks very slowly due to the mass, and we approximate this flow a steady state problem. (a) For the oil flow between the gap, write the complete vertical momentum equation along the y direction. (b) Simplify the momentum equation assuming the flow has no acceleration. (Neglect the body force in the momentum equation). (c) Solve the simplified equation for velocity u in terms D, L, a, u, g, and M. (d) Write the flow rate Q in terms of D, L, a, u, g, and M. (e) Find the ratio between average velocity and peak velocity of the flow. L M a D A large mass M is supported by a piston of diameter D and length L. The piston sits in a cylinder closed at the bottom, and the gap a between the cylinder wall and piston (a<<D) is filled with oil of viscosity μ. The piston sinks very slowly due to the mass, and we approximate this flow a steady state problem. (a) For the oil flow between the gap, write the complete vertical momentum equation along the y direction. (b) Simplify the momentum equation assuming the flow has no acceleration. (Neglect the body force in the momentum equation). (c) Solve the simplified equation for velocity u in terms D, L, a, u, g, and M. (d) Write the flow rate Q in terms of D, L, a, u, g, and M. (e) Find the ratio between average velocity and peak velocity of the flow. L M a D
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SOLUTION a The complete vertical momentum equation along the ydirection can be written as F dVdt where F is the force density is the fluid density and V is the fluid velocity Since the flow is steady ... View the full answer
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