Problem 3: It is possible to join two rotating shafts with a viscous fluid coupling rather...

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Problem 3: It is possible to join two rotating shafts with a viscous fluid coupling rather than a solid connection (this is similar to a torque converter connecting the engine and transmission in an car with an automatic transmission). A simple version of a viscous coupling is shown below. Assume that a flat disk is housed within a cylindrical cavity such that the disk and end of the end of the cavity are perfectly parallel with a narrow gap h between them. The disk rotates with an angular velocity of 2. You can neglect manufacturing tolerances and assume that the disk and cavity both have an outer radius of R. You can also assume the resulting flow is steady, fully developed, and at a low enough Reynolds number that it is laminar and only in the circumferential (0) direction. R r h Q a) Identify the boundary conditions relevant to solving the momentum equation for this problem. Note, the circumferential velocity v is a function of both z and r. b) Write out the momentum equation in the direction and simplify it for this flow. c) In the limit where h < < R, we can assume that the disk and end of the cylindrical cavity form almost infinite plates, where the boundary at r = R can mostly be neglected. In this limit, ave/ar becomes very small. Using this assumption, derive an equation for ve. d) Using your answer, determine the expression for the viscous wall shear stress acting on the surface of the cylindrical cavity and the resulting force acting in the direction. Problem 3: It is possible to join two rotating shafts with a viscous fluid coupling rather than a solid connection (this is similar to a torque converter connecting the engine and transmission in an car with an automatic transmission). A simple version of a viscous coupling is shown below. Assume that a flat disk is housed within a cylindrical cavity such that the disk and end of the end of the cavity are perfectly parallel with a narrow gap h between them. The disk rotates with an angular velocity of 2. You can neglect manufacturing tolerances and assume that the disk and cavity both have an outer radius of R. You can also assume the resulting flow is steady, fully developed, and at a low enough Reynolds number that it is laminar and only in the circumferential (0) direction. R r h Q a) Identify the boundary conditions relevant to solving the momentum equation for this problem. Note, the circumferential velocity v is a function of both z and r. b) Write out the momentum equation in the direction and simplify it for this flow. c) In the limit where h < < R, we can assume that the disk and end of the cylindrical cavity form almost infinite plates, where the boundary at r = R can mostly be neglected. In this limit, ave/ar becomes very small. Using this assumption, derive an equation for ve. d) Using your answer, determine the expression for the viscous wall shear stress acting on the surface of the cylindrical cavity and the resulting force acting in the direction.