Problem $1$: Momentum Transfer between Rotating Walls $(50$ Points$)$

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You solved this problem in HW $7$ and simulated a similar problem in HW $8!$

The space between two coaxial cylinders is filled with an oil at constant temperature $($density: $1000$ kg$/$m$3$ and dynamic viscosity $0\mathrm{.}1$ kg$/$m$-$s$).$

The radius of the inner cylinder is $1$ inch.

The radius of the outer cylinder is $2$ inches.

The inner cylinder is stationary. The outer cylinder is rotating at $($radians per second$)$

$($where $$ corresponds to the last non$-$zero digit of your UND student ID$.$ Example: If your UND ID is: $1234560$ then your value of $$ is $6$ rad$/$s$)$

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Your goal is to determine the velocity distribution in the fluid, the shear stresses on the walls of the two cylinders!

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A$.$ First, draw an illustrative figure that represents the above scenario with a clear depiction of angular velocities that you specified at both cylindrical surfaces $(5$ points$).$

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B$.$ Identify the co$-$ordinate system, the velocity component of interest and the direction along which it is going to exhibit the strongest variation. Pick the appropriate governing equation from the list below and eliminate all the zero terms $(5$ points$).$

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C$.$ Integrate your equation twice and obtain a general solution for the fluid velocity distribution in the annular region $(5$ points$)$

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D$.$ Using the two boundary conditions, solve for the two integration constants C$1$ and C$2$ and report the final expression for the velocity distribution. $(5$ points$)$

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E$.$ You look up a book on Transport Phenomena and the equation for calculating shear stresses in cylindrical co$-$ordinate systems is given as:

Using this equation and the solution for velocity given in Part $($E$),$ calculate the shear stresses on the inner and outer cylindrical surfaces $(10$ x $2=20$ points$)$

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F$.$ The next step is for you to perform ANSYS FLUENT simulations of the hand$-$calculations that you performed above and compare your shear stress predictions against those output by ANSYS FLUENT. Provide screen shots of the shear stresses at the inner and outer cylinders $($these should look like the images below$)(10$ points$)$

Follow the instructions from slide $5$ of HW$8\_$ANSYSFLUENT.ppt and complete these calculations.

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Results $$ Contours $$ Wall Fluxes $$ Wall Shear Stress: Hit $$Display$$ and note the minimum and maximum values $($as shown below$)($The numbers in the image below are for illustrative purposes only and do not represent the correct answers!$).$

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