2021 Fundamentals of Fluid Mechanics B Midterm Exam

Tuesday March 22nd 2021 11:00 — 12:15

INSTRUCTIONS

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CLOSED BOOK EXAM: NO TEXTBOOK OR CLASS NOTES OR ANY NOTES ALLOWED OTHER THAN THE FUNDAMENTALS OF FLUID MECHANICS TABLES.

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ONLY CALCULATORS WITH NO SD CARD CAPABILITIES ARE ALLOWED.

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ALL QUESTIONS HAVE EQUAL VALUE; ANSWER ALL 2 QUESTIONS.

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WRITE YOUR SOLUTIONS IN SINGLE COLUMN FORMAT, WITH ONE STATEMENT FOLLOWING ANOTHER VERTICALLY.

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WRITE YOUR SOLUTIONS NEATLY SO THAT THEY ARE EASY TO READ AND VERIFY.

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DON'T WRITE ONE LINE WITH TWO EQUAL SIGNS.

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HIGHLIGHT YOUR ANSWERS USING A BOX.

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USE A BLACK PEN OR A PENCIL TO WRITE YOUR SOLUTIONS. DO NOT USE NON-BLACK INK.

03.22.21

Question #1

Starting from the non-constant-density and non-constant-viscosity Navier-Stokes equations and the mass conservation transport equation, show that the constant-density and constant-viscosity $y$-momentum equation corresponds to: $$ \rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} \right) = -\frac{\partial P}{\partial y} + \frac{a}{2} \frac{\partial^2 v}{\partial x^2} + \frac{a}{2} \frac{\partial^2 v}{\partial y^2} + \frac{a}{2} \frac{\partial^2 v}{\partial z^2} $$ Also explain why $$ a = 2\mu $$

Question #2

Consider a disc of radius $R$ rotating with the angular velocity $\omega$ interacting with another disc at rest. The two discs are separated by a thin layer of water of thickness $h_{\rm w}$ followed by a thin layer of oil of thickness $h_{\rm o}$, as follows:

The device is built such that the oil and the water are prevented to move out (radially) of the region between the two discs. Given the thicknesses $h_{\rm w}$, $h_{\rm o}$, the viscosity of the water $\mu_{\rm w}$, the viscosity of the oil $\mu_{\rm o}$, the density of the water $\rho_{\rm w}$, the density of the oil $\rho_{\rm o}$, the angular velocity of the left disc $\omega$, find the velocity components within the oil and the water everywhere (at any $x$, $r$, $\theta$). You can neglect gravity forces. Outline clearly your assumptions.