2021 Fundamentals of Fluid Mechanics B Midterm Exam
Tuesday March 22nd 2021
11:00 — 12:15
 INSTRUCTIONS
 • CLOSED BOOK EXAM: NO TEXTBOOK OR CLASS NOTES OR ANY NOTES ALLOWED OTHER THAN THE FUNDAMENTALS OF FLUID MECHANICS TABLES. • ONLY CALCULATORS WITH NO SD CARD CAPABILITIES ARE ALLOWED. • ALL QUESTIONS HAVE EQUAL VALUE; ANSWER ALL 2 QUESTIONS. • WRITE YOUR SOLUTIONS IN SINGLE COLUMN FORMAT, WITH ONE STATEMENT FOLLOWING ANOTHER VERTICALLY. • WRITE YOUR SOLUTIONS NEATLY SO THAT THEY ARE EASY TO READ AND VERIFY. • DON'T WRITE ONE LINE WITH TWO EQUAL SIGNS. • HIGHLIGHT YOUR ANSWERS USING A BOX. • 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.
 $\pi$