Viscous Flow Assignment 2 — Navier Stresses and Stokes Hypothesis
 Question #1
Starting from the angular distortion and volume expansion of a fluid element, show that the shear stresses for a fluid with a linear stress-strain relationship become: $$\tau_{xx}=2 \mu \frac{\partial u}{\partial x}\\ \tau_{yy}=2 \mu \frac{\partial v}{\partial y}\\ \tau_{zz}=2 \mu \frac{\partial w}{\partial z}$$ and $$\tau_{xy}=\tau_{yx}=\mu\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \right)\\ \tau_{xz}=\tau_{zx}=\mu\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x} \right)\\ \tau_{yz}=\tau_{zy}=\mu\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y} \right)$$ where $\mu$ is the viscosity of the fluid.
 06.27.16
 Question #2
Starting from the Navier stresses obtained in the previous section and applying Stokes hypothesis, show that the Navier-Stokes normal shear stresses become: $$\tau_{xx}=\mu\left(\frac{4}{3} \frac{\partial u}{\partial x}-\frac{2}{3}\frac{\partial v}{\partial y}-\frac{2}{3}\frac{\partial w}{\partial z}\right)\\ \tau_{yy}=\mu\left(\frac{4}{3} \frac{\partial v}{\partial y}-\frac{2}{3}\frac{\partial u}{\partial x}-\frac{2}{3}\frac{\partial w}{\partial z}\right)\\ \tau_{zz}=\mu\left(\frac{4}{3} \frac{\partial w}{\partial z}-\frac{2}{3}\frac{\partial u}{\partial x}-\frac{2}{3}\frac{\partial v}{\partial y}\right)$$
 09.26.16
 Question #3
Starting from the Navier-Stokes equations and the mass conservation transport equation, show that the constant-density constant-viscosity $y$-momentum equation corresponds to: $$\rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} \right) = -\frac{\partial P}{\partial y} + \mu \frac{\partial^2 v}{\partial x^2} + \mu \frac{\partial^2 v}{\partial y^2} + \mu \frac{\partial^2 v}{\partial z^2}$$
 Question #4
Recall the normal and shear strain rates: $$S_{xx}= \frac{\partial u}{\partial x}\\ S_{yy}= \frac{\partial v}{\partial y}\\$$ and $$S_{xy}=S_{yx}=\frac{1}{2}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \right)\\$$ Do the following:
 (a) Prove that $S_{xx}$ and $S_{yy}$ become zero for pure translation and no volume distortion. (b) Prove that $S_{xy}$ becomes zero for pure rotation without angular distortion. Hint: start with the angles of distortion with respect to the $x$ and $y$ axes.
 09.24.18
 Due on Monday October 1st at 9:00. Do Questions #1, #3, and #4 only.
 $\pi$