Fundamentals of Fluid Mechanics B Assignment 2 — Navier Stresses and Stokes Hypothesis  
Question #1
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.
Question #2
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: $$ \tau_{xx}=a \frac{\partial u}{\partial x}\\ \tau_{yy}=a \frac{\partial v}{\partial y}\\ \tau_{zz}=a \frac{\partial w}{\partial z} $$ and $$ \tau_{xy}=\tau_{yx}=\frac{a}{2}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \right)\\ \tau_{xz}=\tau_{zx}=\frac{a}{2}\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x} \right)\\ \tau_{yz}=\tau_{zy}=\frac{a}{2}\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y} \right) $$ where $a$ is a constant of proportionality.
Question #3
Starting from the Navier stresses obtained in the previous section and applying Stokes hypothesis, show that the Navier-Stokes normal shear stresses become: $$ \tau_{xx}=\frac{a}{2}\left(\frac{4}{3} \frac{\partial u}{\partial x}-\frac{2}{3}\frac{\partial v}{\partial y}-\frac{2}{3}\frac{\partial w}{\partial z}\right)\\ \tau_{yy}=\frac{a}{2}\left(\frac{4}{3} \frac{\partial v}{\partial y}-\frac{2}{3}\frac{\partial u}{\partial x}-\frac{2}{3}\frac{\partial w}{\partial z}\right)\\ \tau_{zz}=\frac{a}{2}\left(\frac{4}{3} \frac{\partial w}{\partial z}-\frac{2}{3}\frac{\partial u}{\partial x}-\frac{2}{3}\frac{\partial v}{\partial y}\right) $$
Question #4
Starting from the non-constant-density and non-constant-viscosity Navier-Stokes equations and the mass conservation transport equation and the strain rates listed in the tables, show that the Navier-Stokes constant-density $y$-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 #5
Prove that the shear strain rate in $yz$ is equal to: $$ S_{yz}=\begin{array}{c}{\rm lim}\\[-0.5em] \small\Delta t \rightarrow 0 \end{array} \frac{1}{2\Delta t} \left( \frac{y_{\rm A^\prime}-y_{\rm O^\prime}}{\xi}+ \frac{z_{\rm B^\prime}-z_{\rm O^\prime}}{\eta}\right) $$ Make sure to explain clearly the origin of all the terms including the factor $\frac{1}{2}$. Then prove that the latter becomes: $$ S_{yz}=\frac{1}{2}\left( \frac{\partial v}{\partial z}+\frac{\partial w}{\partial y} \right) $$
Do Questions #1 and #5 only. Due on Tuesday February 6th at 11:00.
PDF 1✕1 2✕1 2✕2