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.
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: $$ \tau_{xx}=\mu\left(\frac{4}{3} \frac{\partial u}{\partial x}-\frac{2}{3}\frac{\partial v}{\partial y}-\frac{2}{3}\frac{\partial w}{\partial z}\right)\\ \tau_{yy}=\mu\left(\frac{4}{3} \frac{\partial v}{\partial y}-\frac{2}{3}\frac{\partial u}{\partial x}-\frac{2}{3}\frac{\partial w}{\partial z}\right)\\ \tau_{zz}=\mu\left(\frac{4}{3} \frac{\partial w}{\partial z}-\frac{2}{3}\frac{\partial u}{\partial x}-\frac{2}{3}\frac{\partial v}{\partial y}\right) $$
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: $$ \rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} \right) = -\frac{\partial P}{\partial y} + \mu \frac{\partial^2 v}{\partial x^2} + \mu \frac{\partial^2 v}{\partial y^2} + \mu \frac{\partial^2 v}{\partial z^2} $$
Due on Wednesday September 27th at 13:30.
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