Fundamentals of Fluid Mechanics B Assignment 1 — Helmholtz, Crocco, and the Vortex Force  
Instructions
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. Failure to do this will result in a lower score and fewer comments on my part.
01.20.20
Question #1
Consider the momentum equation in primitive vector form including a simplified form of the viscous terms: $$ \rho \frac{\partial}{\partial t} \vec{v} + \rho (\vec{v} \cdot \vec{\nabla} ) \vec{v} - \mu (\vec{\nabla} \cdot \vec{\nabla} )\vec{v} + \nabla{P} = 0 $$ After taking the curl on both sides, derive the Helmholtz vorticity equation: $$ \frac{\partial}{\partial t} \vec{\omega} +\left( \vec{v} \cdot \vec{\nabla} \right) \vec{\omega} -\left( \vec{\omega} \cdot \vec{\nabla} \right) \vec{v} +\vec{\omega} \left( \vec{\nabla} \cdot \vec{v} \right) -\vec{\nabla} \times \left( \frac{\mu}{\rho} (\vec{\nabla} \cdot \vec{\nabla}) \vec{v} \right) +\vec{\nabla} \times \left( \frac{1}{\rho} \vec{\nabla} P \right) =0 $$ where $\mu$ is the viscosity. Note: the latter does not assume that the density $\rho$ is constant.
Question #2
(a)  Starting from the momentum equation in primitive vector form including a simplified form of the viscous terms: $$ \rho \frac{\partial}{\partial t} \vec{v} + \rho (\vec{v} \cdot \vec{\nabla} ) \vec{v} - \mu (\vec{\nabla} \cdot \vec{\nabla} )\vec{v} + \nabla{P} = 0 $$ Prove that the following is correct: $$ \frac{\partial}{\partial t} \vec{\omega} + \frac{1}{2} \vec{\nabla} \times \left( \vec{\nabla} \left( \vec{v} \cdot \vec{v} \right)\right) +\vec{\nabla} \times \left(\vec{\omega} \times \vec{v}\right) - \vec{\nabla} \times \left( \frac{\mu}{\rho} (\vec{\nabla} \cdot \vec{\nabla}) \vec{v} \right) + \vec{\nabla} \times \left( \frac{1}{\rho} \vec{\nabla} P \right) = 0 $$
(b)  Starting from Gibb's theorem: $$ T ds = de + P d \left(\frac{1}{\rho}\right) $$ Show that $$ T \vec{\nabla} s = \vec{\nabla} h - \frac{1}{\rho} \vec{\nabla} P $$ Outline under which conditions this is valid.
(c)  Starting from the expressions outlined in (a) and (b) prove Crocco's equation: $$ \vec{v} \times \vec{\omega} = \vec{\nabla}H - T \vec{\nabla}s - \frac{\mu}{\rho} (\vec{\nabla}\cdot\vec{\nabla})\vec{v} $$ with $H$ the total enthalpy. Outline all assumptions.
(d)  Based on this equation, if $H$ is constant, what must be true of irrotational flows? What must be true of isentropic flows? The answers to these questions constitute Crocco's theorem.
Question #3
(a)  Starting from the momentum equation for an inviscid fluid, show that it can be expressed in terms of a vortex force as follows: $$ \rho \frac{\partial \vec{v}}{\partial t} + \frac{1}{2} \rho \vec{\nabla}(\vec{v}\cdot\vec{v}) = \rho \vec{v} \times \vec{\omega} - \vec{\nabla}P + \vec{B} $$
(b)  For a constant density flow at steady state where the vortex force and the body force vanish, what property is conserved on each streamline? Is this property constant throughout the flow? Explain.
Due on Thursday January 30th at 11:00. Do all three problems.
01.23.20
Someone came to my office and wondered if the signs are correct in the equations listed in 2a. Yes, they are. Check your solutions again.
01.27.20
However, question 2c had a term missing in the expression to be found. Update your assignment questions.
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