Midterm ONE

1. Euler from Newton’s Second Law

   Derive the Euler momentum equation for an inviscid fluid by applying Newton’s second law to a fixed control volume (fluid parcel). Be explicit about the use of the Reynolds transport theorem and the passage from integral to differential form.

2. (a)

   Prove, in details that

   $$(\nabla \phi \cdot \nabla)\nabla \phi = \nabla \left( \tfrac{1}{2} \vert\nabla \phi\vert^2 \right)$$

   (b)

   Consider an inviscid, incompressible fluid of constant density $\rho$ under gravity. Assume the flow is irrotational, so there exists a velocity potential $\phi(\mathbf{x},t)$ with $\mathbf{u}=\nabla\phi$. Starting from the Euler momentum equation

   $$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} = -\frac{1}{\rho}\nabla p - g\hat{\mathbf{z}},$$
   derive the unsteady Bernoulli equation
   $$\frac{\partial \phi}{\partial t} + \tfrac{1}{2}\vert\nabla\phi\vert^2 + \frac{p}{\rho} + gz = f(t),$$
   where $f(t)$ is a function of time only. Clearly state the assumptions used, show each step, and explain why the arbitrary function $f(t)$ may be absorbed into the potential (i.e., can be set to zero by a gauge choice).

3. Scaling Analysis
   Consider an incompressible flow in a channel of depth $H$ and characteristic velocity $U$.

   (a) Non-dimensionalize the Euler and continuity equations.

   (b) Identify the dimensionless group that measures the importance of the advective term relative to the pressure-gradient term, and interpret its physical meaning for geophysical flows, where

   $$U\propto 0.1 \frac{\rm {meter}}{\rm {second}}.$$

   Hint The Euler number is defined as

   $${\rm {Eu}} = \frac {P}{\rho U^2}$$
   and it might be useful in your analyses

4. Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be three vectors in $\mathbb{R}^3$. Prove that the scalar triple product is invariant under a cyclic permutation of the vectors, i.e., show that

   $$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{b} \cdo... ...athbf{c} \times \mathbf{a}) = \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}).$$