HW 5

Problem 1

Part a

Write the gradient of a scalar $P$ in index notation.

Part b

Write the divergence of a vector $\vec{v}$ in index notation.

Part c

Write the mass, species, momenum, and energy equations in index notation:

$$\frac{\partial\rho}{\partial t} + \nabla\cdot(\rho \vec{v}) = 0$$

$$\frac{\partial\rho y_k}{\partial t} + \nabla\cdot(\rho \vec{v} y_k) + \nabla\cdot\vec{j_k} = \dot{m}_k^{\prime\prime\prime}$$

$$\frac{\partial\rho\vec{v}}{\partial t} + \nabla\cdot{\rho\vec{v}\vec{v}} = -\nabla P - \nabla\cdot\bar{\bar{\tau}} + \rho\vec{g}$$

$$\frac{\partial\rho e_t}{\partial t} + \nabla\cdot{\rho\vec{v}e_t} = -\nabla\cdot\vec{q} - \nabla\cdot(P\vec{v}) - \nabla\cdot(\bar{\bar{\tau}}\cdot\vec{v}) + \rho\vec{g}\cdot\vec{v}$$

Here, $e_t = e + \frac{1}{2}\vec{v}\cdot\vec{v}$, where $e$ is internal energy.

Problem 2

The species equation listed above is in a so-called conservation form. Derive the non-conservation form that was discussed.

Problem 3

Derive the 1D compressible inviscid flow equations for mass, momentum, and energy, neglecting gravity and molecular transport (viscosity, and heat). That is, derive the Euler equations:

$$\frac{\partial\rho}{\partial t} + u\frac{\partial\rho}{\partial x} + \rho\frac{\partial u}{\partial x} = 0$$

$$\rho\frac{\partial u}{\partial t} + \rho u\frac{\partial u}{\partial x} + \frac{\partial P}{\partial x} = 0.$$

$$\rho\frac{\partial e}{\partial t} + \rho u\frac{\partial e}{\partial x} + P\frac{\partial u}{\partial x} = 0$$

Here, $e$ is internal energy.