Problem 1
Write down the Reynolds Transport Theorem (RTT) from memory (review first if you need to, but don’t just copy it from something written).
- What two things does each side of the RTT represent/connect?
- Which side is which?
- What is $B$?
- What is $\beta$?
- Which of $B$ and $\beta$ are extensive/intensive?
- Label the three parts of the RTT in terms of accumulation, in/out, and generation.
- Write down the Gauss Divergence Theorem (GDT) from memory (review first if needed).
- the point of the GDT is to convert between volume and surface integrals.
- We said the system is defined by a marked mass. The velocity appearing in the in/out term is a relative velocity between the system and the control volume. If we have a fixed control volume, then when we apply the RTT to conservation of mass, $v$ is just the fluid velocity (the velocity of the system). What is $v$ is we apply the RTT to a chemical species $i$?
Problem 2
Use the RTT to derive the species transport equation.
Problem 3
Use the RTT to derive the momentum transport equation. The Lagrangian conservation law is simply Newton’s second law, that is, the rate of change of momentum is the sum of the external forces. The forces we consider are gravity, pressure, and viscous. We write these net forces as integrals over a generic body. Pressure and viscous forces act on the surface; gravity forces act volumetrically. Let $\vec{M}$ denote momentum. Denote tensors with double overbars. The pressure force, viscous, and gravity (body) forces are
$$d\vec{F}_P=-P\vec{n}dA=-P\bar{\bar{\delta}}\cdot\vec{n}dA,$$
$$d\vec{F}_v = -\bar{\bar{\tau}}\cdot\vec{n}dA,$$
$$d\vec{F}_g = - \rho\vec{g}dV,$$
where $\bar{\bar{\delta}}$ is the unit tensor.
Problem 4
Use the RTT to derive the total energy transport equation. Total energy is internal plus kinetic: $E=U + \frac{1}{2}m\vec{v}\cdot\vec{v}$. The Lagrangian conservation law is simply the first law of thermodynamics:
$$\frac{dE_{sys}}{dt} = \dot{Q} + \dot{W}.$$
Here, $\dot{Q}$ is the integral of the heat flux vector $\vec{q}$ over the surface, and $\dot{W}$ is the work done on the body, force times distance per time, or, $d\dot{W} = -d\vec{F}\cdot\vec{v}.$