MECENG 106 Notes: Fluid Dynamics

Fall 2024
By Aathreya Kadambi
Expanding on Lectures by Professor Phil Marcus

This fall I’m taking a fluid dynamics class by Professor Phil Marcus at Berkeley! Coincidentally, I actually was really interested in his research on Jupyter’s red spot and these related problems in freshman year, although I ended up going down a different road for a while. Excited to learn fluid dynamics.

Story 1: Dimensions

Relevant lectures: Lecture 1

The lecture notes have anything you need to succeed. You are also strongly recommended to work in teams on the homework. Homework are combination of graded and self-graded. Feel free to go to any discussion section.

When you write down equations, you should keep track of the dimensions of the quanities in the equation. If you are taking the $\sin$ or $\log$ of something, it better be dimensionless.

Dimensional analysis is very important. In fact, people were able to find the energy in the atomic bomb (which was a well-kept secret) based on the speed of the shock wave.

Dimensions are things like length, time, mass, etc.. It is important to distinguish these things from units, which we use to measure these dimensions.

Story 2: Properties of Fluids

Relevant lectures: Lectures 1, 2

Our properties are going to be functions of space and time: $\textbf{x} \in \R^3$ and $t \in \R$. Here are some important properties:

The following properties live in $\text{Maps}(\R^3\times \R, \R)$.

• Temperature: $\text{Temp}$.
• Mass Density: $\rho = \frac{\text{mass}}{\text{Volume}}$. It has dimensions $ML^{-3}$ in MLT units.
• Pressure: $P = \frac{\text{force}}{\text{area}}$. It has dimensions $ML^{-1}T^{-2}$ in MLT units.
• Velocity: $V$. It has dimensions $LT^{-2}$.

In fluid dynamics, we will write the ideal gas law as:

$P = R\rho T$ $R$ is called the gas constant, and the other quantities were discussed above.

In physics we generally use the form $PV = nRT$. So what is the relationship between these two forms? In physics one has to look up the molecular weights and stuff, and in MechE, we have to look up individual gas constants for each gas.

Ideal gas equation of state:

$P = R\rho T$ or in physics, $PV = nR_{\text{physics}}T$ Notice that $\rho = \frac{M}{V} = \frac{\nu_{\text{physics}}n}{V}$ so solving for $n$, $n = \frac{\rho V}{\nu_{\text{physics}}}$. Substituting this into the previous equation, we find that $P = \frac{R_{\text{physics}}}{\nu_{\text{physics}}} \rho T$ so we’ve found that $\boxed{R = \frac{R_{\text{physics}}}{\nu_{\text{physics}}}}$

Story 3: Viscosity

Viscosity is about roughly how well things flow. Physically, it is a diffusion coefficient of momentum.

In fluid dynamics, we have so called boundary conditions. For a Newtonian fluid, the boundary conditions are that the fluid at any surface in the direction that is parallel to the surface must move at the same velocity as the surface. This is called the no-slip boundary condition. The zeroness of momentum diffuses into the middle so that for honey, where that diffusion would happen very quickly for example, the momentum in the middle will also be close to zero. For water on the other hand, the momentum in the middle might be high even if the momentum at the surface is zero.

So far what we have discussed is called the kinematic viscosity. There is another viscosity called dynamic viscosity, which satisfies $\nu_{\text{dyn}} = \rho \nu_{\text{kin}}.$ From now on, $\nu$ will refer to kinematic viscosity. There is a dimensionless quantity called the Reynold’s number, defined: $\text{Re} = \frac{[V][L]}{\nu}$ where we are using brackets here to indicate “characteristic values”. When $\text{Re} \le 10^3$, we call it laminar, when it is $\ge 10^5$ we call it turbulent, and in between this, we call it transitional.

For 3D flows, it is extremely difficult to make calculations in the transitional region without approximations. That’s why you need to take MECENG 106, because that is one of the hardest places to calculate flows.

Story 4: Pressure

Relevant Lectures: Lecture 2

Consider an area. This vector has normal inward vector. We might write: $dF = \hat n P dA$ so that the integral is $\int df = \int \hat n P dA$ A question is, if we change our orientation, then do we get a different force or something? It turns out that the magnitude is independent.

We will show this in the static case, namely where

• $V \equiv 0$
• $\frac{\partial}{\partial t} = 0$

Now let’s imagine a little piece of swiss cheese (proceeds to draw an inclined plane with angle theta).

Story $k$: Gauss’s Law

Our MechE 106 equation which is in many cases more useful than Gauss’s law is:

$-\int_V \nabla P \;d\text{volume} = \int_S P\hat{\mathbf{n}} \; d\text{area}$