### Physics #20: Fluid Dynamics

Mass Density

The mass density of a fluid (liquid or gas) is defined as

$\rho \equiv \frac{m}{V}$

where is the mass of the liquid and is the volume of the liquid.

Pressure

Pressure is defined as the net force per area:

$P \equiv \frac{F}{A}$

Note that this can reduce to

$P \equiv \frac{ma}{A} \equiv \frac{m}{A}\,\frac{\mathrm{d}^2 s}{\mathrm{d}t^2}.$

Now, given $\rho$ it follows that $m \equiv \rho V$, therefore:

$P \equiv \frac{\rho Va}{A}.$

Note that $P \in \mathbb{R}.$ (thus not a vector).

Also note that the units for pressure are $1 \frac{\text{ N}}{\text{ m}^2} \equiv 1 \text{ Pa}$ where the latter stands for Pascal.

Lastly,

$F \equiv PA$

$\therefore\, ma \equiv PA$

$\therefore \frac{\mathrm{d}^2 s}{\mathrm{d}t^2} \equiv \frac{PA}{m}.$

Water Pressure

Given density $\rho,$ depth $h,$ and gravitational acceleration $\mathbf{g},$ the pressure differential between two points underwater is given by

$\Delta P \equiv \rho gh.$

I have realized something:

$\rho g$

is defined in some textbooks as the weight density, but I redefine it as:

$\frac{mg}{V},$

which can be seen as the net gravitational pull (acceleration) per volumetric area. Now, multiplying by causes the following:

$\frac{mg}{lwh}\, h \implies \frac{mg}{lw}$

which can be seen as the net gravitational pull per rectilinear area. Since force is mass times acceleration, the pressure equation literally reduces into “mass times acceleration per area” in the form of gravitational pull over area.

Liquid Pressure

$P_t \equiv P_l + P_{\text{atm}}$

Pascal’s Principle

A perturbation of pressure at one point in a fluid will diffuse to all other points within the fluid, as well as the enclosure. This is mathematically stated as:

$\frac{F_1}{A_1}\equiv\frac{F_2}{A_2}.$

Buoyancy

$F_b \equiv A\Delta P \equiv \rho g hA.$

Rotational Motion

If a fluid is flowing continuously over a surface $A$ with $N$ points of velocity, then the fluid will experience rotational motion if

$\oint_N \mathbf{v} \cdot \mathrm{d}\mathbf{A} = 0.$

Streamline

The streamline $\delta$ for a fluid is defined as:

$\delta \equiv \frac{\partial{s}}{\partial{t}}.$

Fluid Mass Flow Rate: $\Xi \equiv \rho Av.$

Equation of Continuity: $\Xi_1 \equiv \Xi_2.$

Volume Flow Rate: $Q_1 \equiv A_1 v_1 \equiv A_2 v_2 \equiv Q_2$

Bernoulli’s Equation:

Let

$\beta \equiv P + \frac{1}{2}\rho v^2 + \rho g h$

Then

$\beta_1 \equiv \beta_2.$