Streamlines

\[\frac{dx}{v_x} = \frac{dy}{v_y} = \frac{dz}{v_z}\]

Where

• \(dx,\ dy,\ dz\) are components of the fluid location vector
• \(v_x,\ v_y,\ v_z\) are components of the velocity vector

If you receive a vector equation of velocity, treat \(dx,\ dy,\ dz\) as differential elements and integrate to get the streamline equation. The result should have a constant, which creates the multiple beautiful streamlines you see in pictures. If 2D, rewrite as \(y(x)\) if possible.

Pathlines

\[\frac{dx}{dt} = v_x,\quad \frac{dy}{dt} = v_y,\quad \frac{dz}{dt} = v_z\]

Where

• \(dx,\ dy,\ dz\) are components of the fluid location vector
• \(v_x,\ v_y,\ v_z\) are components of the velocity vector

If you need the equation for a pathline, treat \(dx,\ dy,\ dz\) as differential elements like with streamlines. There should be separate equations \(x(t),\ y(t),\ z(t)\) for each coordinate.

Streamlines, pathlines, and streaklines are basically the same in steady state.

Material derivative

Say we have a velocity and pressure fields, where each equation only has coordinate variables but no time. Remember this:

\[a_x = \frac{\partial v_x}{\partial t} + v_x \frac{\partial v_x}{\partial x} + v_y \frac{\partial v_x}{\partial y} + v_z \frac{\partial v_x}{\partial z}\] \[a_y = \frac{\partial v_y}{\partial t} + v_x \frac{\partial v_y}{\partial x} + v_y \frac{\partial v_y}{\partial y} + v_z \frac{\partial v_y}{\partial z}\] \[a_z = \frac{\partial v_z}{\partial t} + v_x \frac{\partial v_z}{\partial x} + v_y \frac{\partial v_z}{\partial y} + v_z \frac{\partial v_z}{\partial z}\]

Then,

\[\frac{d?}{d t} = \frac{\partial ?}{\partial t} + v_x \frac{\partial ?}{\partial x} + v_y \frac{\partial ?}{\partial y} + v_z \frac{\partial ?}{\partial z}\]

Replace the (?) with your param (pressure, temperature, etc).

In short, derivate per variable with respect to all other vars, and multiply them by the corresponding velocity components.

Volume flow rate

\[Q = AV\]

Where

• \(Q\) is volume flow rate
• \(A\) is area (of tube, where the fluid flows)
• \(V\) is velocity (normal to the area A)

Mass flow rate

\[\dot m = \rho Q\]

Where

• \(\dot m\) is mass flow rate
• \(\rho\) is density
• \(Q\) is volume flow rate

If we assume conservation of mass (\(\dot m_{in} = \dot m_{out}\)), we have the following equation:

\[\rho_1 A_1 V_1 = \rho_2 A_2 V_2\]

Where

• \(\rho\) is density
• \(A\) is area (of tube, where the fluid flows)
• \(V\) is velocity (normal to the area A)

In many cases (like a fluid through a pipe), \(\rho_1 = \rho_2\) in which case \(A_1 V_1 = A_2 V_2\). This can be useful along with the equation below.

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Bernoulli Equation

Within a streamline:

\[p + \frac{1}{2} \rho V^2 + \gamma h = k\]

Where

• \(p\) is fluid pressure (static)
• \(\rho\) is density
• \(V\) is velocity
• \(\gamma\) is specific weight
• \(h\) is height
• \(k\) is a constant

Bernoulli Eq assumes 3 conditions:

1. Steady, laminar flow
2. Energy is conserved along streamline (inviscid, frictionless)
3. Fluid is incompressible (constant density through different pressure/temperature)

Consider this equation as a child of the pressure distribution equation, but now we add the term \(\frac{1}{2} \rho V^2\) as the dynamic pressure. Watch this video to understand the equation better.

Stagnation Pressure

\[p_s = p_{\infty} + \frac{1}{2} \rho V_{\infty}^2\]

Where

• \(p_s\) is stagnation pressure
• \(p_{\infty}\) is static pressure
• \(\rho\) is density
• \(V_{\infty}\) is upstream velocity

Stagnation means velocity is 0.