Given a 3D velocity field \(\vec{V}(x, y, z) = u\hat{i} + v\hat{j} + w\hat{k}\):

Property	Equation	Type
Acceleration \(a\)	\(a = \frac{d\vec{V}}{dt}\)	Vector
Vorticity \(\zeta\)	\(\zeta = \nabla \times \vec{V}\)	Vector
Angular Velocity \(\omega\)	\(\omega = \frac{1}{2} \zeta\)	Vector
Angular Deformation \(\Omega\)	\(\Omega_{xy} = \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}\)	2D Scalar
Dilatation rate	\(\nabla \cdot \vec{V}\)	Scalar
Continuity Eq	\(\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0\)	Scalar
Stream Function	\(u = \frac{\partial \Psi}{\partial y},\quad v = -\frac{\partial \Psi}{\partial x}\)	2D Scalar
Navier-Stokes	\(\rho \frac{D\vec{V}}{Dt} = -\nabla P + \rho g + \mu \nabla^2 V\)	Vector

Acceleration

\[a = \frac{d\vec{V}}{dt}\] \[a = \frac{\partial \vec{V}}{\partial t} + u\frac{\partial \vec{V}}{\partial x} + v\frac{\partial \vec{V}}{\partial y} + w\frac{\partial \vec{V}}{\partial z}\]

Where

• \(a\) is acceleration field (vector) in \(m/s^2\)
• \(\frac{d\vec{V}}{dt} = \frac{D\vec{V}}{Dt}\) is the material derivative of \(\vec{V}\), viewed from a Lagrangian POV¹ (vector)
• \(\frac{\partial \vec{V}}{\partial t}\) is the local acceleration (variation of velocity at a fixed point)
• \(u\frac{\partial \vec{V}}{\partial x} + v\frac{\partial \vec{V}}{\partial y} + w\frac{\partial \vec{V}}{\partial z}\) is the acceleration from spatial variations

In steady-state: \(\frac{\partial \vec{V}}{\partial t} = 0\)

Vorticity

\[\zeta = \nabla \times \vec{V}\]

Where

• \(\zeta\) is vorticity (vector) in rad/s
• \(\nabla \times \vec{V} = \text{curl}\ \vec{V}\) (read curl)

We say the fluid is irrotational (does not rotate) when \(\zeta = 0\).

Angular velocity

\[\omega = \frac{1}{2} \zeta\]

Where

• \(\omega\) is angular velocity (vector) in rad/s
• \(\zeta\) is vorticity (vector)

Angular deformation

\[\Omega_{xy} = \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}\] \[\Omega_{xz} = \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}\] \[\Omega_{yz} = \frac{\partial w}{\partial y} + \frac{\partial v}{\partial z}\]

Where \(\Omega\) is the angular deformation (scalar) in rad/s. Note \(\Omega\) is calculated in a 2D plane.

Dilatation rate

\[\text{dilatation rate} = \nabla \cdot \vec{V}\] \[\text{dilatation rate} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\]

Where

• The dilatation rate (scalar) is in Hz (\(s^{-1}\))
• \(\nabla \cdot \vec{V} = \text{div}\ \vec{V}\) (read div)

Dilatation rate tells whether a fluid is expanding or compressing (volume is changing).

If fluid is incompressible: \(\nabla \cdot \vec{V} = 0\)

Continuity Equation

\[\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0\] \[\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0\]

Where \(\rho\) is density

In steady-state: \(\frac{\partial \rho}{\partial t} = 0\)

Stream Function

For incompressible 2D flow, from continuity equation:

\[\frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} = 0\]

We can define a function \(\Psi\) that describes \(u,\ v\) to define streamlines:

\[u = \frac{\partial \Psi}{\partial y},\quad v = -\frac{\partial \Psi}{\partial x}\]

Using some basic 3D calc we can get \(\Psi(x,y)\). We define streamlines as \(\Psi(x,y) = k\), where \(k\) is any constant.

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Navier-Stokes

Given an incompressible fluid (density \(\rho\) is constant) and the velocity field \(V(x,y,z) = u\hat{i} + v\hat{j} + w\hat{k}\), where \(u,\ v,\ w\) represent the velocity field in the \(x,\ y,\ z\) axis respectively:

\[\rho \frac{DV}{Dt} = - \nabla P + \rho g + \mu \nabla^2 V\] \[\rho \left(\frac{\partial V}{\partial t} + V \cdot \nabla V \right) = -\nabla P + \rho g + \mu \nabla^2 V\]

Where

• \(\rho\) is density in \(kg/m^3\)
• \(\frac{DV}{Dt}\) is the acceleration in \(m/s^2\)
• \(\nabla P\) is the gradient of pressure, in Pa/m
• \(\rho g\) is the force due to gravity (may replace if an external force applies), in \(kg/m^3 \cdot m/s^2 = Pa/m\)
• \(\mu \nabla^2 V\) is the viscous diffusion of momentum, in \(Pa\cdot s \cdot m/s \cdot 1/m^2 = Pa/m\)
• \(\nabla^2 V\) is the laplacian of velocity

Per component:

\[u: \quad \rho \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \right) = - \frac{\partial P}{\partial x} + \rho g_x + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right)\] \[v: \quad \rho \left( \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} \right) = - \frac{\partial P}{\partial y} + \rho g_y + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right)\] \[w: \quad \rho \left( \frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} \right) = - \frac{\partial P}{\partial z} + \rho g_z + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right)\]

Also:

Given incompressible flow:

\[\text{div}\ V = 0\] \[\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0\]

Where

• The component \(\frac{\partial V}{\partial t} = \frac{\partial u}{\partial t} + \frac{\partial v}{\partial t} + \frac{\partial w}{\partial t}\)

Diffusion term \(\mu \nabla^2 V\)

Consider a 1D problem about shear stress:

\[\tau_{xy} = \mu \frac{\partial u}{\partial y}\]

Then the force per volume is:

\[\frac{\partial \tau_{xy}}{dy} = \mu \frac{\partial^2 u}{\partial y^2}\]

In 3D we consider forces from all direction, thus

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Simplifying Navier-Stokes

Steady:

All time derivatives are 0.

\[\frac{\partial V}{\partial t} = 0\]

Per component:

\[\frac{\partial u}{\partial t} = \frac{\partial v}{\partial t} = \frac{\partial w}{\partial t} = 0\]

Flow along a specific axis (for example, \(x\)):

\[u \neq 0,\ v = 0,\ w = 0\]

Since flow is incompressible, then also:

\[\text{div}\ V = 0\] \[\frac{\partial^n u}{\partial x^n} = \frac{\partial^n v}{\partial y^n} = \frac{\partial^n w}{\partial z^n} = 0\]

Each component of the div is 0 if flow along a specific axis + incompressible. Then any of its derivatives of order n is also 0.

Navier-Stokes is solvable by hand (analytically) for steady laminar flow between stationary parallel plates.

Reynold’s number

\[\text{Re} = \frac{\rho V L}{\mu}\]

Where

• \(\text{Re}\) is Reynold’s number
• \(V\) is velocity
• \(L\) is
• \(\mu\) is viscosity

The bigger \(\text{Re}\) is, the more turbulent the flow is.

If we have a plane with a very smooth surface, it has less air resistance but it causes the air to be more turbulent, since the viscous forces become smaller.

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