# Governing equations

Notes from my favorite books1 and 2

#### Material, substantial or total derivative

$$\begin{split} \frac{D \phi}{D t} &= \frac{\partial \phi}{\partial t} \frac{d t}{d t}+\frac{\partial \phi}{\partial x} \frac{d x}{d t}+\frac{\partial \phi}{\partial y} \frac{d y}{d t}+\frac{\partial \phi}{\partial z} \frac{d z}{d t}\\ &= \frac{\partial \phi}{\partial t} +\frac{\partial \phi}{\partial x} u+\frac{\partial \phi}{\partial y} v+\frac{\partial \phi}{\partial z} w\\ &= \frac{\partial \phi}{\partial t} +\nabla \centerdot \bold{v}\\ \end{split}$$


The conservation laws like mass, momentum and energy naturally apply to moving material volumes of fluids, i.e., Lagrangian frame of reference . Hence we need to translate those into Eulerian reference frame, using Reynolds Transport Theorem

#### Reynolds Transport Theorem

$$\Bigg(\frac{D B}{D t} \Bigg)_{MV} = \frac{d}{dt} \int_{V} \beta \rho \text{ }dV +\int_{S} \beta \rho v_{r}\cdot n \text{ }dA$$


Where, \beta = \frac{dB}{dm}

\bold v_{r} = \begin{cases}
\bold v   &\text{if CV is fixed}  \\
\bold v - \bold v_{s}  &\text{if CV is moving with constant velocity, non deforming}  \\
\bold v(\bold r,t) - \bold v_{s}(t)  &\text{if CV is moving with variable velocity, non deforming }  \\
\bold v(\bold r,t) - \bold v_{s} (\bold r,t) &\text{if CV is moving with arbitrary velocity and deforming}
\end{cases}

For the fixed control volume, applying Leibniz rule

$$\Bigg(\frac{D B}{D t} \Bigg)_{MV} = \int_{V} \frac{\partial}{\partial t}(\beta \rho) \text{ }dV +\int_{S} \beta \rho \bold v_{r}\cdot n \text{ }dA$$


Applying the divergence theorem to transform the surface integral into a volume integral, for fixed CV,

$$\Bigg(\frac{D B}{D t} \Bigg)_{MV} = \int_{V} \Big[\frac{\partial}{\partial t}(\beta \rho) + \nabla \cdot \beta \rho \bold v\Big] \text{ }dV$$


#### Conservation of Mass

For conservation of mass, substitute B = m and \beta =1 in (4). Noting that RHS is zero, the integrant has to be zero, which results in

$$\frac{\partial \rho}{\partial t} + \nabla \cdot \rho \bold v =0$$


#### Conservation of Linear Momentum

Apply B = \rho \bold v and \beta =\bold v in (4). Noting that RHS becomes force according to Newton’s second law

$$\frac{\partial (\rho \bold v)}{\partial t} + \nabla \cdot \rho \bold {vv} =\bold {f_{s}+f_{b}}$$

Here \bold {vv} is a diadic product resulting in second order tensor.

##### Surface forces

Surface force = surface stress x Area.

Net force on E and W face is

\Bigg[ \Big(  P - \frac{\partial P}{\partial x} \frac{\Delta x}{2} \Big) - \Big(  \tau_{xx} - \frac{\partial \tau_{xx}}{\partial x} \frac{\Delta x}{2} \Big) \Bigg]\Delta y \Delta z + \Bigg[ -\Big(  P + \frac{\partial P}{\partial x} \frac{\Delta x}{2} \Big) + \Big(  \tau_{xx} + \frac{\partial \tau_{xx}}{\partial x} \frac{\Delta x}{2} \Big) \Bigg]\Delta y \Delta z

which simplifies to:

 \Big(  - \frac{\partial P}{\partial x} + \frac{\partial \tau_{xx}}{\partial x} \Big)  \Delta x \Delta y \Delta z

On N and S face:

\frac{\partial \tau_{yx}}{\partial y}   \Delta x \Delta y \Delta z

On Top and Bottom face:

\frac{\partial \tau_{zx}}{\partial z}   \Delta x \Delta y \Delta z

Overall force per unit volume in x direction is

\frac{\partial}{\partial x}(-P+ \tau_{xx}) +\frac{\partial \tau_{yx}}{\partial y}  +\frac{\partial \tau_{zx}}{\partial z}

The x,y and z momentum equations can be written as follows, considering effect of all body forces by one term, S_{Mi}

\rho \frac{Du}{Dt} = \frac{\partial}{\partial x}(-P+ \tau_{xx}) +\frac{\partial \tau_{yx}}{\partial y}  +\frac{\partial \tau_{zx}}{\partial z}  + S_{Mx}
\rho \frac{Dv}{Dt} = \frac{\partial \tau_{xy}}{\partial x}+\frac{\partial}{\partial y}(-P+ \tau_{yy})   +\frac{\partial \tau_{zy}}{\partial z}  + S_{My}

as conserved variables.

Hence, with surface forces, the momentum equation can be written as:

$$\frac{\partial (\rho \bold v)}{\partial t} + \nabla \cdot \rho \bold {vv} =-\nabla P+ \nabla \cdot \bold \tau +f_{b}$$
##### Navier Stokes Equations

For Newtonian fluids stress tensor\tau is a linear function of strain rate (the anti-symmetric part of velocity gradient just rotate the fluid element)\nabla \bold v , given by

\tau = \mu\Big(\nabla \bold v  + (\nabla \bold v )^{T}\Big)+ \lambda(\nabla \cdot \bold v)I

\lambda , the second viscosity relate stresses to the volumetric deformation. \lambda = -\frac{2}{3} \mu is a good approximation for gases. For liquids, it is not needed as they are incompressible. Hence,

$$\frac{\partial (\rho \bold v)}{\partial t} + \nabla \cdot \rho \bold {vv} =-\nabla P+ \nabla \cdot \Big[ \mu\Big(\nabla \bold v + (\nabla \bold v )^{T}\Big)\Big]+ \nabla \Big[\lambda(\nabla \cdot \bold v)\Big] +f_{b}$$

for incompressible fluids with constant viscosity:

$$\frac{\partial (\rho \bold v)}{\partial t} + \nabla \cdot \rho \bold {vv} =-\nabla P+ \mu\nabla^{2} \bold v +f_{b}$$

#### Conservation of Energy

Total energy is sum of internal and kinetic energy. Potential energy is accounted in the body force term.

E = m(\hat u+\frac{1}{2}\bold{v\cdot v})

From first law of thermodynamics,

\dot E = \dot Q - \dot W = \dot Q_{s} + \dot Q_{b} - \dot W_{s} - \dot W_{b}

Work done per unit time is force times velocity

\dot W_{b} = -\int_{v} \bold f_{b} \cdot \bold v \text{ }dV
\begin{split}
\dot W_{s} &= -\int_{S} \bold f_{s} \cdot \bold v \text{ }dS\\
&= -\int_{S} (\bold  \sigma \cdot n) \cdot \bold v \text{ }dS\\
&= -\int_{V} \nabla \cdot(\bold  \sigma \cdot \bold v)  \text{ }dV\\
&= -\int_{V} \nabla \cdot(\bold  [-PI+\tau]\ \cdot \bold v)  \text{ }dV\\
&= -\int_{V} [-\nabla \cdot(P\bold v) +\nabla \cdot (\tau \cdot \bold v)  \text{ }]dV\\

\end{split}
\dot Q_{b} = \int_{v} \dot q_{b} \text{ }dV
\begin{split}
\dot Q_{s} &= -\int_{S} \dot q_{s} \cdot \bold n \text{ }dS\\
&= -\int_{V} \nabla \cdot\dot q_{s}  \text{ }dV\\
&= \int_{V} \nabla \cdot K \nabla T  \text{ }dV\\
\end{split}

Apply Reynolds Transport Theorem with B = \rho e and \beta = e

\int_{V} \Big[\frac{\partial}{\partial t}(\rho e) + \nabla \cdot  \rho e \bold v\Big] \text{ }dV = \dot Q_{s} + \dot Q_{b} - \dot W_{s} - \dot W_{b}
$$\frac{\partial}{\partial t}(\rho e) + \nabla \cdot \rho e \bold v = \nabla \cdot K \nabla T + \dot q_{b} - \nabla \cdot(P\bold v) + \nabla \cdot (\tau \cdot \bold v) \bold + f_{b} \cdot \bold v$$

Energy equation is written in multiple forms in terms of specific internal energy, temperature, specific enthalpy and specific total enthalpy. See Moukalled page no 60-65.

#### Source

1. Moukalled F, Mangani L, Darwish M. The finite volume method in computational fluid dynamics. Berlin, Germany:: Springer; 2016.
2. Versteeg, H. K., & Malalasekera, W. (2007). An introduction to computational fluid dynamics: The finite volume method. Harlow, England: Pearson Education Ltd.
3. Versteeg, H. K., & Malalasekera, W. (2007). An introduction to computational fluid dynamics: The finite volume method. Harlow, England: Pearson Education Ltd.