Model and Simulation: Laws of Conservation

Equation of Continuity:
For 2D incompressible fluid flow from mass conservation,

\[ \frac{\partial v_{x}}{\partial x}+\frac{\partial v_{y}}{\partial y}=0
\]

Thus,

\[ v_{y}=\int_{0}^{y}\frac{\partial v_{x}}{\partial x}dy
\]

Cauchy-Riemann Equation:
Define velocity by new variables, $Psi$, stream function and $phi$, velocity potential,

\[ v_{x}=-\frac{\partial\Psi}{\partial y}
\]

\[ v_{y}=\frac{\partial\Psi}{\partial x}
\]

\[ v_{x}=-\frac{\partial\phi}{\partial x}
\]

\[ v_{y}=-\frac{\partial\phi}{\partial y}
\]


An analytical function $w(z)$ of a complex variable, z, may be chosen such that

\[ w(z)=\phi(z)+i\Psi(z)
\]

satisfying above equations. Velocities can be obtained as the real and imaginary parts of the equation,

\[ \frac{dw}{dz}=-v_{x}+iv_{y}
\]

Equations of continuity, stream function and velocity function in different coordinates are shown below.

Navier-Stokes Equations:

Momentum transport:

\[ \rho\left(\frac{\partial v_{x}}{\partial t}+v_{x}\frac{\partial v_{x}}{\partial x}+v_{y}\frac{\partial v_{x}}{\partial y}+v_{z}\frac{\partial v_{x}}{\partial z}\right)=-\frac{\partial p}{\partial x}+\mu\left[\frac{\partial^{2}v_{x}}{\partial x^{2}}+\frac{\partial^{2}v_{x}}{\partial y^{2}}+\frac{\partial^{2}v_{x}}{\partial z^{2}}\right]+\rho g_{x}
\]

\[ \rho\left(\frac{\partial v_{y}}{\partial t}+v_{x}\frac{\partial v_{y}}{\partial x}+v_{y}\frac{\partial v_{y}}{\partial y}+v_{z}\frac{\partial v_{y}}{\partial z}\right)=-\frac{\partial p}{\partial y}+\mu\left[\frac{\partial^{2}v_{y}}{\partial x^{2}}+\frac{\partial^{2}v_{y}}{\partial y^{2}}+\frac{\partial^{2}v_{y}}{\partial z^{2}}\right]+\rho g_{y}
\]

\[ \rho\left(\frac{\partial v_{z}}{\partial t}+v_{x}\frac{\partial v_{z}}{\partial x}+v_{y}\frac{\partial v_{z}}{\partial y}+v_{z}\frac{\partial v_{z}}{\partial z}\right)=-\frac{\partial p}{\partial z}+\mu\left[\frac{\partial^{2}v_{z}}{\partial x^{2}}+\frac{\partial^{2}v_{z}}{\partial y^{2}}+\frac{\partial^{2}v_{z}}{\partial z^{2}}\right]+\rho g_{z}
\]

Chemical species in incompressible fluids:

\[ \rho\left(\frac{\partial C_{A}}{\partial t}+v_{x}\frac{\partial C_{A}}{\partial x}+v_{y}\frac{\partial C_{A}}{\partial y}+v_{z}\frac{\partial C_{A}}{\partial z}\right)=D_{AB}\left[\frac{\partial^{2}C_{A}}{\partial x^{2}}+\frac{\partial^{2}C_{A}}{\partial y^{2}}+\frac{\partial^{2}C_{A}}{\partial z^{2}}\right]-r_{A}

\]

Heat transfer:

\[ \rho\left(\frac{\partial T}{\partial t}+v_{x}\frac{\partial T}{\partial x}+v_{y}\frac{\partial T}{\partial y}+v_{z}\frac{\partial T}{\partial z}\right)=k\left[\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{\partial z^{2}}\right]-\triangle H_{rxn}
\]

1D momentum balance for an unsteady state model gives,

\[ \rho\left(\frac{\partial v_{x}}{\partial t}+v_{x}\frac{\partial v_{x}}{\partial x}\right)=-\frac{\partial p}{\partial x}+\mu\left[\frac{\partial^{2}v_{x}}{\partial x^{2}}+\frac{\partial^{2}v_{x}}{\partial y^{2}}\right]+\rho g_{x}
\]

Using the incompressible fluid solution of equation of continuity,

\[ \rho\frac{\partial v_{x}}{\partial t}=-\frac{\partial p}{\partial x}+\mu\frac{\partial^{2}v_{x}}{\partial y^{2}}+\rho g_{x}

\]

The solution of 1D model equations are given below.

Boundary Conditions:
Differential equations of second order in spatial coordinates and first order in time require two boundary conditions and one initial condition to describe the system.

BC of the first kind:
The temperature or concentration of chemical species are specified at boundaries. This is also known as Dirichlet condition.

BC of the second kind:
The heat transfer or mass transfer flux is constant, also known as Neumann condition.

\[ q=-k_{f}\frac{\partial T}{\partial x}\mid_{x=0}
\]


BC of the third kind:
The flux is not constant, but the transfer coefficient is constant. Usually described in terms of heat or mass transfer coefficient.

\[ -k_{f}\frac{\partial T}{\partial x}\mid_{x=0}=h(T_{s}-T_{\infty})
\]