CASE (1)

1D Diffusion Energy Equation

$\frac{\partial T}{\partial t}=\alpha \left[\frac{{\partial }^{2}T}{\partial x^2}\right]$ (5)

BCs are taken as: at x = 0, $T\left(x=0\right)=100˚\text{C}$ & at x = 3 m, $T\left(x=3\text{m}\right)=20˚\text{C}$

CASE (2)

2D Energy Equation

$\frac{\partial T}{\partial t}=\alpha \left[\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right]$ (6)

BCs are taken as: at x = 0, $T\left(y=0\right)=100˚\text{C}$ & at x = 3 m, $T\left(y=H\right)=100˚\text{C}$

CASE (3)

Navier Stokes Equation in x &y direction

(Momentum Equation)

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (7)

$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{\rho }\frac{\partial P}{\partial x}+v\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)+g_x$ (8)

$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{\rho }\frac{\partial P}{\partial y}+v\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)+g_y$ (9)

BCs are taken as: Left: u = vRe/D, lower and upper BCs are no-slip boundary conditions.