\( \mathbf{2. Navier-Stokes Equations} \)

\( \frac{\partial u_x}{\partial t}+\frac{\partial u_xu_x}{\partial x}+\frac{\partial u_yu_x}{\partial y} =-\frac{\partial p}{\partial x}+\nu\left(\frac{\partial^{2} u_x}{\partial x^2}+\frac{\partial^{2} u_x}{\partial y^2} \right)+F_x\\ \)

\( \frac{\partial u_y}{\partial t}+\frac{\partial u_xu_y}{\partial x}+\frac{\partial u_yu_y}{\partial y} =-\frac{\partial p}{\partial y}+\nu\left(\frac{\partial^{2} u_y}{\partial x^2}+\frac{\partial^{2} u_y}{\partial y^2} \right)+F_y\\ \)

\( \frac{\partial T}{\partial t}+\frac{\partial u_xT}{\partial x}+\frac{\partial u_yT}{\partial y} =\frac{\lambda}{\rho c}\left(\frac{\partial^{2} T}{\partial x^2}+\frac{\partial^{2} T}{\partial y^2} \right)+S\\ \)

\( \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \)