Incompressible 2D Cubic-Trigonometric MMS

Define the functions

u={beta * Cos[t]+x^3+y^3, beta*Sin[t]+2*x^3-3*x^2y}

p=3/2*x^2+3/2*y^2-1

T =beta*Cos[t]+1/2*x^2+1/2*y^2

Density Equation

Qden=Div[rho*u,{x,y}]

Temperature Equation

HSQ=rho*Cp*(S*D[T,t]+Dot[u,Grad[T,{x,y}]])-1/P*Div[k*Grad[T,{x,y}],{x,y}]

Momentum Equations

epsilon=0.5*(Grad[u,{x,y}]+Transpose[Grad[u,{x,y}]])

Qmom=rho*S*D[u,t]+Dot[rho*u,Transpose[Grad[u,{x,y}]]]+Grad[p,{x,y}]-1/R*Div[2*mu*epsilon,{x,y}]

In C Form for MMS

CForm[Qden]

CForm[HSQ]

CForm[Qmom]