Question: You need to do analytical solution of following Navier-Stokes equation

Navier-Stokes Equation

Applying the mass, momentum and energy conservation, we can derive the continuity equation, momentum equation and energy equation as follows.

Continuity Equation:

Dρ/Dt + ρ∂U_i/∂x_i = 0

Momentum Equation

ρ∂U_j/∂t + ρU_i∂U_j/∂x_i = -∂P/∂x_j - ∂τ_ij/∂x_i + ρg_j

    I             II              III          IV        V

I: Local change with time
II: Momentum convection
III: Surface force
IV: Molecular-dependent momentum exchange (diffusion)
V: Mass force

Energy Equation

ρc_μ∂T/∂t + ρc_μU_i∂T/∂x_i = -P∂U_i/∂x_i + λ∂²T/∂x_i² - τ_ij∂U_j/∂x_i

    I                II                III           IV              V

I : Local energy change with time
II: Convective term
III: Pressure work
IV: Heat flux (diffusion)