(Solved): 3- A Newtonian fluid with constant viscosity, density and conductivity, initially at temperature \ ...

3- A Newtonian fluid with constant viscosity, density and conductivity, initially at temperature \( T_{0} \) enters a rectangular duct extends in the \( Z \) direction with length \( L \) and a rectangular cross section \( (x=a, y=b) \). The flow is transient. The entering fluid temperature is \( T_{1} \). Two walls of the duct at \( (x=0 \) and \( x=a) \) exchange heat with surroundings at temperature \( T_{a} \) with heat transfer coefficient \( h_{a} \). The wall at \( y=b \) is completely insulated. The wall at \( y=0 \) is kept at temperature \( T_{2} \). Gravitational force is in the \( \mathrm{Z} \) direction. a) Assume flow is fully developed, write down energy equation for this flowing system. How many boundary conditions are needed? Write down all needed boundary and initial conditions b) If the flow is steady state and is not fully developed, write down energy equation. Write down all needed boundary conditions