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Figure: A fluid in fully developed laminar flow between two parallel plates at temperature T1 in a region with z0, where the two plates are maintained at a constant temperature T0. Question: A Newtonian fluid is in steady laminar flow between two parallel plates a distance 2B apart as shown in figure above. In the inlet region z0, the wall temperature is maintained at T0. [Hint: Use function diff(y,x,1) to write xy or diff(y,x,2) to write x22y ] 1. By carrying out a suitable shell energy balance, one can show that the temperature distribution in the film can be described by, CP[tT+vzzT]=k[x22T+z22T]+(xvz)2 CP[tT+vzzT]=k[x22T+z22T]+(xvz)2 Give the simplified form of the equation to determine the governing partial differential equation for the temperature of the fluid T(x,z) in the region z>0 by neglecting terms in the equation diff(T,z,1)_povz=diff(T,x,2)k Your last answer was interpreted as follows: dzdTcpvz=dx2d2Tk The variables found in your answer were: [T,cp,k,,vz,x,z] Write the boundary conditions for T(x,z) in the region z>0. Note that one can prescribe two ts of boundary conditions for the dependence of temperature on x : (i) two Dirichlet boundary Inditions, or (ii) one Neumann and one Dirichlet boundary condition, where the latter exploits the mmetry of the temperature profile at the mid-plane. Write both sets of boundary conditions. a. at z=0, Your last answer was interpreted as follows: T=T1 The variables found in your answer were: [T,T1] Set 1: two Dirichlet boundary conditions b. at x=+B Set 1: two Dirichlet boundary conditions b. at x=+B Your last answer was interpreted as follows: T=T0 The variables found in your answer were: [T,T0] c. at x=B Your last answer was interpreted as follows: T=T0 The variables found in your answer were: [T,T0] Set 2: one Neumann and one Dirichlet boundary condition d. at x=0, Your last answer was interpreted as follows: dxdT=0 The variables found in your answer were: [T,x] e. at x=+B at x=0, Your last answer was interpreted as follows: dxdT=0 The variables found in your answer were: [T,x] at x=+B Your last answer was interpreted as follows: dxdT=0 The variables found in your answer were: [T,x] 3. It is desired to determine T(x,z) for small values of z, when the temperature change at the wall has not penetrated very far into the fluid. If s is the distance measured from the wall, i.e., s=Bx, this assumption implies that the fluid may be regarded as extending from the wall s=0 to s=. Rewrite the governing differential equation, and the boundary conditions in terms of s. Note that for the boundary conditions at s=, it can be assumed that the temperature is unchanged from the value at the inlet. a. at z=0, b. at s=0, C. at s=, 4. The fully developed velocity profile for flow between two parallel plates is given by the expression, vz(x)=23va[1(Bx)2] where, va=3L(P0PL)B2 is the mean velocity of the fluid. Rewrite the velocity field in terms of the variable s. vz(s)= Your last answer was interpreted as follows: 2L(P0PL)(S22BS) The variables found in your answer were: [B,L,P0,PL,S,]