Thermodynamic Optimization of Flow Over an Isothermal Moving Plate
930 WordsJun 24, 20184 Pages
Boundary layers are thin regions next to the wall in the flow where viscous forces are important. The above-mentioned wall can be in various geometrical shapes. Blasius  studied the simplest boundary layer over a flat plate. He employed a similarity transformation which reduces the partial differential boundary layer equations to a nonlinear third-order ordinary differential one before solving it analytically. The boundary layer flow over a moving plate in a viscous fluid has been considered by Klemp and Acrivos , Hussaini et al. , Fang and Zhang  and recently Ishak et al.  and Cortell which is an extension of the flow over a static plate considered by Blasius. A large amount of literatures on this problem has been cited…show more content…
u=∂ψ/∂y and v=-∂ψ/∂x and ν is the kinematic viscosity of the fluid.
Substituting Eqs.(5) and (6) into Eq.(2) we obtained the following ordinary differential equation. f^''' (η)+f(η)f''(η)=0 (7)
With these boundary conditions:
f(0)=0 ,f^' (0)=λ (8) lim┬(η→∞)〖f^' (η)〗 = 1
Where λ=u_w/u_∞ is the plate velocity ratio that represents the direction and magnitude of the moving plate.
The skin friction coefficient C_f can be defined as:
C_f=τ_w/(ρ〖〖 u〗_∞〗^2 ) (9)
Where τ_w is the surface shear stress which is given by: τ_w=├ μ(∂u/∂y)┤| y=0 (10)
Substituting Eqs.(5),(6) into Eqs. (9) and (10) we obtain:
√(2〖Re〗_x ) C_f=f^'' (0) (11)
Where 〖Re〗_x is the local Reynolds number.
Looking for Similarity solution for energy equation, Eq.3, we obtained: θ^'' (η)+Pr f(η) θ^' (η)=0 (12)
Where θ=(T-T_∞)/(T_w-T_∞ ) (13)
Is dimensionless temperature and Pr=ν/α .
The boundary conditions are:
At η=0: θ(0)=1 (14) lim┬(η→∞)θ(η) = θ(∞)=0
The local Nusselt number〖 Nu〗_x, is defined as:
〖Nu〗_x=(x q_w)/(k (T_w-T_∞)) (15)
Where q_w is the surface heat flux which is: q_w=-k├ ∂T/∂y┤|_(y=0 ) (16)
Using Eq.(5), (6),(15) and (16) we obtain:
〖[〖Re〗_x/2]〗^( -1/2) 〖 Nu〗_x=-θ^' (0) (17)
 H. Blasius, Grenzschichten in FlüssigkeitenmitkleinerReibung. Z. Math. Phys. 56 (1908) 1-37.
 J.B. Klemp, A. Acrivos, A method for integrating the