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Thermodynamic Optimization of Flow Over an Isothermal Moving Plate

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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 [1] 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 [2], Hussaini et al. [3], Fang and Zhang [4] and recently Ishak et al. [5] and Cortell[6] 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)

Works Cited

[1] H. Blasius, Grenzschichten in FlüssigkeitenmitkleinerReibung. Z. Math. Phys. 56 (1908) 1-37.
[2] J.B. Klemp, A. Acrivos, A method for integrating the

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