Mixed convection flow in vertical channel with boundary conditions of third kind in presence of heat source/sink
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Abstract
The effects of viscous dissipation and heat source/sink on fully developed mixed convection for the laminar flow in a parallel-plate vertical channel are investigated. The plate exchanges heat with an external fluid. Both conditions of equal and different reference temperatures of the external fluid are considered. First, the simple cases of the negligible Brinkman number or the negligible Grashof number are solved analytically. Then, the combined effects of buoyancy forces and viscous dissipation in the presence of heat source/sink are analyzed by a perturbation series method valid for small values of the perturbation parameter. To relax the conditions on the perturbation parameter, the velocity and temperature fields are solved by using the Runge-Kutta fourth-order method with the shooting technique. The velocity, temperature, skin friction, and Nusselt numbers at the plates are discussed numerically and presented through graphs.
Key words
mixed convection viscous fluid perturbation method Runge-Kutta shooting method heat source/sinkNomenclature
- Bi1,Bi2
Biot numbers
- Br
Brinkman number
- cp
specific heat at constant pressure
- g
acceleration due to gravity
- Gr
Grashof number
- h1,h2
external heat transfer coefficients
- k
thermal conductivity
- L
channel width
- Nu1,Nu2
Nusselt numbers
- p
pressure
- P
difference between the pressure and the hydrostatic pressure
- Pr
Prandtl number
- Q
rate of internal heat absorption/generation
- Re
Reynolds number
- RT
temperature difference ratio
- T
temperature
- T1,T2
reference temperatures of the external fluid
- T0
reference temperature
- u
dimensionless velocity in the
- X
direction
- ū
mean value of u
- U
velocity component in the X-direction
- U0
reference velocity
- x
dimensionless streamwise coordinate
- X
streamwise coordinate
- y
dimensionless transverse coordinate
- Y
transverse coordinate.
Greek letters
- α
thermal diffusivity
- β
thermal expansion coefficient
- ɛ
dimensionless parameter as defined by (54)
- φ
dimensionless parameter of heat absorption/generation coefficient
- λ
dimensionless parameter as defined by (19)
- µ
viscosity
- ν
kinematic viscosity
- θ
dimensionless temperature
- θb
dimensionless bulk temperature
- ρ
mass density
- ρ0
value of the mass density when T = T 0
Chinese Library Classification
O357.12010 Mathematics Subject Classification
76D50Preview
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