Abstract
The classical problems of natural and forced convection flows are reformulated and combined into a single model by introducing new variables for the field quantities. The proposed model is characterized by additional features of physical interest. A generalized problem of force convection flow and heat transferred is solved for a vertical stretching (shrinking) and porous (impermeable) sheet of variable thickness. The generalization gives rise to new problems, which are obtained in the next fourth section. It is assumed that the stretching/shrinking and porous velocities are variable such that both suction and injection can take place through the porous surface. The variable surface velocity may have linear, nonlinear, exponential and power law forms. Analysis of viscous flow and heat transfer is accomplished by considering force convection boundary layer flow over a stretching (shrinking) and porous surface of variable thickness. The nonlinear problem of partial differential equation is simplified by considering the boundary layer approximations and reduced it into nonlinear ODEs. The ODEs are obtained by introducing unusual and generalized similarity transformations for the stream function and similarity variables. Final ODEs are characterized by suction (injection), stretching (shrinking), convection parameters and Prandtl number. The ODEs are solved numerically, and effects of all existing parameters are studied on flow and heat transfer characteristics. Comparisons of the new problem and its solutions are established with the studies to demonstrate the applicability, validity and high accuracy of the present approach.
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Abbreviations
- \(V\) :
-
Velocity of the flow
- \(T\) :
-
Temperature field
- \(\eta\) :
-
Apparent viscosity
- \(\alpha\) :
-
Thermal diffusivity
- \(U_{\text{w}} (x)\) :
-
Sheet stretching (shrinking) velocity
- \(x , { }y\) :
-
Cartesian coordinates
- Pr:
-
Prandtl number
- \(\kappa\) :
-
Thermal diffusivity
- \(\delta_{3}\) :
-
Suction (injection) parameter
- \(c_{\text{p}}\) :
-
Specific heat
- \(k ,\,k_{1} ,\,c ,\,c_{2}\) :
-
Controlling parameters
- \(\beta\) :
-
Thermal expansion
- \(T_{\infty }\) :
-
Free stream temperature
- \(\tau_{\text{w}}\) :
-
Wall shear stress
- \(\mu\) :
-
Dynamic viscosity
- \(\rho\) :
-
Density of the fluid
- \(\nu\) :
-
Kinematic velocity
- \(\psi\) :
-
Stream function
- \(V_{\text{w}} \left( x \right)\) :
-
Suction (injection) velocity
- \(\nabla T\) :
-
Variation between wall temperature and local temperature
- \(B\) :
-
Stretching(shrinking) parameter
- \(\gamma\) :
-
Controlling parameter
- \({\text{Gr}}\) :
-
Modified Grashof number
- \(A_{1}\), \(A_{2}\), \(d_{0} ,\,d_{1}\) :
-
Controlling parameters
- \(A,B\) :
-
Surface controlling parameters
- \(T_{\text{w}} \left( x \right)\) :
-
Heat transfer coefficient
- \(T_{0}\) :
-
Constant temperature
- \(d_{ 2}\) :
-
Controlling parameter
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Ali, A., Marwat, D.N.K. Force convective flow over a porous and stretching (shrinking) sheet of variable thickness. J Therm Anal Calorim 143, 3559–3567 (2021). https://doi.org/10.1007/s10973-020-09266-x
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DOI: https://doi.org/10.1007/s10973-020-09266-x