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Three-dimensional heat transfer in nonlinear flow: a FEM computational approach

  • U. Nazir
  • S. SaleemEmail author
  • M. Nawaz
  • A. A. Alderremy
Article
  • 29 Downloads

Abstract

Finite element simulations for the dynamics of Casson fluid flow over time-dependent two-dimensional stretching sheet subjected to magnetic field and variable time and space-dependent temperature are studied numerically through Galerkin finite element method implementation. For this, weak form of the governing boundary value problems is derived through their residuals. Domain is discretized using two nodes per element, and assembly process is performed. The system of algebraic nonlinear equations is linearized through Picard’s linearization algorithm. Linear system of algebraic equations is solved iteratively with computational tolerance \(10^{ - 8}\). The independent variable is searched through several computational experiments, and code is tested by comparing the results for special case with already published benchmarks. After the validation of code, simulations are performed in order to capture the dynamics of the physical situation against the variation of the pertinent parameters. Behavior of stresses and heat flux for different values of the physical parameters is studied. The temperature decreases when the intensity of radiation in the form of electromagnetic waves is increased. Boundary layer thickness for the Casson fluid is less than the boundary layer thickness of Newtonian fluid. However, opposite trend of thermal boundary layer thickness is noted. The magnetic is responsible for producing a hindrance to flow. Consequently, wall shear stress increases. Heat flux at the surface of stretching sheet increases when the values of unsteadiness parameter are increased, whereas there is a decreasing trend in the rate of heat transfer when the value of Eckert number is increased. Shear stresses are increasing function of the temperature. However, there is an increasing trend in the rate of heat transfer.

Keywords

GFEM 3D simulations Dissipation Thermal radiation Casson rheology 

List of symbols

t

Time

\(f, g\)

Dimensionless velocities

\(x, y, z\)

Space coordinates

\(T_{{\infty }}\)

Ambient temperature

\(T_{0}\)

Reference temperature

\(T_{\text{w}}\)

Wall temperature

\(\varvec{V},\varvec{V}_{{\mathbf{w}}}\)

Wall velocity field

a, c, L

Constants

\(\frac{\text{d}}{{{\text{d}}t}}\)

Material derivative

\(P\)

Pressure field

J

Current density

L

Velocity gradient tensor

B

Magnitude induction

\({\text{Nu}}\)

Nusselt number

\({\text{Re}}\)

Reynolds number

\(k\)

Thermal conductivity

\(C_{{{\text{f}}_{\text{x}} }} , C_{{{\text{g}}_{\text{y}} }}\)

Skin friction coefficients

\(M\)

Hartmann number

\(C_{\text{p}}\)

Specific heat

q

Radiative heat flux vector

V

Velocity field

\(T\)

Temperature field

E

Electric field

\(k^{*}\)

Stefan–Boltzmann constant

\(N_{\text{R}}\)

Radiation parameter

\(u, v, w\)

Velocity components

\({ \Pr }\)

Prandtl number

Greek symbols

\(\tau\)

Tensor field

\(\sigma^{*}\)

Mean absorption coefficient

\(\nu\)

Kinematic viscosity

\(\eta\)

Similarity variable

\({\text{Ec}}\)

Eckert number

\(\lambda^{*} , \lambda\)

Unsteadiness parameter, stretching rate ratio

\(\sigma\)

Electrical conductivity

\(\rho\)

Fluid density

\(\mu\)

Dynamic viscosity

\(\beta\)

Casson fluid parameter

\(\theta\)

Dimensionless temperature

\(\nabla\)

Vector differential operator

\(\psi\)

Stream function

\(\beta_{0}\)

Strength of magnetic field

Notes

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under Grant No. R.G.P.1/64/40.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics and StatisticsInstitute of Space TechnologyIslamabadPakistan
  2. 2.Department of Mathematics, College of ScienceKing Khalid UniversityAbhaSaudi Arabia

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