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

  • U. Nazir
  • S. SaleemEmail author
  • M. Nawaz
  • A. A. Alderremy


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.


GFEM 3D simulations Dissipation Thermal radiation Casson rheology 

List of symbols



\(f, g\)

Dimensionless velocities

\(x, y, z\)

Space coordinates

\(T_{{\infty }}\)

Ambient temperature


Reference temperature


Wall temperature


Wall velocity field

a, c, L



Material derivative


Pressure field


Current density


Velocity gradient tensor


Magnitude induction


Nusselt number


Reynolds number


Thermal conductivity

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

Skin friction coefficients


Hartmann number


Specific heat


Radiative heat flux vector


Velocity field


Temperature field


Electric field


Stefan–Boltzmann constant


Radiation parameter

\(u, v, w\)

Velocity components

\({ \Pr }\)

Prandtl number

Greek symbols


Tensor field


Mean absorption coefficient


Kinematic viscosity


Similarity variable


Eckert number

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

Unsteadiness parameter, stretching rate ratio


Electrical conductivity


Fluid density


Dynamic viscosity


Casson fluid parameter


Dimensionless temperature


Vector differential operator


Stream function


Strength of magnetic field



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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