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

- 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.

## References

- 1.Mustafa M, Hayat T, Ioan P, Hendi A. Stagnation-point flow and heat transfer of a Casson fluid towards a stretching sheet. Z Nat. 2012;67:70–6.Google Scholar
- 2.Nawaz M, Naz R, Awais M. Magnetohydrodynamic axisymmetric flow of Casson fluid with variable thermal conductivity and free stream. Alex Eng J. 2018;57(3):2043–50.CrossRefGoogle Scholar
- 3.Mukhopadhyay S, Moindal IC, Hayat T. MHD boundary layer flow of Casson fluid passing through an exponentially stretching permeable surface with thermal radiation. Chin Phys B. 2014;23(10):104701.CrossRefGoogle Scholar
- 4.Mehmood Z, Mehmood R, Iqbal Z. Numerical investigation of micropolar Casson fluid over a stretching sheet with internal heating. Commun Theor Phys. 2017;67:443.CrossRefGoogle Scholar
- 5.Iqbal Z, Mehmood R, Azhar E, Mehmood Z. Impact of inclined magnetic field on micropolar Casson fluid using Keller box algorithm. Eur Phys J Plus. 2017;4:132–75.Google Scholar
- 6.Dash RK, Mehta KN, Jayaraman G. Casson fluid flow in a pipe filled with a homogeneous porous medium. Int J Eng Sci. 1996;34:1145–56.CrossRefGoogle Scholar
- 7.Shaw S, Gorla RS, Murthy PV, Ng CO. Pulsatile Casson fluid flow through a stenosed bifurcated artery. Int J Fluid Mech. 2009;36:43–63.CrossRefGoogle Scholar
- 8.Li Z, Saleem S, Shafee A, Chamkha AJ, Du S. Analytical investigation of nanoparticle migration in a duct considering thermal radiation. J Therm Anal Calorim. 2019;135:1629–41.CrossRefGoogle Scholar
- 9.Saleem S, Nadeem S, Rashidi MM, Raju CS. An optimal analysis of radiated nanomaterial flow with viscous dissipation and heat source. Microsyst Technol. 2019;25:683–9.CrossRefGoogle Scholar
- 10.Ali A, Sulaiman M, Islam S, Shah Z, Bonyah E. Three-dimensional magnetohydrodynamic (MHD) flow of Maxwell nanofluid containing gyrotactic micro-organisms with heat source/sink. AIP Adv. 2018;8(8):085303.CrossRefGoogle Scholar
- 11.Shah Z, Bonyah E, Islam S, Gul T. Impact of thermal radiation on electrical MHD rotating flow of carbon nanotubes over a stretching sheet. AIP Adv. 2019;9:015115.CrossRefGoogle Scholar
- 12.Sadiq MA, Khan AU, Saleem S, Nadeem S. Numerical simulation of oscillatory oblique stagnation point flow of a magneto micropolar nanofluid. RSC Adv. 2019;9:4751–64.CrossRefGoogle Scholar
- 13.Raei B, Shahraki F, Jamialahmadi M, Peyghambarzadeh SM. Experimental study on the heat transfer and flow properties of γ-Al
_{2}O_{3}/water nanofluid in a double-tube heat exchanger. J Therm Anal Calorim. 2017;127:2561–75.CrossRefGoogle Scholar - 14.Khan LA, Raza M, Mir NA, Ellahi R. Effects of different shapes of nanoparticles on peristaltic flow of MHD nanofluids filled in an asymmetric channel. J Therm Anal Calorim. 2019. https://doi.org/10.1007/s10973-019-08348-9.CrossRefGoogle Scholar
- 15.Shirvan KM, Mamourian M, Mirzakhanlari S, Ellahi R. Numerical investigation of heat exchanger effectiveness in a double pipe heat exchanger filled with nanofluid: a sensitivity analysis by response surface methodology. Powder Technol. 2017;313:99–111.CrossRefGoogle Scholar
- 16.Rashidi S, Akar S, Bovand M, Ellahi R. Volume of fluid model to simulate the nanofluid flow and entropy generation in a single slope solar still. Renew Energy. 2018;115:400–10.CrossRefGoogle Scholar
- 17.Asadollahi A, Esfahani JA, Ellahi R. Evacuating liquid coatings from a diffusive oblique fin in micro-/mini-channels. J Therm Anal Calorim. 2019;12:1–9.Google Scholar
- 18.Ali FM, Nazar R, Arifin NM, Pop I. Unsteady flow and heat transfer past an axisymmetric permeable shrinking sheet with radiation effect. Int J Numer Methods Fluids. 2011;67:1310–20.CrossRefGoogle Scholar
- 19.Ishak A. Thermal boundary layer flow over a stretching sheet in a micropolar fluid with radiation effect. Meccanica. 2010;45(3):367–73.CrossRefGoogle Scholar
- 20.Akbar NS, Nadeem S, Haq RU, Khan ZH. Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface with convective boundary condition. Chin J Aeronaut. 2013;26(6):1389–97.CrossRefGoogle Scholar
- 21.Hayat T, Qasim M, Abbas Z. Radiation and mass transfer effects on the magnetohydrodynamic unsteady flow induced by a stretching sheet. Z Nat. 2010;65(3):231–9.Google Scholar
- 22.Hayat T, Qasim M. Radiation and magnetic field effects on the unsteady mixed convection flow of a second grade fluid over a vertical stretching sheet. Int J Numer Methods Fluids. 2011;66(7):820–32.CrossRefGoogle Scholar
- 23.Bhattacharyya K, Layek GC. Effects of suction/blowing on steady boundary layer stagnation-point flow and heat transfer towards a shrinking sheet with thermal radiation. Int J Heat Mass Transf. 2011;54:302–7.CrossRefGoogle Scholar
- 24.Nawaz M, Zubair T. Finite element study of three dimensional radiative nano-plasma flow subject to Hall and ion slip currents. Results Phys. 2017;7:4111–22.CrossRefGoogle Scholar
- 25.Turkyilmazoglu M. Buongiorno model in a nanofluid filled asymmetric channel fulfilling zero net particle flux at the walls. Int J Heat Mass Transf. 2018;126:974–9.CrossRefGoogle Scholar
- 26.Reddy JN. An introduction to the finite element method. New York: McGraw-Hill; 1993.Google Scholar
- 27.Reddy JN. An introduction to the nonlinear finite element analysis. Oxford: Oxford University Press; 2005.Google Scholar
- 28.Saleem S, Shafee A, Nawaz M, Dara R, Tlili I, Bonyah E. Heat transfer in a permeable cavity filled with a ferrofluid under electric force and radiation effects. AIP Adv. 2019;9:095107.CrossRefGoogle Scholar
- 29.Sheikholeslami M, Saleem S, Shafee A, Li Z, Hayat T, Alsaedi A, Khan MI. Mesoscopic investigation for alumina nanofluid heat transfer in permeable medium influenced by Lorentz forces. Comput Methods Appl Mech Eng. 2019;349:839–58.CrossRefGoogle Scholar
- 30.Sheikholeslami M, Jafaryar M, Saleem S, Li Z, Shafee A, Jiang Y. Nanofluid heat transfer augmentation and exergy loss inside a pipe equipped with innovative turbulators. Int J Heat Mass Transf. 2018;126:156–63.CrossRefGoogle Scholar
- 31.Din SM, Zubair T, Usman M, Hamid M, Rafiq M, Mohsin S. Investigation of heat and mass transfer under the influence of variable diffusion coefficient and thermal conductivity. Ind J Phys. 2018;92(9):1109–17.CrossRefGoogle Scholar
- 32.Khan JA, Mustafa M, Hayat T, Alsaedi A. On three-dimensional flow and heat transfer over a non-linearly stretching sheet: analytical and numerical solutions. PLoS ONE. 2014;9:107287.CrossRefGoogle Scholar