# MHD free convective heat transfer in a Walter’s liquid-B fluid past a convectively heated stretching sheet with partial wall slip

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## Abstract

The prime aim of the present investigation is to capture the mechanism of Navier’s velocity slip and convective thermal boundary condition on the flow of MHD viscoelastic fluid over a stretching surface. Additionally, the analysis also includes the effect of natural convection and thermal radiation. The governing boundary layer equations are transformed into a set of highly non-linear ordinary differential equations using suitable similarity transforms. Galerkin Finite element method is used to solve this boundary value problem. Effects of pertinent flow parameters on the Skin friction coefficient, Nusselt number, velocity and temperature, are described graphically. Numerical results obtained in this paper are compared with earlier published results and are found to be in excellent agreement. Significant findings of the present article are the conjugate effect of partial velocity slip and viscoelasticity of the fluid on Skin friction, Nusselt number, velocity and temperature. The analysis shows that presence of partial velocity slip changes the behavior of Nusselt number and skin friction coefficients significantly in comparison to the no slip condition. The present problem has potential to serve as a model for many industrial processes such as cooling and/or drying of paper and textile, rolling sheet drawn from a die, manufacturing of polymeric sheets, sheet glass and crystalline materials, etc.

### Keywords

MHD Viscoelastic fluid Stretching sheet Finite element method### Abbreviations

- \(a\)
Constant parameter (s

^{−1})- \(A\)
Rate of strain tensor (s

^{−1})- \(b\)
\(= \left( {b_{x} ,b_{y} ,0} \right)\) body force vector (N)

- \(B\)
Magnetic field (Kg s

^{−2}A^{−1})- \({\text{Bi}}\)
Biot number

- \({\text{Cf}}_{x}\)
Local skin friction coefficient

- \(f\)
Dimensionless stream function

- \(g\)
Gravitational acceleration (ms

^{−2})- \(Gr_{x}\)
Grashof number

- \(h\)
Element size (m)

- \(I\)
Unit tensor

- \(k\)
Thermal conductivity (Wm

^{−1}K^{−1})- \(k^{*}\)
Rosseland mean absorption coefficient (m

^{−1})- \(k_{0}\)
First order coefficient of short relaxation (kg m

^{−1})- \(M\)
Magnetic parameter

- \(N\)
Velocity slip factor (m)

- \(p\)
Pressure (Pa)

- \(Nu_{x}\)
Local Nusselt number

- \(\Pr\)
Prandtl number

- \(\Pr_{\text{eff}}\)
Effective Prandtl number

- \(q_{r}\)
Radiative heat flux (Wm

^{−2})- \(q_{s}\)
Wall heat flux (Wm

^{−2})- \(R\)
Radiation parameter

- \(\text{Re}_{x}\)
Local Reynolds number

- \(T\)
Temperature (K)

- \(T_{s}\)
Temperature of the left side of surface (K)

- \(T_{\infty }\)
Temperature in free stream (K)

- \(t\)
Time (s)

- \(V\)
\(= \left( {u,v,0} \right)\) velocity vector (ms

^{−1})- \(u_{s}\)
Stretching sheet velocity (ms

^{−1})- \(u_{\text{slip}}\)
Slip velocity (ms

^{−1})

### Greek Symbols

- \(\alpha\)
Viscoelasticity parameter

- \(\alpha_{m}\)
Thermal diffusivity (m

^{2}s^{−1})- \(\beta\)
Thermal expansion coefficient (K

^{−1})- \(\gamma\)
Velocity slip parameter

- \(\varUpsilon\)
Cauchy stress tensor (Pa)

- \(\sigma\)
Electrical conductivity (Sm

^{−1})- \(\rho\)
Density (kg m

^{−3})- \(\lambda\)
Thermal buoyancy parameter

- \(\left( {\rho C_{p} } \right)\)
Specific heat capacity of the fluid (JK

^{−1})- \(\theta\)
Dimensionless temperature

- \(\upsilon\)
Kinematic coefficient of viscosity (m

^{2}s^{−1})- \(\psi\)
Stream function (m

^{2}s)- \(\eta\)
Similarity variable

- \(\mu\)
Dynamic viscosity (kg m

^{−1}s^{−1})- \(\sigma^{*}\)
Stefan Boltzmann constant (Wm

^{−2}K^{−4})- \(\tau_{s}\)
Wall shear stress (kg m

^{−1}s^{−2})

## Notes

### Acknowledgements

We are grateful to learned reviewers for their valuable suggestions which helped us to improve the quality of this research paper.

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