An analysis of Maxwell fluid through a shrinking sheet with thermal slip effect: a Lie group approach

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ORIGINAL PAPER

An analysis of Maxwell fluid through a shrinking sheet with thermal slip effect: a Lie group approach M N Tufail1, M Saleem1,2*

and Q A Chaudhry2,3

1

Department of Mathematics, University of Management and Technology Lahore, Sialkot Campus, Sialkot 51310, Pakistan 2

Department of Mathematics, University of Engineering and Technology, Lahore 54890, Pakistan 3

Department of Mathematics, College of Science, University of Hail, Hail, Saudi Arabia Received: 17 April 2019 / Accepted: 06 December 2019

Abstract: Two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid passing through the shrinking sheet is considered. With the impact of thermal slip, thermal radiation and heat sourcesink conditions, the UCM fluid model is integrated. The method of the Lie scaling group is used to transform the strongly nonlinear governing partial differential equations (PDEs) into the ordinary differential equations (ODEs). The transformed ODEs are numerically solved using NDSolve command of MATHEMATICA and graphically presented with their results. The Deborah number’s influence on the velocity profile f 0 ðgÞ is studied for different values and different behavior observed. The Hartmann number M and the mass transfer parameter S have decreased the boundary layer thickness. The Prandtl number has increased the temperature profile hðgÞ. In contrast, the thermal boundary layer thickness was decreased by the heat source-sink parameter Q , the radiation parameter R and the thermal slip parameter L. Table 1 shows the verification of the results. Keywords: Upper-convected Maxwell fluid; Shrinking sheet; Scale analysis; Thermal slip; Magnetic field PACS Nos.: 0.2.20.Sv; 05.70.-a; 47.65.Md; 47.10.ab; 47.15.-x; 44.40.?a List B0 D1 k L1 Qo 0 R T Tw T1 Uw u Vr V0 v x y

of symbols Uniform magnetic field strength (T) Shrinking sheet constant Thermal conductivity [W/(m K)] Slip constant Heat source-sink parameter Rate of chemical reaction Temperature of the fluid (K) Temperature at the wall (K) Free stream temperature (K) Shrinking sheet velocity (m/s) x velocity component (m/s) Free stream velocity of the fluid (m/s) Free stream velocity constant (m/s) y velocity component (m/s) Direction along the sheet Direction perpendicular to the sheet

*Corresponding author, E-mail: [email protected]

Greek symbols k Relaxation time (s) m Kinematic viscosity (m2 /s) r Electric conductivity r1 Stefan–Boltzmann constant (W/m2 K4 Þ q Density (kg/m3 Þ g Similarity parameter k Mean absorption coefficient (1/m) CP Specific heat capacity c Thermal radiation parameter

1. Introduction The UCM fluid model’s boundary layer flow passing through the shrinking sheet has several engineering system implementations with heat transfer characteristics. The flow of the boundary layer changed anomalies through the shrinking sheet rather than through the stretching sheet

Ó 2020 IACS

M N Tufail et al.

flow. In the occurrence of a shrinking sheet, the external body force is not confined to the bounda