J. Appl. Comput. Mech., 7(2) (2021) 638-645 DOI: 10.22055/JACM.2020.35880.2753

ISSN: 2383-4536 jacm.scu.ac.ir

Published online: December 03 2020

Non-linear Radiation and Navier-slip effects on UCM Nanofluid

Flow past a Stretching Sheet under Lorentzian Force

P. Sreenivasulu1 , T. Poornima2 , B. Vasu3, Rama Subba Reddy Gorla4, N. Bhaskar Reddy5

1

Department of Mathematics, SVEC, Tirupati-517502, India

2 Department of Mathematics, SAS, VIT University, Vellore, T.N., India

3 Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, Uttar Pradesh- 211004, India

4 Department of Aeronautics and Astronautics, Air Force Institute of Technology, Wright Patterson Air Force Base, Dayton, Ohio 45433, USA

5 Department of Mathematics, SV University, Tirupati, India

Received August 23 2020; Revised November 27 2020; Accepted for publication December 02 2020.

Corresponding author: R.S. Reddy Gorla (r.gorla@yahoo.com)

© 2020 Published by Shahid Chamran University of Ahvaz

Abstract. In the present article, the novel contributions are modelling of Upper convected Maxwell nanoflow under Lorentzian influence over a stretching surface and investigating it using bvp4c procedure with MATLAB software. The boundary is set fixed with axial slip. Non-linear energy distribution is incorporated. Similarity variables are utilized to transmute non-linear PDEs of the basic fluid model to coupled system of ODEs. Computed numerical results are better compared with the past literature work to evidence its efficacy. The nanoflow momentum, energy, species diffusion are visualized graphically and analyzing the performance of proficient physical quantities on shear stress, energy dispersion coefficient, mass diffusion coefficient scatter of the system are seen through tables. Presence of magnetic field reduces friction at the wall and acts as a cooling agent. Navier slip increases the friction factor near the wall. Non-linear radiation transfers more heat from the system. Energy transfer coefficient is high in linear thermal rather than non-linear thermal distribution.

Keywords: Lorentzian force, Non-linear radiation, Maxwell fluid, Heat transfer, Navier slip.

1. Introduction

In modern fluid mechanics, the study of non-Newtonian nanofluid models was attracted by many researchers due to its wide rage applications in food processing industry and chemical engineering systems such as fermentation, polymer depolarization, boiling, plastic foam processing and others. Under stress, the fluid which changes their flow behavior or viscosity is term as non-Newtonian. Most of the fluids in biomedicine and industries do not obey the Newton’s law of viscosity which includes blood, dyes, honey, ketchup, mud, toothpaste, printer inks, clay coating, egg whites, shampoos, fruit juices, slurries, certain oils and many others have some complicated relation between stress and strain. The constitutive equations governing these fluids are much more complicated and of higher order than the Navier–Stokes equations. Therefore, several models of non-Newtonian fluids have been proposed. Among the non-Newtonian models, the most popular rheological model for non-Newtonian fluid is the power-law model. However the power-law model is incapable of explaining the visco-elastic effects in the flow. Two visco-elastic fluid models have been consistently used by the researchers namely the (i) second grade model and (ii) the upper-converted Maxwell (UCM) model. While second grade fluid model emphasizes on the normal stress differences, the UCM model can adequately address the characteristics of fluid relaxation time. The governing equations for two-dimensional flow of Maxwell fluid was initiated and formulated by Harris [1]. Heat transfer and rotating flow upper converted Maxwell fluid using Cattaneo-Christov modal was developed by Mustafa [2]. Cattaneo-Christov heat flux model for the MHD radiating upper converted Maxwell fluid in the presence of Joule heating was analyzed by Shah et al. [3]. Mahanthesh et al. [4] studied the effects of double stratification and exponential space dependent internal heat source on UCM nanofluid flow past a melting surface using Cattaneo-Christov heat flux model. Khan et al. [5] presented the chemically reactive flow of UCM using Cattaneo-Christov heat flux model. Mushtaq et al. [6] analyzed the stagnation point flow of Maxwell fluid utilizing non-Fourier heat flux. The effects of variable thermal conductivity and convective surface boundary conditions on magneto-hydrodynamic boundary layer slip flow of EG based Cu nanofluid was investigated by Reddy et al. [7]. Thermophoresis and Brownian motion effect in heat transfer nanofluid was investigated by Arun Kumar das and Chatterjee [8]. Ghalambaz et al. [9] examined the suspension of nano–encapsulated phase change materials in an inclined porous cavity and they work with in an annulus of a porous eccentric horizontal cylinder [10].

Hydrodynamic flows with magnetic field and slip has received much attention of researchers because of its varied and wide applications in industrial and manufacturing processes. Applications of MHD flows exist in numerous industrial fields like

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electric propulsion for space exploration, crystal growth in liquids, cooling of nuclear reactors, electronic packages, microelectronic devices etc. [11]. In most cases, no-slip boundary condition exhibits erroneous reports, like in systems like MEMs thee no-slip is no longer applicable. Bhaskar Reddy et al. [12] studied MHD slip flow effects on nanofluid flow. Waini et al. [13] studied the inclined MHD effect on heat transfer of UCM fluid past a stretching/ shirking surface. Tamoor [14] investigated the effects of MHD on slip flow past a cylinder in a thermally stratified medium. Nayak et al. [15] examined the influence of variable magnetic field on 3D flow with radiation. Takhar et al. [16] studied the unsteady flow under aligned magnetic field. Later he continued three dimensional flow of impulsive motion[17]. Krishna and Chamkha [18] elucidated the nanoflow on an inclined MHD boundary layer past and infinite vertical plate. Basha et al. [19] discussed non-radiation influence on diamond-EG nanoflow under induced magnetic field. Kumar et al. [20] examined induced magnetic field on convective dissipation fluid. Gorla et al. [21] elucidated Newtonian boundary layer flow of nanofluid past a non-isothermal vertical plate. Ganesh Kumar et al. [22] worked on transverse magnetic field under Cattaneo-Christov heat diffusion model. Ali et al. [23] investigated on MHD heat transfer of nanofluid using FEM method. Benkhedda et al. [24] studied the different shapes of nanoparticles in a hybrid nanoflow through a horizontal pipe. Chamkha et al. [25] deliberated unsteady energy and species transfer on a stretching surface.

Thermal radiation effect in the heat transfer has scope in various thermal engineering such as industrial processes which includes nuclear power plants, gas turbines and the various propulsion devices for aircraft, missiles, satellites and space vehicles. Fascinatingly, it is worthwhile to see that both linear-radiation is not enough for manufacturing thermal equipment with much difference in temperature, so non-linear radiation given by Rosseland approximation is being used widely for numerous technological processes. Archana et al. [26] worked out and studied the effect of radiation effect on 3D Maxwell flow. Influence of radiation on Burgers’ nanofluid flow was investigated by Khan et al. [27]. Awais et al. [28] suggested the utilization of nanoparticles and non-linear thermal radiation properties. Radiation effect on Carreau fluid was carried out by Lu et al. [29]. Shoaib et al. [30] studied and discussed the MHD hybrid nanofluid with radiation dispersion. MHD unsteady nanoflow with ramped velocity and temperature with suction/blowing was examined by Anwar et al. [31].

Since flow-through permeable boundaries have enormous importance from many decades due to their tremendous applications in bio-sciences and engineering, such as processes like membrane filtration, desalination processes using reverse osmosis, transpiration cooling, blood flow, renal proximal tubule flow within a kidney and filtration of blood in hemodialysis of an artificial kidney are the key examples related to flows in permeable boundaries. For many decades, authors have been interested with slip condition due to its practical implications in the polymer industry (polymer melts), which shows a macroscopic wall slip and also it extends from technological applications to medical applications as it used in polishing artificial heart valves. In view of the above applications, the present paper analyzes the effects of aligned magnetic field and non-linear thermal radiation on boundary layer slip flow of an upper convected Maxwell nanofluid over a stretching sheet.

2. Formulation of the Problem

A steady two-dimensional laminar incompressible boundary layer flow of an upper-convected Maxwell nanofluid past a stretching surface is considered. The flow is assumed along the x-axis, which is taken along the stretching surface and the y - axis is normal to it (see Fig. 1). The velocity of the stretching surface is considered as =( )wu x ax , where a is positive constant. The magnetic field of strength B0 is imposed perpendicular to the flow. Assumed that the flow is steady and the magnetic Reynolds number is very small so that the induced magnetic field can be neglected in comparison with the applied magnetic field. Under the above assumptions, the boundary layer equations can be expressed in dimensional form

∂ ∂+ =

∂ ∂0

u v

x y (1)

σν λ

ρ

∂ ∂ ∂ ∂ ∂ ∂ + = − + + − ∂ ∂ ∂ ∂ ∂ ∂ ∂

22 2 2 22 2 0

2 2 22nf

nf

Bu u u u u uu v u v uv u

x y y x y x y (2)

α τρ ∞

∂ ∂ ∂ ∂ ∂ ∂ ∂ + = − + + ∂ ∂ ∂ ∂ ∂ ∂ ∂

22

2

1 r TB

p

T T T q T C D Tu v D

x y y C y y y T y (3)

∞

∂ ∂ ∂ ∂+ = +

∂ ∂ ∂ ∂

2 2

2 2T

B

C C C D Tu v D

x y y T y (4)

The boundary conditions of these equations are

2( ) , , ( ) , ( ) 0

0, ,

w w w w

uu u x N v v T T x T bx C C x at y

y

u T T C C as y

ν ∞

∞ ∞

∂= + = = = + = =

∂

→ → → →∞

(5)

The radiative heat flux qr is described by Rosseland approximation is such that

σ ∂=−

∂

44

3s

r

e

Tq

k y (6)

Optically thick fluid model needs Rosseland approximation for its study and hence equation (6) is non-linear in T, now it can be linearized by considering the temperature differences within the flow is small such that expanding T4 into the Taylor series about T and neglecting higher order terms to take the form ( ∞ ∞≅ −4 3 44 3T T T T ).

σ∞

∂=−

∂34

3s

r

e

Tq T

k y (7)

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Fig. 1. Physical model and Coordinate system

In view of equations (6) and (7), equation (3) becomes

στ

ρ ∞

∂ ∂ ∂ ∂ ∂ ∂ ∂ + = + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂

2

3161

3s T

B

e

T T k T T C D Tu v T D

x y Cp y k k y y y T y (8)

we introduce the following non-dimensional variables:

( ) ( )

σ µη ψ ν η θ φ β λ

ν ρ

τ τσ να ν

ν νν

∞ ∞

∞ ∞

∞ ∞∞

− −= = = = = = =

− −

− −= =− = = = =

20

3

, ( ), , , , , Pr ,

4, , , , ,

3

w w

B W T Ws w

e B

Ba T T C C Cpy a xf a M

T T C C a k

D C C D T TT vR S Nt Nb Le N a

k k Da

(9)

We now introduce the stream function , which is defined by u = /y and v = −/x, the continuity equation is satisfied identically. Then equations (2), (3) and (8) can be transformed to the following non-dimensional form:

( )β ′′ ′′′′′′ ′′ ′ ′ ′+ − + − − =2 2( ) 2 0f ff f ff f f f Mf (10)

( )( )( )( )θ θ θ θ θ θ φ θ′ ′ ′ ′ ′ ′ ′+ + − + − + + =

3 211 1 1 2 0

Pr wR f f Nb Nt (11)

φ φ θ′′ ′ ′′− + =Le 0Nt

fNb

(12)

Corresponding boundary conditions (4) become

α θ φ

θ φ

′′′ = + = = =

′ ∞ → ∞ → ∞ →

(0) 1 (0), (0) , (0) 1, (0) 1

( ) 0, ( ) 0, ( ) 0

f f f S

f (13)

The quantities of practical interest in this study are the skin friction Cf, the local Nusselt number Nux and Sherwood number Shx which are shown below

µ

ρ ∞ ∞=

∂ = = = ∂ − − 2

0

2, ,

( ) ( )w w

x xfw w B wy

xq xmuC Nu Sh

u y k T T D C C (14)

where ( )=

∂ =− + ∂ 0

w r w

y

Tq k q

yand

=

∂ =− ∂ 0

w B

y

Cq D

y

Using the similarity transformation (12), the skin friction coefficient, local Nusselt number and Sherwood number can be expressed as

( )( )θ θ θ

φ

−

−

′′=

′=− + + −

′=−

1 /2

31/2

1/2

Re 2 (0),

Re 1 1 ( 1) (0) (0),

Re (0)

f

x w

x

C f

Nu R

Sh

(15)

where ν

=Re wu xis the local Reynolds number.

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Table 1. Compared values of surface skin friction for different values of for some reduced cases.

Waini et al.[13] Present Study

0 −1.00000005 −1.0000048

0.2 −1. 05188989 −1.0529922

0.4 −1.10190327 −1.1005412

0.6 −1.15013734 −1.1567735

0.8 −1.19671125 −1.1988518

1.2 −1.28536326 −1.2894539

3. Results and Discussion

The coupled non liner boundary layer equations (10) - (12) subjected to the boundary values (13) is numerically solved by bvp4c code using MATLAB software. As a matter of first importance higher order non-linear conditions are changed over into concurrent first order straight differential equations. The various values of the parameters are given here: �=0.1; Rd=0.5; Pr=0.71; �=0.1; α=0.1; �=6; Nt=0.3; Nb=0.3; S=0.1; θw=1.3; Le=10. Here Prandtl number is taken 0.71 which corresponds to majority fluids and particularly air at room temperature. A delegate set of numerical outcomes are introduced graphically in Figs. 2 – 11. We found that an excellent agreement with the existing literature for some reduced cases that of Waini et al. [13].

From Fig. 2, it is noticed that, the velocity of the Maxwell nanofluid decreases with an increase in magnetic field parameter. Larger the value of magnetic parameter causes decrease in fluid velocity, creates drag force that opposes the fluid motion, causing the momentum boundary layer thickness to decrease and this phenomenon is termed as Lorentz force which generates a resistive force to decelerate the fluid motion. This force resists momentum boundary layer thickness ascend. As magnetic field augments tends to decrease the fluid velocity and increase fluid temperature also within the boundary layer, consequently, the skin friction factor in terms of shear stress are decreased. The effect of Maxwell parameter on momentum boundary layer is depicted in Fig. 3. It is noticed that the velocity of the fluid descends for higher values of Maxwell parameter (). The velocity curves show that the rate of transport is considerably decreasing with augmenting values. The thickening of the solute boundary layer occurs due to increase in the elasticity stress parameter. It can also be seen from that the momentum boundary layer thickness decreases as increases, and hence induces an increase in the absolute value of the velocity gradient at the surface.

Fig. 4 explains the effect velocity slip parameter on momentum boundary layer. It is evident that, the velocity of the fluid decreases with an increase in slip parameter α. With the increasing α, the stream wise component of the velocity is found to decrease. When slip occurs, the flow velocity near the sheet is no longer equal to the stretching velocity of the sheet. With the increase in α, such slip velocity increases and consequently fluid velocity decrease because under the slip condition, the pushing of the stretching sheet can be only partly transmitted to the fluid. It is noted that α has a substantial effect on the solutions.

Fig. 2. Impact of M on η′( )f Fig. 3. Impact of on η′( )f

Fig. 4. Impact of α on η′( )f Fig. 5. Impact of S on η′( )f

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Fig. 6. Impact of R on θ η( ) Fig. 7. Impact of Nt on θ η( )

Fig. 8. Impact of Nt on φ η( ) Fig. 9. Impact of Nb on θ η( )

The influence of suction or injection S (< 0 for injection and > 0 for suction) parameter on velocity is presented in Fig.5. It is

clear that the velocity of the fluid decreases with an increase in suction/ injection parameter S. It is known that the effect of suction is to bring the fluid closer to the surface and, therefore, to reduce the momentum boundary layer thickness.

The influence of nonlinear thermal radiation on temperature is presented in Fig. 6. It can be seen that as R rises, the thermal boundary layer thickness increases due to reduce the absolute value of the temperature gradient at the surface. Thus, the heat transfer rate at the surface decreases with increase in �, thereby causing the temperature to increase for fluid. Also it is observed that the thermal boundary layer for nonlinear thermal boundary layer is thicker than the linear thermal radiation. This is because as the radiation parameter augmented, the absorption of radiated heat from the heated plate releases more heat energy released to the fluid and the resulting temperature increases the buoyancy forces in the boundary layer which also increases the fluid motion and the momentum boundary layer thickness accelerates. This is expected, because the considered radiation effect within the thermal boundary layer increases the energy of the fluid increases. And it can be seen from the figures 6, 7, 9, non-linear thermal radiation transfers more energy form the system thus cooling the system.

Figs. 7-8 respectively, represents the temperature and nanoparticles volume fraction for various values of thermophoresis parameter Nt. It is evident that, the temperature of the fluid upshoots with an increase in thermophoretic parameter Nt. Due to an increase in thermophoretic force allows the nanoparticles to penetrate slighter in the fluid. Also it is observed that the nanoparticles volume fraction dispersion also increases with an increase in Nt. The larger values of Nt coincide with the stronger thermophoretic diffusion which blows the nanoparticles away from the hot surface towards the cold ambient fluid. The influence of Brownian motion parameter on temperature and volume fraction function are presented in Figs. 9 - 10. It is noted that a rising impression of species concentration as Brownian motion parameter progresses. Higher Brownian motion induces the random acceleration of the fluid particles. Extra energy is generated because of this random acceleration. Hence energy rise is observed (Fig.9) as the thermophoresis phenomenon, tiny fluid particles are forced back from those in the warmer to the cold surface. As a result, the fluid particles returned from those in the warmed surface and the thermal curve then increased.

The effect of Lewis number Le on the concentration profile is depicted in Fig. 11. Lewis number is a mass transfer analog of Prandtl number. Le is the correlation of mass diffusion to fluid thermal conductivity. Increase in Le leads to decrease the Brownian diffusion coefficient which restricts the nanoparticles to infiltrate deeper into the fluid. For this reason, penetration depth of nanoparticles volume fraction decreases when Le is increased.

Variations in surface skin friction for various governing physical parameter is shown in Table 2. It is evident that the skin friction rate at the surface decreases with an increase in magnetic parameter, Maxwell parameter

and suction/injection parameter, while shows an opposite behavior for an increase in slip parameter. Table 3 shows the rate heat transfer at the surface for various physical parameters like Prandtl number, radiation parameter,

Brownian motion and thermophoretic. It is noticed the rate of heat transfer increases with an increase in Prandtl number Pr while it shows an opposite behavior for increasing values of radiation parameter R since the radiation effect within the boundary layer increases the motion of the fluid which increases the surface frictions. As the temperature of the fluid increases for increasing R, consequently, the temperature difference between the plate and the ambient fluid reduces which turns to decrease the heat transfer rate in flow region. Brownian motion parameter Nb and thermophoretic parameter Nt decreases the heat transfer coefficient. Also it is noticed that the heat transfer has higher values in linear radiation than the nonlinear radiation.

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Fig. 10. Impact of Nb on φ η( ) Fig. 11. Impact of Le on φ η( )

The rate of nanoparticles volume fraction function at the surface for different values of Lewis number, Brownian motion and thermophoretic is shown in Table 4. It is observed that the nano particle volume fraction rate increases with an increase in Lewis number Le and thermophoretic parameter Nt while it decreases with an increase in Brownian motion parameter Nb.

Table 2. Values of wall shear stress for various physical parameters.

M S f (0)

0.5 0.1 0.1 0.1 −0.978231

1.0 −1.261890

2 −1.496421

0.5 0.2 −1.137862

0.3 −1.158383

0.4 −1.178764

0.2 −0.978825

0.3 −0.874162

0.4 −0.791456

−0.3 −0.983326

0.0 −1.069621

0.3 −1.120464

Table 3. Values of Nusselt number for various material parameters

Table 4. Values of Sherwood number for various physical parameters

Le Nb Nt - (0)

3 0.3 0.3 0.809005

5 1.372874

7 1.850331

10 0.1 2.696042

0.2 2.586153

0.4 2.376182

0.1 1.716855

0.3 2.289373

0.5 2.574221

Pr R Nb Nt - (0)

Linear thermal radiation Nonlinear thermal radiation (w = 1.3)

0.71 0.5 0.3 0.3 0.674303 0.580494

3 1.255841 1.132552

5 1.439041 1.327254

1 0.586807 0.475944

2 0.488084 0.381272

3 0.433992 0.337634

0.1 0.692312 0.593766

0.2 0.683182 0.587051

0.4 0.665663 0.57409

0.1 0.718842 0.610507

0.2 0.696199 0.595295

0.4 0.653131 0.566092

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4. Conclusions

In this work, the influences of magnetic field, non-linear thermal radiation on free convection flow and heat transfer of UCM nanofluid over a stretching surface have been analyzed using bvp4c method with Rosseland approximation. The accuracies of the developed analytical solutions were verified with the results generated by some other methods as presented in the past works. The developed analytical solutions were used to investigate the effects of fluid parameter, thermal radiation parameter, Prandtl number, nanoparticles size and shapes on the flow and heat transfer behaviour of various UCM nanofluids. From the parametric studies, the following observations were established.

Momentum descends as the magnetic parameter ascends. The speed of the flow lowers for fluid parameter (Maxwell). Suction/blowing stood as obstacle for free flow of the fluid. Presence of Navier slip descends the momentum of nanoflow. Energy profiles elevates as radiation dispersion is more. Pedesis movement of the particles decrease the heat transfer rate. Thermophoresis factor in the fluid system increases the nanoparticle diffusion rate . Diffuso-thermo effects on energy seems to be elevated whereas for species diffusion, it seems to be demoted. Skin friction at the surface rises for higher values of Maxwell parameter β. Energy transfer rate is less with an increase in radiation and Pedesis movement also. Brownian motion decreases the energy transfer rate of the UCM fluid. Mass transfer rate falls down with an increase Nb.

Author Contributions

The paper was planned, formulated, written through the equal contribution of all authors. All authors discussed the results, reviewed and approved the final form of the manuscript.

Acknowledgments

The authors would like to thank the Editor and anonymous Reviewers for their careful reading and constructive comments and suggestions, which help a lot for improvement of the manuscript.

Conflict of Interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and publication of this article.

Funding

The authors received no financial support for the research, authorship and publication of this article.

Nomenclature

a,b B0 C Cw C DB hf k ke p T Tw T qr u,v uw x,y

f

w

Dimensional Variables Constants Uniform magnetic field strength(Tesla) Concentration (kg/m3) Hot fluid concentration Ambient fluid concentration Diffusion coefficient Coefficient of heat transfer Thermal conductivity (W/(m.K)) Mean absorption coefficient Pressure (Pa) Temperature (K) Hot fluid temperature Ambient fluid temperature Radiative heat flux(W/m2) Velocity components (m/s) Stretching velocities (m/s) Cartesian co-ordinates Dimensionless symbols Independent variable Axial velocity Temperature Non-linear radiation parameter Species concentration

c Cf Le R Nb Nt Nu Pr Re S

t Cp s ρ

Stretching ratio parameter Skin friction Lewis number Radiation parameter Brownian motion parameter Thermophoresis parameter Local Nusselt number Prandtl number Local Reynolds number Suction/injection Greek symbols Thermal diffusivity (m2/s) Dynamic Viscosity(Pa.s or kg/ms) Kinematic viscosity of basefluid (m2/s) Heat capacitance (J/(kg.K)) Electric conductivity (S/m) Stefan-Boltzmann constant(W/m2K4) Density (kg/m3) Axial slip Maxwell parameter Thermal Biot number Solutal Biot number Fluid parameter

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ORCID iD

P. Sreenivasulu https://orcid.org/0000-0002-3949-2730

T. Poornima https://orcid.org/0000-0002-0077-0545

© 2020 by the authors. Licensee SCU, Ahvaz, Iran. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0 license) (http://creativecommons.org/licenses/by-nc/4.0/).

How to cite this article: Sreenivasulu, P., Poornima T., Vasu, B., Reddy Gorla, R.S., Bhaskar Reddy N. Non-linear Radiation and Navier-slip effects on UCM Nanofluid Flow past a Stretching Sheet under Lorentzian Force, J. Appl. Comput. Mech., 7(2), 2021, 638–645. https://doi.org/10.22055/JACM.2020.35880.2753