Variable diffusion and conductivity change in 3D rotating Williamson fluid flow along with magnetic field and activation energy

Mair Khan (Department of Mathematics, Quaid-I-Azam University, Islamabad, Pakistan)
T. Salahuddin (Department of Mathematics, Faculty of Science, Mirpur University of Science and Technology, Mirpur, Pakistan)
Muhammad Malik Yousaf (Department of Mathematics, College of Sciences, King Khalid University, Abha, Saudi Arabia)
Farzana Khan (Department of Mathematics, Quaid-I-Azam University, Islamabad, Pakistan)
Arif Hussain (Department of Mathematics, Quaid-I-Azam University, Islamabad, Pakistan)

International Journal of Numerical Methods for Heat & Fluid Flow

ISSN: 0961-5539

Publication date: 2 August 2019



The purpose of the current flow configurations is to bring to attention the thermophysical aspects of magnetohydrodynamics (MHD) Williamson nanofluid flow under the effects of Joule heating, nonlinear thermal radiation, variable thermal coefficient and activation energy past a rotating stretchable surface.


A mathematical model is examined to study the heat and mass transport analysis of steady MHD Williamson fluid flow past a rotating stretchable surface. Impact of activation energy with newly introduced variable diffusion coefficient at the mass equation is considered. The transport phenomenon is modeled by using highly nonlinear PDEs which are then reduced into dimensionless form by using similarity transformation. The resulting equations are then solved with the aid of fifth-order Fehlberg method.


The rotating fluid, heat and mass transport effects are analyzed for different values of parameters on velocity, energy and diffusion distributions. Parameters like the rotation parameter, Hartmann number and Weissenberg number control the flow field. In addition, the solar radiation, Joule heating, Prandtl number, thermal conductivity, concentration diffusion coefficient and activation energy control the temperature and concentration profiles inside the stretching surface. It can be analyzed that for higher values of thermal conductivity, Eckret number and solar radiation parameter the temperature profile increases, whereas opposite behavior is noticed for Prandtl number. Moreover, for increasing values of temperature difference parameter and thermal diffusion coefficient, the concentration profile shows reducing behavior.


This paper is useful for researchers working in mathematical and theoretical physics. Moreover, numerical results are very useful in industry and daily-use processes.



Khan, M., Salahuddin, T., Yousaf, M.M., Khan, F. and Hussain, A. (2019), "Variable diffusion and conductivity change in 3D rotating Williamson fluid flow along with magnetic field and activation energy", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 30 No. 5, pp. 2467-2484.



Emerald Publishing Limited

Copyright © 2019, Mair Khan, T. Salahuddin, Muhammad Malik Yousaf, Farzana Khan and Arif Hussain.


Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at

1. Introduction

The study of heat and mass transfer is very beneficial due to its practical applications in engineering, ground water pollution, temperature controlled reactor, geophysical, electrical machinery, flat-plate solar collector, microelectronic device, thermal insulation buildings, etc. (Jonnadulaa et al., 2015) demonstrated heat and mass transfer characteristics of chemical reaction on magnetohydrodynamics (MHD) flow induced by a permeable stretching surface. The transformed equations were solved by using an effective numerical scheme known as implicit finite difference scheme. Pal and Mandal, (2017) exemplified the magnetic field of heat and mass transport close a stagnation point due to a shrinking/stretching sheet. Mishra and Bhatti (2017) analyzed chemical response on heat and mass assignment flow induced by a linear stretching surface with Ohmic heating. The thermal and concentration diffusions in nanofluid flow due to linear and nonlinear stretching surface was presented by Sreedevi et al. (Eid and Mahny, 2017). Sui et al. (2018) discussed the unsteady heat and mass transfer non-Newtonian nanoliquid flow over a permeable stretching surface. It was found that the concentration and thermal boundary layer becomes large foe higher values of heat generation and magnetic field. Sui et al. Khan et al. (2018a) analyzed heat and mass transfer micropolar fluid over a stretching surface along with slip effect. Khan et al. Gireesha et al. (2018) deliberated the effects of heat and mass transfer on Williamson nanofluid induced by a nonlinear stretching surface. Viscosity was assumed to be temperature dependent. Gireesha et al. Khan et al. (2018b) performed the thermal and concentration diffusions in Oldroyed-B nanofluid flow due to a stretching surface with convective boundary condition. It was found that concentration profile diminishes for improving values of Lewis number. Khan et al. Sarojamma et al. (2018) analyzed heat and mass diffusion in Jeffery nanofluid flow persuaded by inclined stretching sheet in the presences of magnetic field. Sarojamma et al. Azizian et al. (2014) discussed the thermal and solutal stratification effects in non-Newtonian micropolar fluid flow due to stretched sheet along with transport of heat energy and mass species.

The work on MHD for electrical conducting fluid due to a heat surface has increased great attention of numerous researchers in view of its significant applications in several problems such as petroleum industries, plasma study, glass manufacturing, cooling of nuclear reactors and crystal growth. The impact of external magnetic field on heat diffusion coefficient of magnetite nanoliquid in laminar flow regime was presented by Azizian et al. (Makinde et al., 2016). Sreedevi et al. (2017) presented the concept of variable viscosity on MHD nanofluid flow due to a radially stretched surface. Khan et al. (2017) exemplified the chemical reaction change in concentration diffusion on MHD Williamson nanofluid flow due to an isothermal cone and plate in preamble surface. Joule heating was also considered in heat equation. Nourazar et al. (2017) investigated the MHD flow of nanofluid induced by a preamble stretching cylinder and calculated the solution by Optimal collection scheme. Ramay et al. (2018) investigated the effect of thermal wall slip on MHD nanofluids flow over a nonlinear stretching surface. Turkyilmazoglu (2018) illustrated the effect of magnetohydrodynamic viscous fluid over a nonlinearly deforming porous surface. (Chen et al., 2018) examined the hydromagnetic flow of a viscoelastic fluid induced by a stretching surface.

In fluid mechanics, fluid is divided into three classes, namely, integral type, differential type and rate type. The Williamson fluid is one of the simplest model that describes the characteristics of pseudoplastic materials. This model was presented by (Williamson, 1929). Nadeem et al. (2013) investigated the two-dimensional boundary layer equations for Williamson liquid induced by a stretching surface. Kumaran and Sandeep (2017) exemplified the heat and mass diffusions in magnetohydrodynamic Casson and Williamson fluid flow due to a parboiled surface. It was seen that mass and heat transfer in Casson fluid was comparatively large than that of Williamson fluid. Khan et al. (2018c) discussed the computational aspect of temperature based variable viscosity on Williamson nanoliquid over a nonlinear stretching surface. Soomro et al. (2018) analyzed the numerical simulation of MHD stagnation point flow on Williamson nanoliquid due to a vertical moving surface. Temperature and concentration of nanoliquid was reduced due to increase in thermal and solutal slip parameters. Some recent literature to relevant of nanofluid flow are illustrated (Farooq et al., 2018; Hayat et al., 2016a; Hayat et al., 2015; Khan et al., 2018d; Khan et al., 2018e; Hayat et al., 2016b).

In the current article, the main objective is to examine the transport of MHD Williamson fluid flow over an accelerated/decelerated rotating stretchable surface along with heat and mass diffusions. The modeling includes the similarity transformations in order to convert the partial differential equations into ordinary differential equations and then the problem is solved by numerically by fifth order RK-Fehlberg technique. Graphs and tables are constructed in order to examine the velocity, heat and mass diffusions for different values of dimensionless parameters.

2. Mathematical formulation

2.1 Flow formulation

Let us assume the variable diffusion coefficients (mass diffusion coefficient and thermal conductivity) in Williamson rotating incompressible fluid flow due to elastic stretching surface placed in xy–plane, where fluid is positioned at z 0. It is assumed that the surface is stretched with velocity uw = cx in x–direction, where c >0. Let Cw and Tw be the concentration and temperature at the surface. Therefore, C and T are the concentration and temperature far away from the surface as illustrated in Figure 1. The solar radiation and Joule heating are added in the problem. In addition, the magnetic field Ho is applied perpendicular to the stretched plan.

2.2 Rheological model for Williamson model

Cauchy stress tensor for Williamson fluid model is defined as:

(1) S=PI+τ,
(2) τ=(μ+μ0μ1Γγo)A1,
(3) γo=12π,π=tra(A12),
in which S* is the extra tensor, I is the identity tensor, P is the Hydrostatic pressure, τ is the Cauchy tensor, μ0 denotes limiting viscosity at zero shear rate, μ represents limiting at infinite shear rate, Γ > 0 is a time constant, A1 = (∇V∗)+(∇V∗)t = L+Lt  is the first Rivlin–Erkicsen tensor.

We assume that μ = 0 and Γγo < 1. equation (2) becomes:

(4) τ=(μ01Γγo)A1,
using binomial expansion we get:
(5) τ=μ0(1+Γγo)A1,

2.3 Binary chemical reaction and activation energy

Fist order chemical reaction and activation energy are:

(6) KA=KB(TT)nexp[EakT],
where KA denotes the rate of chemical reaction ( 1sec), Ea is the activation energy (eV), KB represents the pre-exponential constant ( Kwsec), n is the fitted rate constant lies in the range –1 < n <1, T0 is known as reference temperature, the expression KB(TT)nexp[EakT] denotes the modified Arrhenius function and k = 8.61×10−5  denotes the Boltzmann constant (eVK).

2.4 Problem formulation

Basic equations of heat and mass transfer 3 D Williamson rotating fluid flow are

2.4.1 Continuity equation

(7) div V=0,

continuity equation can be written in component form

(8) xu+yv+zw=0,

2.4.2 Momentum equation

(9) ρdVdt(Ω×Ω×r)+(2Ω×V)=P+div S+J×Ho,

where J is the current density, H = [0, Ho, 0] is the external magnetic field, Ω = [0,0,Ω] angular velocity and ∇ represents operator.

After boundary layer approximations, the Williamson model can be written as:

(10) uxu+vyu+wzu2Ωv=z(1+Γγo)zuσHo2ρu,
(11) uxv+vyv+wzv2Ωu=z(1+Γγo)zvσHo2ρv,

2.5 Variable diffusion coefficients

Studies considering temperature dependent linear thermal conductivity (Qureshi et al., 2018) is:

(12) K(T)=k[1+ε1(TTTwT)],
here ε1 is very small parameter. For ε1 = 0 the above equation converts into constant thermal conductivity.

Now we consider the concentration diffusion coefficient as:

(13) D(C)=k[1+ε2(CCCwC)],
here ε2 is very small parameter. Equation (13) is converted into constant mass diffusion coefficient for ε2 = 0.

2.5.1 Energy and concentration equations

(14) dTdt=1(ρc)p.(K(T)T)1(ρc)pzqr+1(ρc)pσHo2ρ(u2+v2),
(15) dCdt=.(D(c)C)KA(CC),

after applying the boundary layer approximations, energy and concentration equations become

(16) uxT+vyT+wzT=kρcpz(K(T)zT)+1(ρc)p(16T3σ3k)zzT+1(ρc)pσHo2ρ(u2+v2),
(17) uxC+vyC+wzC=z(D(T)zT)KB(TT)nexp[EakT](CC),
the specified boundary conditions are:
(18) u=Uw=cx,v=0,T=Tw,C=Cwat z=0,u0,v0,TT, CCas z.

Where ρ is the density, Tw is denotes temperature at the wall, σ is electrical conductivity of the liquid and cp denotes the specific heat.

The main PDEs are reduced into ODEs by using following similarity transformations:

(19) u=cxf(η),v=cxg(η),w=cνf(η),η=cvz,θ(η)(TwT)=(TT),ϕ(η)(CwC)=(CC).

Continuity equation (8) is satisfied, while the boundary layer equations (10), (11), (16) and (17) are transformed into ODEs as:

(20) Wef(f+g)+Wef((f)2+(g)2)+((f)2+(g)2)(f+ff(f)2+2λgHaf)=0,
(21) Weg(f+g)+Weg((f)2+(g)2)+((f)2+(g)2)(gfg+gf2λfHag)=0,
(22) θ(1+ϵ1θ+R)+ϵ1(θ)2+Prθf+HaEcPr(f+g)=0,
(23) ϕ(1+ϵ2ϕ)+ϵ2(ϕ)2+PrLefϕPrLeK(1+δθ)nexp[E1+δθ]ϕ=0,
Using equation (18), the boundary conditions become:
(24) dfdη=1,f=0,g=0,θ=0,ϕ=1,atη=0,dfdη0,θ0,ϕ0,asη.
where λ=Ωc denotes the rotation parameter, We=Γcxνc is known as Weissenberg number, Ha=σHo2cρ represents the Hartmann number, Pr=ucpk is Prandtl number, K=KBc denotes the dimensionless reaction rate, E=EakT represents the non-dimensional activation energy, δ=TwTT is the temperature difference parameter, Ec=Uw2TwT is the Eckert number, Le=νD is the Lewis number and R=16σT33kk3 is thermal radiation parameter.

2.6 Physically interests

To get physical interests near the wall i.e. the friction factor coefficient, heat transfer rate and mass transfer rate, we use the following relations:

(25) Cfx=τxzρUw2,Cfy=τyzρUw2,Nu=xqwk(TwT),Sh=xJwk(CwC).

Here τw denotes the shear stress or skin friction, qw and Jw denotes the heat and mass diffusion near the wall, where τxz, τyz, qw and Jw are defined as:

(26) τxz=μ[zu+(Γ2(zu)2+(zv)2)zu]z=0,τyz=μ[zv+(Γ2(zu)2+(zv)2)zv]z=0,qw=K[Ty]z=0,Jw=Db[Cy]z=0.

After using the scaling variables, friction factor, Nusselt and Sherwood numbers transformed into:

(27) Cfx(Rex12)=[f(η)+Wef(η)f(η)+g(η)]η=0,Cfy(Rex12)=[g(η)+Weg(η)f(η)+g(η)]η=0,NuRex12=(1+ϵ1θ(0)+R)θ(0),ShRex12=(1+ϵ2ϕ(0))ϕ(0).

3. Numerical procedure

In the procedure of numerical simulation, the technique used to solve the set of equations play an important role in view of its order and accuracy in yielding the solutions. A highly nonlinear differential equations (21)-(24) with corresponding boundary conditions (25) are solved using the RK-Fehlberg fifth order scheme along with Cash and Karp. The highly nonlinear ODEs are of 3rd order f, 2nd order θ and ϕ are then converted into simultaneous ordinary equations. With the initial conditions, we can take the values of f′(η), θ(η), and ϕ(η). Thus, we may achieve unknown initial conditions at η using shooting method. The most important step of this technique is to choose the appropriate finite values of η. We choose ηmax⁡ = 5 which is adequate to obtain far field boundary conditions asymptotically for all values of the parameters considered.

4. Results and discussion

We expose the results to keep up the examine of various non-dimensional parameters such as Hartman number, activation energy, dimensionless reaction rate, thermal conductivity, mass diffusion coefficient and temperature difference parameter on velocity, skin friction coefficient, temperature, heat transfer rate, concentration and mass transfer rate. Figures 2 and 3 the radial and azimuthal velocity profiles for different values of Ha. It is found that the momentum flow field declines for increment values of Hartmann number Ha. This would happen because the application of transverse magnetic field sets Lorentz force, which retards the radial velocity. On the other hand, the azimuthal velocity profile is increased due to increase in Hartmann number Ha. Figure 4 influence of Weissenberg number We on radial velocity profile. It is noticed that the radial velocity profile reduces with increasing values of Weissenberg number We. Physically, deduced as the relaxation time of fluid advances for higher stander of Weissenberg number We causing reduction in radial velocity profile. Figure 5 depicts the nature of We on azimuthal velocity profile. It is found that an oscillatory pattern encountered for growing values of We assist the flow in negative y-direction. Figure 6 illustrates the impact of rotation parameter λ on radial velocity profile. It is found that radial velocity increases for large values of λ. For λ = 0, the frame is non-rotating. With gradually improvement in λ, rotation rate gets stronger than the stretching rate and offer more conflict to the liquid motion. Eventually, a thinner boundary layer with improve in radial velocity profile is observed. Figure 7 influence of rotation parameter λ on azimuthal velocity. It is fact that an oscillatory pattern confront for improving values λ which accommodate the flow is negative direction. Figure 8 is plotted for several values of Prandtl number Pr against temperature profile. In this figure we can analyze that an increment in Pr generates a marked decrease in the temperature distribution. For increasing values of Pr lower thermal diffusivity causes reduction in temperature. Such kind of lower thermal diffusivity signifies the decrement in temperature profile. Figure 9 displays the behavior of radiation parameter R on temperature profile. It is found that temperature profile enhances for enhancing values of solar radiation parameter R. This is because an increase in the solar radiation parameter R leads to enhance boundary layer thickness. Figure 10 presents the effect of variable thermal conductivity ε1 on temperature distribution. It is also clear that the temperature of the liquid for the case of constant thermal conductivity (ε1 = 0) is less than that of the liquid having variable thermal conductivity i.e, ε1≠0. Thermal boundary layer thickness enhances when thermal conductivity parameter ε1 is improved. Figure 11 is plotted to explain the influence of Eckert number Ec on temperature profile. It is noticed that the temperature profile enhances with enhancing Eckert number Ec, this is due to the action of friction heating. For low-speed liquids, the viscous heating can be ignored. Figure 12 presents the effect of concentration diffusion coefficient ε2 on concentration profile. It can be analyzed that concentration diffusion has improving rate under diffusion coefficient. Accordingly, the concentration boundary layer thickness enhances when concentration diffusion coefficient ε2 is improved. In Figure 13, the behavior of Lewis number Le on concentration distribution has been displayed. It can be examined that the higher values of Lewis number Le reduces the mass concentration. Therefore, the concentration profile reduces by enhancing values of Lewis number Le. Figure 14 is plotted to depict the effect of temperature difference parameter δ on concentration profile. Figure 15 represents the nature of dimensionless activation energy E on concentration profile. Physically, improve in values of activation energy E lower the temperature. Table I gives the numerical results of skin friction coefficient with respect to behavior of various physical parameters. It is observed that for enhancing values of Hartmann number and rotation parameter, the skin friction coefficient enhances. Moreover, as expected, skin friction coefficient reduces with enhancing values of Weissenberg number We. Table II computational values of heat transfer rate with respect to variation of different physical parameters. It can be analyzed that increasing values of Prandtl number Pr, thermal conductivity parameter ε1 and solar radiation parameter R increases the heat transfer rate. Therefore, as expected, rate of heat transfer reduces with increment values of Eckert number Ec. Table III depicts that the concentration diffusion coefficient ε2 increases the mass transfer rate. This is obvious form the date of this table that reduction is noticed for large values of Lewis number Le, chemical reaction rate K and dimensional difference rate parameter δ.

5. Concluding remarks

The mathematical model including heat and mass diffusions in Williamson fluid flow along a rotating stretching surface has been analyzed in existence of transverse magnetic field and solar radiation. The numerical solution has been achieved and the effects of several dimensionless parameters on diffusion phenomena have been analyzed. Some of the main points are:

  • For increasing values of Hartmann number Ha and Weissenberg number, We the radial velocity profile reduces. Whereas the converse trend is noticed in the case of Ha and We.

  • It is found that increase in values of rotation parameter λ reduces the radial velocity profile.

  • Eckert number Ec and Radiation parameters R have important role in the improvement of heat phenomena.

  • Rising values of Prandtl number Pr reduces the temperature distribution. Whereas the opposite trend is noticed for thermal conductivity ε1.

  • Concentration profile enhances for increasing values of concentration diffusion coefficient ε2, dimensional difference parameter δ and activation energy parameter E.


Geometry of the problem

Figure 1.

Geometry of the problem

f′(η) against Ha

Figure 2.

f′(η) against Ha

g(η) against Ha

Figure 3.

g(η) against Ha

f′(η) against We

Figure 4.

f′(η) against We

g(η) against We

Figure 5.

g(η) against We

f′(η) against λ

Figure 6.

f′(η) against λ

g(η) against λ

Figure 7.

g(η) against λ

θ(η) against Pr

Figure 8.

θ(η) against Pr

θ(η) against R

Figure 9.

θ(η) against R

θ(η) against ε1

Figure 10.

θ(η) against ε1

θ(η) against Ec

Figure 11.

θ(η) against Ec

ϕ(η) against ε2

Figure 12.

ϕ(η) against ε2

ϕ(η) against Le

Figure 13.

ϕ(η) against Le

ϕ(η) against δ

Figure 14.

ϕ(η) against δ

ϕ(η) against E

Figure 15.

ϕ(η) against E

Numerical results for wall friction factor (CfRex12) for unlike values of Ha, We and λ. When L e = Pr ⁡ = 1.1 ,   E = δ = Ec = K = ε 1 = ε 2 = 0.1

Ha We λ C fxRex12 C fyRex12
0.1 1.1275 0.1177
0.3 1.2245 0.1043
0.5 1.3148 0.0944
0.1 1.1275 0.1177
0.3 1.0697 0.1151
0.5 1.0237 0.1131
0.10 1.1275 0.1177
0.11 1.1288 0.1299
0.12 1.1303 0.1422

Computational results of heat diffusion rate (NuRex12) for different values of Pr⁡, ε1, R  and Ec. When Le = 1.1 ,   E = δ = λ = H a = K = W e = ε 2 = 0. 1

Pr ε1 R Ec NuRe x12
1.1 −0.6903
1.2 −0.7261
1.3 −0.7610
0.1 −0.6903
0.2 −0.7449
0.3 −0.7933
0.1 −0.6903
0.2 −0.7115
0.3 −0.7323
0.1 −0.6903
0.2 −0.6835
0.3 −0.6772

Values of mass diffusion rate (ShRex12) for unlike values of Le, ε2, K, E  and Ec = 0.3. When Pr ⁡ = 1.1,   E = δ = K = ε1 = H a = λ = W e = 0. 1

Le ε2 K E δ ShRe x12
1.1 −0.3375
1.2 −0.3092
1.3 −0.2750
0.1 −0.3375
0.2 −0.3514
0.3 −0.3641
0.1 −0.3375
0.2 - 0.0433
0.3 - 0.3978
0.1 −0.3375
0.2 −0.3742
0.3 −0.4077
0.1 −0.3375
0.2 −0.3572
0.3 −0.3118


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The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia for funding this work through research groups program under grant number R.G.P-2/29/40.

Conflict of interest: The corresponding author on behalf of all authors declares “no conflict of interest”.

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