# Lie Group Analysis of Double Diffusive MHD Tangent Hyperbolic Fluid Flow over a Stretching Sheet

Authors: Salman Zeb; Suliman Khan; Zakir Ullah; Muhammad Yousaf; Ilyas Khan; Nawa Alshammari; Nur Alam; Nawaf N. Hamadneh
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this study, we have used Lie group analysis procedure to propose a novel model for transforming the governing equations of double diffusive MHD hyperbolic tangent fluid flow model into a system of nonlinear ordinary differential equations (ODEs). The solution of these equations is then investigated numerically by employing Shooting method. We also reported and presented our results graphically illustrating the results and analysis of physical parameters on concentration, velocity, and temperature profiles and on other physical quantities present in the flow model. The results show that fluid temperature increases with rise in the modified Dufour and velocity slip parameters whereas opposite behavior is observed for thermal slip parameter. Moreover, the Nusselt number declines with enhanced values of modified Dufour parameter whereas its opposite effect has been observed for Dufour-solutal Lewis number and Prandtl number.


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## Body

## 1. Introduction

The magnetohydrodynamic (MHD) flow over a stretching sheet has numerous industrial and manufacturing applications, which include polymer extrusion, continuous casting of metals, petroleum industries, and electrical power generators. Due to its large-scale applications, many researchers attempted to study and analyze the solutions of differential equations describing boundary layer and MHD fluid flow problems along a stretching sheet. The pioneering work in this regard is carried out by Sakiadis [1], who studied the boundary layer equations over continuous solid surfaces. Later on, Crane [2] analyzed laminar boundary layer steady flow caused by a linear stretching sheet. Pavlov [3] obtained exact solution of momentum equation for MHD fluid flow along a stretching sheet. Since then, a number of researchers studied the stretching sheet problems by considering various aspects [4–8]. The investigation of numerically solving the heat transfer of MHD fluid flow in a stretching cylinder is attributed to Ishak et al. [9]. Mukhopadhyay [10] discussed the slip flow influence of the magnetic field towards a stretchable cylindrical surface. Zaimi and Ishak [11] presented the analysis of heat transfer along a stretching vertical sheet by considering partial slip conditions on stagnation-point flow. Bhojappa and Zeb [12] proposed the aspect of chemical reaction on steady 3D Maxwell nanofluid through a stretched sheet. They noticed that growing values of chemical reaction enhance the concentration distribution. Some other researchers recently studied and reported the dynamics of MHD boundary layer flow problems over the cylindrical surfaces. Interested readers may refer to [13–17] for more details.The applications of non-Newtonian fluids in different fields of science and engineering have gained considerable attention of researchers for investigating their thermo-physical aspects. These fluids are appropriate in chemical, pharmaceutical, and other industrial sectors. However, the study of non-Newtonian fluids poses a number of challenges. These include the existence of highly nonlinear governing equations in comparison with equations for Newtonian fluids and the nonexistence of a single constitutive equation, which explicitly describes these fluids due to their diverse nature. Keeping in view these issues, researchers have suggested a variety of fundamental equations for evaluating the physical aspects of non-Newtonian fluids. An important non-Newtonian model is tangent hyperbolic fluid, which is utilized in a number of industrial processes and laboratory experiments. The tangent hyperbolic model has been studied from different perspectives by various researchers [18–24]. The authors in [25] explored the peristaltic MHD tangent hyperbolic fluid flow by taking into account an inclined asymmetric channel having partial slip conditions. The effect of double diffusive convection on stagnation-point flow for a model of hyperbolic tangent nanofluid was inspected in [26] where the authors revealed that Brownian motion variable decreases the concentration curves and at the same time increases the temperature. Rehman et al. [27] deliberated the influence of heat generation and magnetic parameter for a stretching cylindrical surface by considering a model of thermally stratified tangent hyperbolic fluid. Zakir et al. [28] appraised the impact of stretching parameter on the tangent hyperbolic fluid model with suction/injection. They reported the enhancement of velocity distribution for growing values of stretching parameter.Lie group analysis is a technique valuable for investigation of nonlinear partial differential equations. The development of this method can be seen in Lie and Ackerman’s work [29] whereas for the unification of known exact integration techniques considering differential equations, refer to [30]. Many researchers studied various fluid flow problems under different conditions using the Lie group analysis method [31–36]. Kandasamy et al. [37] used the Lie group transformation method for analyzing an electrically conducting steady incompressible flow along a vertical stretchable sheet. In [38], the authors explored the free convective flow of a nanofluid by considering a porous media on a chemically reacting horizontal plate and obtained its Lie group solution. Ullah and Zaman [39] studied Lie group analysis of the MHD hyperbolic tangent slip flow model along a stretchable sheet. Ahmad et al. [40] deliberated characteristics of heat transfer and flow behavior along a shrinking sheet considering the effects of thermal slip conditions and magnetic parameter.In this work, we investigated numerical simulation of the double diffusive MHD hyperbolic tangent fluid flow model. We found new similarity transformations using Lie group analysis procedure, which transformed the nonlinear governing equations into a system of nonlinear ODEs. Its numerical solution is then investigated by employing the Shooting method in Mathematica software. We also carried out the analysis of different parameters including Weissenberg number, Lewis number, Prandtl number, Brownian motion, Hartmann number, power law index, source/sink parameter, thermophoresis on temperature, and velocity/concentration profiles. All the important findings are illustrated graphically.We organized this paper as follows. In Section2, the problem is formulated under certain assumptions. In Section 3, similarity transformations are obtained using Lie group analysis procedure, which gives us nonlinear ODEs. In Section 4, the numerical results, graphical illustrations, and discussion are presented. In Section 5, conclusion of the proposed work is drawn.


## 2. Mathematical Model

In this work, we consider a double diffusive convective MHD incompressible flow for a tangent hyperbolic fluid model and assume that the flow generation is anticipated from linear stretching. Further, we assume that the flow is bounded to the regiony>0 and coincident with the plane y=0 (see Figure 1). We define the extra stress tensor for hyperbolic tangent fluid model as shown in following equation [25, 41].(1)S¯=η∞+η0+η∞tanhΓγ˙¯nγ˙¯,where n is the flow behavior index, η∞ stands for infinite shear rate viscosity, Γ represents time dependent material constant, and η0 appears for zero shear rate viscosity. The term γ˙¯ in (1) is defined as given in the following equation:(2)γ˙¯=12ΣiΣjγ˙¯ijγ˙¯ji=12Π,whereΠ=12trgradV+gradVT2.Figure 1
Flow configuration.IncorporatingΓγ˙¯<1 and η∞=0 due to shear thinning characteristics of the fluid flow, the modified form of (1) implies(3)S¯=η01+nΓγ˙¯−1γ˙¯.Under the assumptions mentioned above, we have the continuity equation, i.e., as follows:(4)∂u∂x+∂v∂y=0.The equation of momentum as of [28] is given as follows:(5)u∂u∂x+v∂u∂y=ν1−n∂2u∂y2+2νnΓ∂u∂y∂2u∂y2−σB2ρu.The energy equation as of [42] is given as follows:(6)u∂T∂x+v∂T∂y=kρcp∂2T∂y2+Q0ρcpT−T∞+DTC∂2C∂y2.The concentration equation as of [42] is expressed as follows:(7)u∂C∂x+v∂C∂y=Dsm∂2C∂y2+DCT∂2T∂y2.In these governing equations, the termsDsm, DCT, and B represent the solutal diffusivity of porous medium, diffusivity of Soret type, and applied uniform magnetic field, respectively. Similarly, ν stands for kinematic fluid viscosity, ρ denotes the fluid density, and σ stands for electrical conductivity of the fluid. Also, velocity components along x,y coordinate system are represented by u,v. The term cp represents the specific heat, Q0 stands for volumetric rate of the heat generation, and k represents thermal conductivity. Moreover, the terms T, Tw, T∞, and C represent the temperature, temperature at the wall, the free stream temperature, and concentration, respectively. The appropriate boundary conditions associated with Equations (4)–(7) are identified as follows:(8)aty=0,u=ax+L0∂u∂y,v=0,T=Tw+K0∂T∂y,C=Cw,asy⟶∞,u⟶0,T⟶T∞,C⟶C∞,where K0 stands for thermal slip factor, L0 stands for velocity slip factor, and a stands for the stretching rate. For requirements of nondimensionalized form of Equations (4)–(7) and (8), we consider the following:(9)u¯=uaν,x¯=aνx,v¯=vaν,y¯=aνy,ϕ=C−C∞Cw−C∞,θ=T−T∞Tw−T∞,which shows dimensionless quantities. After simplification, (4) implies the following:(10)∂u∂x+∂v∂y=0.Similarly, (5) implies the following:(11)u∂u∂x+v∂u∂y=1−n∂2u∂y2+2nΓa∂u∂y∂2u∂y2−σB2ρau.Likewise, (6) implies the following:(12)u∂θ∂x+v∂θ∂y=kμcp∂2θ∂y2+Q0ρcpaθ+DTCνCw−C∞Tw−T∞∂2ϕ∂y2,and (7) becomes(13)u∂ϕ∂x+v∂ϕ∂y=Dsmν∂2ϕ∂y2+DCTνTw−T∞Cw−C∞∂2θ∂y2.From boundary conditions in (8), we have(14)aty=0,u=x+aνL0∂u∂y,v=0,θ=1+aνK0∂θ∂y,ϕ=1,asy⟶∞,u⟶0,θ⟶0,ϕ⟶0,where for brevity, we removed bars from the equations above.


## 3. Scaling Transformations

Before applying scaling transformations, we first introduce the stream functionψ as follows:(15)u=∂ψ∂y,v=−∂ψ∂x,and put it into (10)–(13) and (14) so Equation (10) is satisfied automatically and Equation (11) is transformed into Equation (15).(16)∂ψ∂y∂2ψ∂x∂y−∂ψ∂x∂2ψ∂y2=1−n∂3ψ∂y3+2nΓa∂2ψ∂y2∂3ψ∂y3−σB2ρa∂ψ∂y.Similarly, (12) becomes the following:(17)∂ψ∂y∂θ∂x−∂ψ∂x∂θ∂y=kμcp∂2θ∂y2+Q0ρcpaθ+DTCνCw−C∞Tw−T∞∂2ϕ∂y2.Equation (13) becomes the following:(18)∂ψ∂y∂ϕ∂x−∂ψ∂y∂ϕ∂y=Dsmν∂2ϕ∂y2+DCTνTw−T∞Cw−C∞∂2θ∂y2,and its associated boundary conditions become as shown in the following:(19)aty=0,∂ψ∂x=0,∂ψ∂y=x+aνL0∂2ψ∂y2,θ=1+aνK0∂θ∂y,ϕ=1,asy⟶∞,∂ψ∂y⟶0,θ⟶0,ϕ⟶0.Letε be the small parameter of the scaling transformation. Then, transformation Ϝ (a special form of Lie group analysis) is considered as given in the following:(20)Ϝ:x∗=xeεr1,y∗=yeεr2,ψ∗=ψeεr3,θ∗=θeεr4,ϕ∗=ϕeεr5,Γ∗=Γeεr6,where r1,r2,…,r6 are arbitrary real numbers. The point-transformation given in (20) transformed the co-ordinates x,y,ψ,θ,ϕ,Γ into x∗,y∗,ψ∗,θ∗,ϕ∗,Γ∗.Therefore, utilizing Equation (20), Equation (16) results in the following:(21)eεr1+2r2−2r3∂ψ∗∂y∗∂2ψ∗∂x∗∂y∗−∂ψ∗∂x∗∂2ψ∗∂y∗2=eε3r2−r31−n∂3ψ∗∂y∗3+eε5r2−2r3−r62nΓ∗a∂2ψ∗∂y∗∂3ψ∗∂y∗3−σB2ρaeεr2−r3∂ψ∗∂y∗,Equation (17) implies the following:(22)eεr1+r2−r3−r4∂ψ∗∂y∗∂θ∗∂x∗−∂ψ∗∂x∗∂θ∗∂y∗=eε2r2−r4kμcp∂2θ∗∂y∗2+Q0ρcpae−εr4θ∗+eε2r2−r5DTCνCw−C∞Tw−T∞∂2ϕ∗∂y∗2,and (18) is transformed into the following:(23)eεr1+r2−r3−r5∂ψ∗∂y∗∂ϕ∗∂x∗−∂ψ∗∂x∗∂ϕ∗∂y∗=eε2r2−r5Dsmν∂2ϕ∗∂y∗2+eε2r2−r4DCTνTw−T∞Cw−C∞∂2θ∗∂y∗2.If the exponent of these transformed systems of equations satisfies the following equations:(26)(24)r1+r2−2r3=3r2−r3=5r2−2r3−r6=r2−r3,(25)r1+r2−r3−r4=2r2−r4=−r4=2r2−r5,(26)r1+r2−r3−r5=2r2−r5=2r2−r4,then Equations (21)–(23) will remain invariant under the transformation Ϝ.Next, by solving Equations (24)–(26) simultaneously, we get(27)r1=r1,r2=0,r3=r1,r4=0,r5=0,r6=−r1.Putting (27) into the scaling transformations in (20), we obtain(28)Ϝ:x∗=xeεr1,y∗=y,ψ∗=ψeεr1,θ∗=θ,ϕ∗=ϕ,Γ∗=Γe−εr1.By utilizing Taylor’s series expansion on transformations in (28), we get the characteristic equations as(29)dxxr1=dy0=dψψr1=dθ0=dϕ0=dΓ−Γr1.From (29), one can easily obtain new similarity transformations as given in the following:(30)y=η,ϕ=ϕη,θ=θη,ψ=xfη,Γ=x−1Γ0.Putting quantities given in (30) into equations (16)–(18) and in boundary conditions specified in (19), we obtain(31)1−nf′′′+nWef″f‴+f″f−f′2−M2f′=0,where n stands for power law index, M2=σB2/ρa stands for Hartmann number, and We=2Γa is the Weissenberg number.(32)θ′′+Prfθ′+Qθ+PrNdϕ″=0,where Pr=μcp/k is the Prantdl number, Nd=DCT/νCw−C∞/Tw−T∞ is modified Dufour parameter, and Q=Q0/ρcpa is the source/sink parameter.(33)ϕ′′+PrLefϕ′+Ldθ″=0,where Le=k/ρcpDsm represents Lewis number, Ld=DCT/DsmTw−T∞/Cw−C∞ stands for Dufour-solutal Lewis number, and(34)atη=0,f′=1+αf″0,f=0,ϕ=1,θ=1+bθ′0,asη⟶∞,f′⟶0,ϕ⟶0,θ⟶0,where α=a/νL0 stands for velocity slip parameter and b=a/νK0 represents thermal slip parameter. The prime in these equations denotes differentiation with respect to η.Hence, it is required to solve equations (31)–(33) subject to the conditions in (34).Here, we define equations (35)–(37).(35)Cf=τwρax2,Shx=xhmDsmCw−C∞,Nux=xqwkTw−T∞,where Cf stands for skin friction coefficient, Shx is the local Sherwood number, Nux stands for the local Nusselt number, and(36)hm=−Dsm∂C∂y,qw=−k∂T∂y,τw=μ1−n∂u∂y+nΓ2∂u∂y2,where hm is the mass flux, qw stands for the heat transfer from the plate, and τw represents the skin friction. Substituting (9), (30), and (36) into (35), we obtain its dimensionless form as follows:(37)Re1/2Cf=1−nf″0+n2Wef″02,Re−1/2Nux=−θ′0,Re−1/2Shx=−ϕ′0,where the local Reynolds number is represented as Re=ax2/ν.


## 4. Numerical Results and Discussion

Here, the Shooting method is utilized to numerically solve the governing nonlinear system of ODEs (30)–(33). For coding purpose, MATHEMATICA 10 software is used. A small step sizeΔη=0.0001 along with convergence criteria is taken upto 10−7. We used a variety of values of the controlling parameters describing the flow problem for numerical simulations. We have also compared our proposed scheme in Table 1 for the skin friction coefficient with Zakir and Zaman [39] and Noreen Sher Akbar et al. [25]. We carried out the comparison by keeping α (the velocity slip parameter) absent. The results show excellent agreement and demonstrate accuracy for each considered value.Table 1
Comparison of skin friction coefficient values.
n↓M↓Akber et al. [21]Zakir and Zaman [39]Results of the present studyWe=0We=0.3We=0.5We=0We=0.3We=0.5We=0We=0.3We=0.50.001111111.00141.00141.00140.100.948680.942480.938260.948680.942480.938260.9496740.9434560.9392270.200.894420.880230.870260.894420.880230.870260.8950920.8808710.8708910.30.51.024720.988040.960011.024720.988040.960011.024721.074821.060110.31.01.183221.134541.096161.183221.134541.096161.183221.237641.096160.31.51.322881.261931.212351.322881.261931.212350.332881.261931.21235The graphical representations of the numerical results for different governing parameters influencing the heat transfer and flow behavior of the proposed model are depicted in Figures2–9.Figure 2
Variations in velocity profiles forM, n, We, and α.
(a)(b)(c)(d)Figure 3
Variations in temperature profiles forPr, Q, b, and α.
(a)(b)(c)(d)Figure 4
Variations in temperature profiles forNd and n.
(a)(b)Figure 5
Variations in concentration profiles forLe, Ld, n, and b.
(a)(b)(c)(d)Figure 6
Variations in concentration profiles forα and M.
(a)(b)Figure 7
Influence of governing parameters on skin friction coefficient.
(a)(b)Figure 8
Influence of several parameters on local Nusselt numbers.
(a)(b)(c)(d)Figure 9
Influence of several parameters on Sherwood number.
(a)(b)(c)(d)Figures2(a)–2(d) depict the behavior of We, α, M, and n on velocity distribution, respectively. It is evident that a decline in velocity and momentum boundary layer thickness occurs with growing intensity of M. This behavior is owing to the occurrence of Lorentz force, which causes retardation and opposes the fluid flow. The thickness of the hydrodynamic boundary layer declines with growing n values. The velocity profile reduces showing more resistance to the flow for each increasing value of We due to increase in relaxation time. In addition to this, we also observe that velocity profile reduces with increase in α.Figures3(a) and 4(b) show the effects of Pr, Q, α, b, Nd, and n on temperature profile, respectively. Here, the temperature profile reduces and a rapid decline occurs in thermal boundary layer for growing Pr values. The skin parameter Q has the same behavior on temperature function as of Figure 3(a). The temperature distribution reduces with rising b values. The qualitative behavior of α on the temperature profile is opposite to the quantities that are shown in Figure 3(c). Moreover, the qualitative behavior of Figures 4(a) and 4(b) is similar.In Figures5(a) and 6(b), we can see the effects of Le, Ld, n, b, α, and M on concentration profile of the flow. The concentration profile reduces with increase in Le. It is attributed to the reason that mass diffusivity reduces by increasing Le. The concentration function ϕη and the concentration boundary layer thickness increase by enhancing Ld. The concentration profile increases for increasing values of n and α whereas for b it has an opposite effect. Moreover, the concentration function and thickness of boundary layer observe a strict rise by increasing M, which are depicted in Figure 6(b).The skin friction coefficient reduces for augmented values ofn and α as shown in Figure 7(a). Moreover, less resistance to the flow is depicted for increasing values of We whereas it shows an increasing trend for M as illustrated in Figure 7(b).The heat transfer rate enhances for increasingPr whereas Nd has an opposite effect on it as shown in Figure 8(a). The variations in local Nusselt number against Le for distinct values of Q are shown in Figure 8(b). It is same qualitatively as of Figure 8(a). Further, it is observed that Ld and α have opposite behavior on the local Nusselt number as illustrated in Figure 8(c). The decreasing behavior of heat transfer rate with increase in n and b is highlighted in Figure 8(d).Sherwood number reduces with increasing values ofNd and Pr as depicted in Figure 9(a). Figure 9(b) shows that Sherwood number enhances for b and Le whereas reduces for α and M as shown in Figure 9(c). Sherwood number with Dufour-solutal Lewis number for distinct values of n is shown in Figure 9(c). Moreover, the qualitative behavior of Figures 9(d) and 9(c) is similar.


## 5. Conclusion

We presented numerical solutions for the double diffusive MHD hyperbolic tangent fluid flow model by developing similarity transformations through implementing Lie group procedure. The transformed nonlinear ODEs solution is then obtained by utilizing the Shooting method. Furthermore, we conclude the present study with the following remarks:(i)
Intensity in the flow for governing parameters including velocity slip, Hartmann number, Weissenberg number, and power law index causes the velocity to decrease(ii)
The temperature profile reduces for enhancing values of Prandtl number, thermal slip, source/sink, and Dufour parameter(iii)
The power law index, modified Dufour, and velocity slip parameters show similar behavior on temperature functionθη(iv)
Enhancement of the concentration profile is observed for increasing power law index, thermophoresis parameter, Hartmann number, and velocity slip(v)
Increasing Lewis number has reducing effect on concentration profile(vi)
Reduction in skin friction coefficient shows less resistance to the flow for velocity slip parameter, Weissenberg number, and power law index but opposite behavior for the Hartmann number(vii)
The local Nusselt number of the flow shows increasing behavior for source/sink parameter, Prandtl number, and Dufour-solutal Lewis number whereas decreasing behavior for power law index, modified Dufour parameter, Lewis number, and slip parameters(viii)
The thermal slip parameter and Lewis number enhance the Sherwood number whereas they reduce the power law index, Dufour parameter, Prandtl number, Dufour-solutal Lewis number, velocity slip parameter, and Hartmann number



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