# Effects of the Magnetohydrodynamic Flow within the Boundary Layer of a Jeffery Fluid in a Porous Medium over a Shrinking/Stretching Sheet

Authors: Shaila S. Benal; Jagadish V. Tawade; Mahadev M. Biradar; Haiter Lenin Allasi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The consequences of magnetohydrodynamic flow inside the boundary layer of a Jeffery fluid in a porous material across a shrinking/stretching sheet are discussed in this paper. The Runge–Kutta fourth-order technique is used to turn partial differential equations into nonlinear ordinary differential equations and solve them using similarity transformation. On the velocity and temperature profiles, the effects of key factors such as “thermal stratification”e1, λ1 “Jeffery parameter,” Pr “Prandtl number”, M “Magnetic field,” “Porous parameter” λ2, and “heat generation/absorption” have been visually described. In terms of heat transmission, the Jeffrey nanofluid beats other fluids such as Oldroyd-B and Maxwell nanofluids, according to the findings. According to our findings, the thickness of the boundary layer is explored in both stretching and shrinking. When the “thermal stratification” e1 parameter is increased, fluid velocity and temperature rise, while the “heat generation/absorption” γ parameter has the opposite effect.


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## Body

## 1. Introduction

Non-Newtonian fluids such as printing inks, polymer solutions, ketchup, glues, detergent slurries, and pastes are basically nonlinear and regularly show elastic as well as viscous properties. Such constitutive equations are more advanced than standard (Navier Stokes) Newtonian fluid. Preeminent non-Newtonian type Walters-B short memory models, Jeffery models, and Oldroyd-B models have different degrees of clarification of the classical momentum preserve these equations. Jeffrey most is the most straightforward non-Newtonian liquid which displays shear diminishing properties, excessive shear viscosity, and yield pressure. Many engineering researchers are fascinated because of its variety of applications and simplicity in engineering and science. Crane (1) studied flow overstretching plate and gave a precise resembling solution and solved analytically for peripheral layer flow of incompressible viscous fluid.“Wang [1] conferred the concept of the flow around the shrinking sheet in his study of unsteady film. The existence and uniqueness of the solution of steady viscous flow over a shrinking sheet were proved by Milkovich and Wang [2]. Miklavcic and W [3] first studied the MHD flow over a stretching surface in an electrically conducting fluid. The authors of [4, 5] studied stretching sheets. Makinde [6] explored heat and mass transfer with mixed convection in presence of a stagnation point. Shateyi and Makinde [7] discussed on convectively heated disk with hydromagnetic stagnation point flow towards a radically stretching. Ellahi et al. [8] discussed the peristaltic flow pf Jeffery in a rectangular duct. [9] Yakubu Seini and Oluwole Makinde traced on boundary layer flow near stagnation points on a vertical surface in the presence of the transverse magnetic field. The third-grade nanofluid flow generated by sheet stretching was influenced by Khan et al. [10]. With viscous dissipation and Joule heating, the MHD stagnation point flow of Jeffery fluid radically was studied by Hayat et al. [11]. El-Aziz [12] studied the dual solutions in hydromagnetic stagnation point flow and heat transfer towards a stretching/shrinking sheet with a nonuniform heat source/sink and variable surface heat flux. Turkyilmazoglu [13] analysed the flow of a micropolar fluid due to a porous stretching sheet and heat transfer.” Babu and Narayana [14] studied on mixed convection of a Jeffrey fluid over a stretching sheet with power-law heat flux. Shahzad et al. [15] reported on unsteady axisymmetric flow and heat transfer over a time-dependent radically stretching sheet. Eswara Rao and Sreenadh [16] discussed on boundary layer flow of Jeffery fluid over a shrinking/stretching sheet. Mishra et al. [17] proposed a non-Newtonian convective flow in two dimensions in the presence of a heat source and sink. Eswara and Krishna Murthy [18] proposed the flow of Jeffrey fluid through a stretching/shrinking sheet over porous material was studied using MHD stagnation point flow. The authors of [19, 20] studied on stagnation point. The heat transport properties of an incompressible non-Newtonian Jeffery liquid over a stretching/shrinking surface with polluted radiation and a heat source were investigated by Babu et al. [21]. A number of authors have freshly investigated non-Newtonian fluid flow models incorporating a variety of heat transfer effects.The effect of MHD flow on various fluid models was later addressed by multiple writers [22–26] with the convective boundary flows through the periphery layer. Peripheral layer flow with convective boundary conditions is used in manufacturing and ecological technologies, as well as energy storage, gas turbines, geothermal reservoirs, and nuclear reactors. Transmission has gained a lot of traction. The relevant studies comprise Afzal et al. [27], Afzal et al. [28], and Tayyaba et al. [29].Motivated by previous literature, we study the effects of thermal stratification, Jeffery parameter, Prandtl number, magnetic field, porous parameter, and heat generation/absorption for Jeffery fluid by taking into account the later wall being impermeable. The major observation is the rise in the porosity parameter of the fluid is caused by an increase in the viscosity of the fluid, whereas a drop in the porosity at the wall results in a progressive reduction in the fluid's flow velocity, as observed. The results are obtained with the help of bvp4c and are presented in form of tables and figures.


## 2. Formulation Problem

A two-dimensional (x, y) MHD stream of incompressible Jeffrey liquid flows across a y = 0 shrinking/stretching sheet. In standard notation, the MHD Jeffery fluid flow and temperature equations are expressed as (ref [28] and [29])(1)∂u∂x+∂u∂y,(2)u∂u∂x+v∂u∂y=ν1+λ1∂2u∂y2−σB02ρu−νku,(3)u∂T∂x+v∂T∂y=kρCp∂2T∂y2+QρCpT−T∞.Velocity componentsu & v, “ν=μ/ρ” denotes kinematic fluid viscosity, “ρ” denotes fluid density, µ is the coefficient of fluid viscosity, λ1 represents Jeffery parameter, Cp and k represent specific heat and thermal conductivity at constant pressure, and Q denotes heat generation/absorption. The boundary conditions for the present study are(4)u=Uw,v=−vw,T=Tw=T0+b1xaty=0u⟶0⇒T⟶T∞=T0+b2x,y⟶∞,asy⟶∞,where Uw=cx represents for stretching sheet case, and Uw=−cx represents for the situation of shrinking sheet case with e > 0 being shrinking/stretching constant. vw wall mass transfer velocity with vw>0 for mass suction and vw<0 for mass injection.Similarity transformations are(5)ψ=cνxfηandη=ycvθη=T−T∞Tw−T0⇒T=Tw−Toθη+T∞,where ψ is stream function, and η is similarity variable. Stream functions are defined as(6)u=∂ψ∂yandv=−∂ψ∂x.Also,(7)T=b1xθη&T∞=T0+b2.Using equation (5), the equations (2) and (3) take the forms(8)11+λ1f′′′+ff′′−f′2−M+λ2f′=0,(9)θ′′+Prθ′f−Prθf′−Pre1f′+Prγθ=0.The corresponding boundary condition for the stretching sheet is as follows:(10)fη=S,f′η=1,θ0=1−e1atη=0;f′η⟶0,θ∞⟶0,asη⟶∞.For shrinking sheet,(11)fη=S,f′η=−1,θ0=1−e1atη=0;f′∞⟶0,θ∞⟶0,asη⟶∞,S=vw/cν1/2. With S > 0 (i.e., vw>0) S < 0 (i.e., vw<0) for wall mass suction and wall mass injection, S represents wall mass parameter.whereM=σB02/ρc magnetic parameter, λ2=v/ck porous parameter, pr=μCp/K. Thermal stratification parameter e1=b2/b1 and γ=Q/ρCpa heat generation/absorption parameter.


## 3. Methodology

To solve the BVPs, the governing equation is solved using the firing system with the Runge–Kutta fourth-order technique. Equations (8) and (11) are used to translate the altered ODEs into the following system.(12)df0dη=f1,df1dη=f2,1+λdf2dη=f12+M+λ2f1−f0f2,(13)dθ0dη=θ1,dθ1dη=Prf1θ0+e1f1+γθ0−θ1fo.Following that, the boundary conditions, equations ten and eleven, take the form of stretching and shrinking.fo= S, f10=−1,f1∞=0,θ00=1−e1,θ0∞=0fo=fη,andθ0=θη By correctly estimating the omitted slopes, the aforementioned BVP is first turned into an IVP. The MATLAB bvp4c package is used to solve the generated IVPs.


## 4. Results & Discussion

Figure1 shows the impact of Prandtl no “Pr” on f′ (velocity profile) & “θ” temperature profile for both cases stretching and shrinking. An increase in Prandtl no composes the fluid more dense, which causes a decline in the velocity profile. In temperature profile for both the cases, for different values of Pr no decreases, because the dimensionless number is inversely related to thermal conductivity, it follows that increases in Prandtl no hold weak energy diffusion. Upgrading Pr causes a significant drop in fluid temperature, resulting in a smaller thermal boundary layer. While shrinking a sheet, an immense Prandtl number fluid induces thermal unsteadiness at the superficial (i.e., a negative value of Nu), but this is not the case when stretching a sheet. This is shown in Figure 2. Thermal stratification e1 is plotted in Figures 3 and 4 for stretching and shrinking on the velocity profile and temperature profile. Thermal stratification e1 because the convective potential among the sheet surface and the ambient temperature is reduced, the fluid’s velocity is reduced. “θ” temperature decreases enhances in the stratification parameter. An effect of heat generation/absorption γ in Figures 5 and 6 demonstrates how the temperature changes as the heat emission/immersion parameter is changed. It has been discovered that as γ grows, so does the temperature. The existence of a transverse magnetic field induces the Lorentz force, which results in a velocity field retarding force. The retarding force increases as the value of M increases, and as a result, the velocity decreases as the temperature and concentration profiles increase. The thickness of the thermal boundary layer thickens as the fluid temperature rises. The presence of an external heat source has a considerable effect on the fluid’s temperature gradient, resulting in an increase in both the temperature distribution and the fluid’s thermal state. The thermal boundary layer thickness grows to a greater extent as a considerable amount of heat energy is created among fluid particles. When the parameter M shrinks, different values of M cause the velocity to rise, as shown in Figure 7. The influence of the magnetic field parameter M on flow temperature profiles is shown in Figure 8 for both cases. Figure 9 demonstrates, for different values of the Jeffrey parameter λ1, the fluctuation in velocity f in the stretching situation, the velocity reduces as the Jeffrey parameter λ1 grows, and the viscosity of the boundary layer decreases, whereas, in the shrinking case, the velocity decreases as the Jeffrey parameter λ1 increases, and the thickness of the peripheral layer decreases. Figure 10 demonstrates the impact of λ1 on the temperature profile for stretching and shrinking cases, and because of the greater temperature and profuse thermal boundary layer, the temperature profile reduces as the Jeffery parameter λ1 enhances, resulting in a rise in moderation time and a drop-in obstruction time. The reason behind the decrease in velocity profile is that, as we increase the values of the Jeffrey fluid parameter, the boundary layer momentum thickness will rise. Hence, the velocity distribution declines as the values rise up. Figures 11 and 12 are plotted the graphs for the porous parameter λ2 for velocity f′ and temperature profile “θ” for both the cases stretching and shrinking. λ2 porous parameter increases as velocity decreases for both the cases. In temperature profile for different values of λ2, porous parameter increases as temperature profile decreases in stretching case, but in shrinking case λ2, porous parameter increases as temperature decreases. The rise in the porosity parameter of the fluid is caused by an increase in the viscosity of the fluid, a drop in the permeability at the edge, or a decrease in the stretching rate of the accelerating surface, which results in a progressive reduction in the fluid’s flow velocity, as observed.Figure 1
The impact of Prandtl number Pr on the velocity profile.Figure 2
“Pr” on the temperature profile.Figure 3
Thermal stratificatione1 for velocity profile.Figure 4
The influence of “e1” thermal stratification on the temperature profile.Figure 5
Effects of heat generation/absorptionγ on the velocity profile.Figure 6
Temperature profile effects of heat generation/absorptionγ.Figure 7
The velocity profile on which the magnetic parameterM has an effect.Figure 8
Magnetic parameterM on the temperature profile.Figure 9
The influence of the Jeffery parameterλ1 on the velocity profile.Figure 10
Plots Jeffery parameterλ1 on the temperature profile.Figure 11
Porous parameterλ2 on the velocity profile.Figure 12
Plots porosity parameterλ2 on the temperature profile.The Nusselt number is provided in tabular form for various values of specified physical parameters. The following conclusion can be drawn from the current investigation as shown in Table1.Table 1
The rate of heat transfer−θ′ for different values of b/a.
b/aPrMSBhattacharyya [5]Dash [17]Eswara Rao [21]Present values−1.240.1000.1282970.1281660.1181980.118077−1.240.5000.0983720.0958860.0958480.095330−1.240.51−10.6537250.5981040.6740000.672103−10.7100—0.2282800.2282790.227102−10.7110—0.3249630.3249630.323102−10.7110.2—0.1782890.1782890.164122−0.50.7110.2—0.2997880.2997880.29978600.7110.2—0.4028400.4028400.40284010.7110.2—0.5740880.5740880.574087−1710.2—−0.685150−0.685150−0.685150


## 5. Conclusion

The main focus of this research is on the momentum and heat transfer of boundary layer fluid flow of a Jeffrey fluid in a porous material over a shrinking/stretching sheet. The concept of dimensionless velocity and temperature is also investigated. From the current analysis, we may derive the following conclusions for various values of the stated physical parameters, the Nusselt number, and skin friction [30, 31].(i)
The effects of the Prandtl number on velocity, temperature, and concentration have been observed; the rise in the fluid’s Prandtl number is related to increased viscosity.(ii)
The flow for different values of Jeffrey fluid parameterλ1, on the velocity profile f′(η), it is observed that, an increase in the Jeffrey fluid parameter, increases the velocity in the boundary region.(iii)
The thermal stratification parametere1’s strength can aid in fluid velocity and temperature control. For increasing stratification parameters, the temperature θ (η) of the flowing fluid drops. As e1 decreases, the temperature differential between the surrounding fluid and the fluid on the surface decreases, lowering the temperature as illustrated.(iv)
For both cases, theλ2 porous parameter increases as velocity decreases. In the stretching scenario, the λ2 porous parameter grows as the temperature profile lowers, but in the shrinking situation, the λ2 porous parameter increases as the temperature decreases.(v)
Finally, raising the absolute value of the heat absorption parameter raises the local Nusselt number whereas increasing the magnetic parameter and the heat generation parameter decreases it.(vi)
The Oldroyd-B fluid model is used to investigate the behavior of blood flow across an abdominal aortic segment in real life (hemodynamics).



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