Analysis of MHD Flow on Exponentially Stretching Sheet with Slip Conditions

Analysis of MHD Flow on Exponentially Stretching Sheet with Slip Conditions

Preeti Kaushik

Assistant Professor, Department of Applied Sciences and Humanities, Ajay Kumar Garg Engineering College, Ghaziabad

DOI: https://doi.org/10.51584/IJRIAS.2025.10060015

Received: 09 June 2025; Accepted: 13 June 2025; Published: 28 June 2025

ABSTRACT

This work examines radiative heat transfer and magnetohydrodynamic (MHD) boundary layer flow with second-order slip condition. The fluid flow in the direction of an exponentially stretched sheet is taken into consideration. Thermal conductivity and the impact of the magnetic field are considered. By using similarity transformation, the nonlinear mathematical expression of the flow that is acquired is converted into ordinary differential equations. Numerical solutions are found for the linked higher order nonlinear ordinary differential equations. Analysis is done on the velocity and temperature profile solution in relation to the first and second order slip parameters. Graphs for several relevant metrics are used to explain the flow’s features. As the Prandtl number rises, the rate of heat transmission falls, but the radiation parameter increases.

Keywords: Exponentially stretching sheet, MHD flow, Second order slip condition, Heat transfer, Finite difference method.

INTRODUCTION

An exponentially stretched sheet plays an important role in boundary layer fluid flow problems, this phenomenon has developed interest in several researchers. The exponentially expanding sheet idea has been used in several industrial engineering and technological applications, including the manufacturing of polymer extrusion, electric heaters, and stretching of metal, where the flow dynamics are influenced by the sheet’s deformation. The primary benefit of MHD flow is that it makes cooling rate estimate possible. Gas turbines, various aviation systems, and several space spacecraft are produced using the energy equation of a viscous fluid over an exponentially stretched sheet in conjunction with the magnetohydrodynamic flow principle. Crane et al. [1] originally looked at the distance-dependent velocity of the boundary layer flow toward the linear stretching sheet issue.  Ibrahim et al. [2] developed the new concept for solving stagnation point flow problem across the stretching sheet. Irfan et al. [3] used the application of exponential stretching sheet for solving two dimensional incompressible flow. Ishak et al. [4] proposed the fluid flow of nanofluid due chemical reaction with effect of porous medium. Majeed et al. [5] investigated suction and injection impact on the velocity of Casson fluid flow. Nandeppanavar et al. [6] provided the numerical solution of nanofluid flow towards exponentially stretching surface. Ene et al. [7] analyzed influences of the heat generation and absorption on the 2D viscous flow of nanofluid. Kumar et al. [8] proposed the unsteady flow of MHD nanofluid due to exponentially extending surface. Bhattacharyya et al. [9,10] explored the condition of velocity slip applied on the carreau nanofluid with porous medium effects. Aurangzaib et al. [11] examined the Newtonian heating condition on the nanofluid over a stretched surface. Bhattacharyya et al. [12] presented consequences of the velocity slip conditions on the Casson fluid flow. Mukhopadhyay et al. [13] provided the Brownian motion and Thermophoresis parameters influences on the flow passing through vertical sheet. Waini et al. [14] investigated the properties of various variable impacting the flow over an exponential sheet. Nadeem et al. [15] developed the scheme to solve the issue of viscous fluid model in the presence of porous media. Reddy et al. [16] studied the incompressible fluid flow with chemical reaction impact across the stretching surface. Mabood et al. [17] MHD flow of Non-Newtonian fluid with thermal radiation. Nandeppanavar et al. [18] provided the solution of Maxwell fluid flow issue with velocity slip condition. Bhattacharyya et al. [19,20,21] explored several problems related to fluid flow using analytical and numerical methods.  Mandal et al. [22] analyzed the radiative heat transfer in nanofluid due to heat source and sink effect. Mabood et al. [23] determined the characteristics of nanofluid with over a nonlinear stretching sheet. Ghosh et al. [24] implemented the advanced finite element method to get the solution of nanofluid problem. Adegbie et al. [25] examined the joule heating effect on the heat transfer on Maxwell fluid flow. Sheikholeslami et al. [26] inspected the homogeneous-heterogeneous effect on velocity distribution of nanofluid flow. Afterwards, several researchers [27,28,29,30,31] presented the work on nano fluid flow past an exponentially extended sheet. Zeeshan et al. [32] studied the axisymmetric flow of casson fluid with the condition of Brownian motion and Thermophoresis effect. Xun et al. [33] illustrated the consequences of viscous dissipation on MHD flow.

 The purpose of this investigation is to examine the effect of second order slip on MHD flow over an exponentially stretching sheet. The effect of first and second order slip, magnetic parameter, radiation parameter explored in this study. The study offers important insights into the boundary layer flow problems in a porous medium.  The mathematical equations are solved numerically by bvp4c through Matlab. The fluid velocity and temperature profile for pertinent parameters are presented graphically. The results validate with the previously published work and found to be in good agreement.

Mathematical Formulation

We are considering two-dimensional magnetohydrodynamic fluid flow towards an exponentially stretching sheet. The stretching sheet is taken along x axis at y=0. A strong magnetic field of strength B_0 applied, where B_0 is constant and it is applied normal to the x axis. The induced magnetic field is not considered because of the low value of the Reynolds number. Here it is considered that thermal conductivity varies linear with temperature. u_w=-Ce^(x⁄L)is velocity at boundary , where C is constant, v=v_w= v_0 e^(x⁄2L) is a special kind of velocity, where v_0is a constant, v_0is considered as velocity suction for v(x)>0,when v(x)<0, v_0 is the velocity blowing, 〖T=T〗_w=T_∞+T_0 e^(x⁄2L)+N ∂T/∂y is fluid temperature. T_w is fluid temperature at wall, T_∞ is ambient temperature, , T_0 is reference temperature, N is thermal slip coefficient. The governing equations of continuity, momentum, and energy for boundary layer flow problems are represented as.

Figure 1 Physical model of the flow

where u and v are fluid velocity components along x and y axis respectively. ν is kinematic viscosity, k is permeability for Porous medium, ρ is fluid density, σ is chemical reaction rate constant, c_p is specific heat, T is fluid temperature, k_fis thermal conductivity, q_r is radiative heat flux, B〖=B_0 e〗^(x⁄2L) is magnetic field where B_0 is magnetic field constant.

The corresponding boundary conditions are:

Similarity Transformation

The Thermal conductivity of the fluid varies linearly with temperature

k_f=k_∞ (1+ϵθ)                     (15)

Where ϵ is thermal conductivity variation parameter. Using similarity transformation and boundary layer approximation Eqs. (10) and (12) become

Numerical Solution

The highly nonlinear system of ordinary differential equations (16) to (17) with the corresponding boundary conditions (18) and (19) are solved numerically using MATLAB bvp4c software. The software is using finite difference method to solve the boundary value problems. The transformed equations converted into initial value problem by taking the following transformation.

Finite value for boundary condition η→∞, i.e. η_max taken as 40. The step size is taken as ∇η=0.01, with the tolerance limit up to 10^(-5). Trial values of f”(0),θ'(0) were adjusted to get a better approximation to satisfy the corresponding boundary condition

RESULTS AND DISCUSSION

An analysis of a two-dimensional flow have been figured. Various physical quantities of interest like skin friction, Nusselt number functions are being calculated mathematically and the outcomes are traced out for the same in terms graphs. Some important quantities are presented in tabular form. The variations in velocity and temperature through the pertinent quantities are clear explained in the study. The solution of modified equations (16) to (17) together with boundary conditions (18) to (19) are acquired numerically. Graphs for many relevant inputs have been created to depict the fluid boundary layer flow and temperature distribution. In the whole inquiry, the non-dimensional numbers γ=0,δ=-1,M=0.2,R=0.7,Pr=0.5,α=-2 are taken commonly

Figure 2 Variation of f^’ (η) for the different values of δ for γ=0,M=0.2,R=0.7,Pr=0.5,α=-2

Figure 3 Variation of f^’ (η) for the different values of γ for δ=-1,M=0.2,R=0.7,Pr=0.5,α=-2

The consequence of second-order slip parameter δ on velocity profile f^’ (η) elluciated in Figure 2. The intriguing outcome seen that velocity is dropping down, when the value of δ boost up. To see the performance of velocity function f^’ (η) for variety of γ values is highlighted in Figure 3. The findings further explain that f^’ (η) degrades, as γ enhances. The resistance to fluid flow increases as a result of the first order slip parameter (γ) which results in a decrease in the velocity rofile f^’ (η).

Figure 4 Variation of f^’ (η) for the different values of M for values γ=0,δ=-1,R=0.7,Pr=0.5,α=-2

Figure 5 Variation of θ(η) for the different values of M for γ=0,δ=-1,R=0.7,Pr=0.5,α=-2.

The velocity distribution f^’ (η) owing to variations in the magnetic parameter (M) is portrays in Figure 4. It is noticed that, the upshot of escalating magnetic parameter M is seen in terms of declining behavior of velocity profile f^’ (η). For the hydrodynamic flow condition M is considered to be zero while for the hydro-magnetic flow, M is considered to be nonzero. It is clearly revealed in the graph that velocity flow field diminishes as higher values of M. Resistance is caused by a resistive force that is produced as a result of a physical magnetic field. Because of this resistance, the fluid moves more slowly. With respect to changing magnetic parameter values M. The variation in the strength of the temperature field is seen in Figure 5. It is evidence in the graph For rising amount of M.

Figure 6 Variation of θ(η) for the different values of Pr for γ=0,δ=-1,M=0.2,R=0.7,α=-2

Figure 7 Variation of θ(η) for the different values of R for γ=0, δ=-1, M=0.2, Pr=0.5, α=-2

Figure 6 demonstrate the effect of Prandtl number (Pr) on temperature profile θ(η). The graph elucidate that temperature profile θ(η) gradually decreases with the increasing value of Prandtl number (Pr).It is due to the property of Prandtl number that momentum diffusivity is higher than thermal diffusivity for large vale of Pr. Figure7 shows that the impact of Radiation parameter (R) on temperature profile θ(η). Graph elaborate that temperature profile θ(η) enhances against the Radiation parameter (R). Due to the phenomenon of electromagnetic radiation, Thermal energy generates in the fluid flow because of radiation parameter. That causes to increase in the temperature profile. 

Concluding Remark

In this study, we investigated the impact of second-order velocity slip condition on the two-dimensional flow of the fluid towards an exponentially stretching sheet. The effect of pertinent parameters on the velocity profile and temperature profile has been investigated.

The major findings are summarized as follows:

• In comparison to the second-order slip parameter (δ=-2), Velocity profile f^’ (η) is a decreasing function for second-order slip parameter (δ=-4). This is due to an enhancement in resistance of fluid flow, which results in a decrement in the velocity profilef^’ (η).
• Velocity profile f^’ (η) is maximum for the first order slip parameter(γ=0) while it is lower for the first order slip parameter(γ=6). This shows a significant decrement in the velocity profile.
• Velocity profile f^’ (η) is higher for the low magnetic parameter (M=0) but it is lower for high magnetic parameter (M=0.6).This reduction in velocity profile occurs because of resistive force generated by magnetic field.
• Temperature profile θ(η) is lesser for magnetic parameter(M=0), but for magnetic parameter(M=0.6), it displays a higher value of temperature profile θ(η). The increasing variation was observed in temperature profile for the magnetic parameter.
• Temperature profile θ(η) is maximum for Prandtl number (Pr=0.1) but it decreases for Prandtl number (Pr=0.7).The opposite behavior observed in Temperature profile against the Radiation parameter.

REFERENCES

1. LJ Crane. Flow past a stretching plate. Z. Angew. Math. Phys. 1970; 21: 645-655.
2. W Ibrahim. Magnetohydrodynamics (MHD) flow of a tangent hyperbolic fluid with nanoparticles past a stretching sheet with second order slip and convective boundary condition. Results in Physics 2017; 7, 3723-3731.
3. M Irfan and M Asif Farooq. Thermophoretic MHD free stream flow with variable internal heat generation/absorption and variable liquid characteristics in a permeable medium over a radiative exponentially stretching sheet. Journal of Materials Research and Technology, 2020; 9, Issue 3,4855-4866.
4. A Ishak. MHD Boundary Layer Flow Due to an Exponentially Stretching Sheet with Radiation Effect. Sains Malaysiana 2011; 40(4), 391–395.
5. A Majeed, FM Noori, A Zeeshan, T Mahmood, SU Rehman and I. Khan. Analysis of activation energy in magnetohydrodynamic flow with chemical reaction and second order momentum slip model. Case Studies in Thermal Engineering. 2018; 12,765–773.
6. M Mahantesh, Nandeppanavar, K Vajravelu, MS Abel and MN Siddalingapp. Second order slip flow and heat transfer over a stretching sheet with non-linear Navier boundary condition. International Journal of Thermal Sciences 2012; 58, 143-150.
7. RD Ene and V marinca. Approximate solutions for steady boundary layer MHD viscous flow and radiative heat transfer over an exponentially porous stretching sheet. Applied Mathematics and Computation, 2015; 269, 389-401.
8. R Kumar, S Sood, S A Shehzad and M Sheikholeslami. Numerical modeling of time-dependent bio-convective stagnation flow of a nanofluid in slip regime. Results in Physics. 2017; 7, 3325-3332.
9. K. Bhattacharyya, S Mukhopadhyay and G. C. Layek. Slip effects on an unsteady boundary layer stagnation-point flow and heat transfer towards a stretching sheet. Chinese Physics Letters.2011; 28, 9.
10. K. Bhattacharyya and G. C. Layek. Slip effect on diffusion of chemically reactive species in boundary layer flow over a vertical stretching sheet with suction or blowing. Chem. Eng. Comm.2011; 198, 1354-1365.
11. Aurangzaib, Md S Uddin, K Bhattacharyya and S Shafie. Micropolar fluid flow and heat transfer over an exponentially permeable shrinking sheet. Propulsion and Power Research. 2016; 5, Issue 4, 310-317.
12. K Bhattacharyya, MS Uddin and GC Layek. Exact solution for thermal boundary layer in Casson fluid flow over permeable shrinking sheet with variable wall temperature and thermal radiation. Alexandria Engineering Journal. 2016; 55, Issue 2, 1703-1712.
13. S Mukhopadhyay. Slip effects on MHD boundary layer flow over an exponentially stretching sheet with suction/blowing and thermal radiation. Ain Shams Engineering Journal. 2013; 4, Issue 3, 485-491.
14. I Waini, A Ishak and I Pop. Mixed convection flow over an exponentially stretching/shrinking vertical surface in a hybrid nanofluid. Alexandria Engineering Journal. 2020; 59, Issue 3, 1881-1891.
15. S Nadeem, MN Khan and N Abbas. Transportation of slip effects on nanomaterial micropolar fluid flow over exponentially stretching. Alexandria Engineering Journal. 2020; 59, Issue 5, 3443-3450.
16. NN Reddy, VS Rao and BR Reddy. Chemical reaction impact on MHD natural convection flow through porous medium past an exponentially stretching sheet in presence of heat source/sink and viscous dissipation. Case Studies in Thermal Engineering. 2021; 25, 100879.
17. F Mabood, A Shafiq, T Hayat and S Abelman. Radiation effects on stagnation point flow with melting heat transfer and second order slip. Results in Physics. 2017; 7, 31-42.
18. MM Nandeppanavar, S Vaishali, MC Kemparaju and N Raveendra. Theoretical analysis of thermal characteristics of Casson nano fluid flow past an exponential stretching sheet in Darcy porous media. Case Studies in Thermal Engineering. 2020; 21, 1007
19. K. Bhattacharyya and G. C. Layek. Chemically reactive solute distribution in MHD boundary layer flow over a permeable stretching sheet with suction or blowing. Chemical Engineering Communications.2010; 12.
20. K. Bhattacharyya, S Mukhopadhyay and G. C. Layek. Unsteady MHD boundary layer flow with diffusion and first-orde chemical reaction over a permeable stretching sheet with suction or blowing. Chemical Engineering Communications.2013; 200, Issue 3,379 397.
21. K. Bhattacharyya, T. Hayat and A. Alsaedi. Exact solution for boundary layer flow of Casson fluid over a permeable stretching/shrinking sheet. ZAMM · Z. Angew. Math. Mech. 2014; 94, No. 6, 522 – 528.
22. IC Mandal and S Mukhopadhyay. Nonlinear convection in micropolar fluid flow past an exponentially stretching sheet in an exponentially moving stream with thermal radiation. Mechanics of advanced materials and structures. 2018; doi 10.1080/15376494.2018.1472325.
23. F Mabood and Das K. Melting heat transfer on hydromagnetic flow of a nanofluid over a stretching sheet with radiation and second order slip. Eur Phys J Plus, 2016; 131:3.
24. S Ghosh and S Mukhopadhyay. Flow and heat transfer of nanofluid over an exponentially shrinking porous sheet with heat and mass fluxes. Propuls. Power Res. 2018; 7, 268–275.
25. SK. Adegbie, OK Koriko, and IL Animasaun. Melting heat transfer effects on stagnation point flow of micropolar fluid with variable dynamic viscosity and thermal conductivity at constant vortex viscosity. Nigerian Math. Soc.2015; 35, 1, 34–47, doi:10.1016/j.jnnms.2015;06.004.
26. M Sheikholeslami and DD Ganji. Nanofluid convective heat transfer using semi analytical and numerical approaches, A review. J. Taiwan Inst. Chem. Eng. 2016; 65, 43–77.
27. K. Bhattacharyya and G. C. Layek. Magnetohydrodynamic Boundary Layer Flow of Nanofluid over an Exponentially Stretching Permeable Sheet. Physics Research International.2014; Article ID 592536, 12 pages.
28. I Ullah, Bhattacharyya, S shafie and I Khan. Unsteady MHD mixed convection slip flow of Casson fluid over nonlinearly stretching sheet embedded in a porous medium with chemical reaction, thermal radiation, heat generation/absorption and convective boundary conditions. PLOS ONE, 2016; https://doi.org/10.1371/journal.pone.0165348.
29. GS Seth, AK Singha, MS Mandal and A Banerjee, K MHD stagnation-point flow and heat transfer past a non-isothermal shrinking/stretching sheet in porous medium with heat sink or source effect. International Journal of Mechanical Sciences.2017; 134, 98-111.
30. K. Bhattacharyya, S Mukhopadhyay and G. C. Layek. Reactive solute transfer in magnetohydrodynamic boundary layer stagnation-point flow over a stretching sheet with suction/blowing. Chem. Eng. Comm.2012; 199, 368-383.
31. B Mandal, Bhattacharyya, A Banerjee. MHD mixed convection on an inclined stretching plate in Darcy porous medium with Soret effect and variable surface conditions. Nonlinear Engineering, 9, Issue-1.
32. A Zeeshan, N Shehzad and R Ellahi. Analysis of activation energy in Couette-Poiseuille flow of nanofluid in the presence of chemical reaction and convective boundary conditions, Results in 2018; 8,502–512.
33. S Xun, J Zhao, L Zheng and X Zhang. Bioconvection in rotating system immersed in nanofluid with temperature dependent viscosity and thermal conductivity. Int J Heat Mass Transfer. 2017; 111, 1001–6.