Numerical investigation of mhd casson nanofluid flow towards a non linear stretching sheet in presen

Abstract

Keywords: Non-Linear Stretching Sheet; Brownian Motion; Thermophoresis; Casson Fluid; MHD; Viscous Dissipation; Ohmic Dissipation

Introduction

In various fields of science and technology rapid progress has urged the researchers to extend their study towards the regime of boundary layer flow over a stretching sheet. The boundary layer flow behavior towards a linearly or nonlinearly stretching sheet plays a significant role for solving engineering problems and possess vast applications in manufacturing and production processes including metal spinning, rubber sheet manufacturing, production of glass fibers, wire drawing, extrusion of polymer sheets, petroleum industries, polymer processing etc. In these cases, the final product of desired characteristics depends on the rate of cooling in the process and the process of stretching. The dynamics of the boundary layer fluid flow over a stretching surface originated from the pioneering work of Crane [1] and he examined the incompressible steady boundary layer fluid flow caused by stretching sheet which moves in its own plane with linear velocity due to the uniform stress applications. This problem is particularly interesting as Crane [1] obtained the exact solution of 2D Navier-Stokes equations. Afterthat, Gupta and Gupta [2] extend Crane [1] work over different mathematical geometries. Yoon et al. [3] studied the theoretical and experimental results using Coulomb friction model by considering punch-sheet interface. Also, Sarma and Rao [4] examined the viscoelastic fluid flow by considering stretched sheet. In view of this, Vajravelu [5] studied flow and heat transfer in a viscous fluid over a nonlinear stretching sheet without using the impact of viscous dissipation. Cortell [6] examined heat and fluid flow transportation over a nonlinear stretching sheet for two different types of thermal boundary conditions, prescribed surface temperature (PST) and constant surface temperature (CST). The influence of heat transfer on the stagnation point flow of a third-order fluid over a shrinking surface has been studied by Nadeem et al. [7]. Recently, Prasad et al. This paper was recommended for publication in revised form by Regional Editor Muslum Arici 1 School of Basic and Applied Sciences, GD Goenka University, Gurgaon-122103, Haryana, India. 2 Department of Mathematics, KLP College, Rewari-123401, Haryana, India. *E-mail address: vini252011makkar@gmail.com, vikaspoply@gmail.com, rangoligoyal@gmail.com, naresh.sharma2006@gmail.com Orcid id: 0000-0001-6377-0377, 0000-0002-1573-3210, 0000-0002-6631-493X, 0000-0001-8398-927X Manuscript Received 04 May 2020, Accepted 17 July 2020

Journal of Thermal Engineering, Research Article, Vol. 7, No. 2, Special Issue 13, pp. 1-17, February, 2021 [8] examined the mixed convection heat transfer aspects with variable fluid flow properties over a non-linear stretching surface. Fluid heating and cooling are important in many industries such as power, manufacturing and transportation. Effective cooling techniques are greatly needed for cooling any sort of high energy device. Common heat transfer fluids such as water, ethylene glycol, and engine oil have limited heat transfer capabilities due to their low heat transfer properties. In contrast, metals have thermal conductivities up to three times higher than these fluids, so it is naturally desirable to combine the two substances to produce a heat transfer medium that behaves like a fluid, but has the thermal of a metal. Since last two decades, study of nanofluid has urged the researcher’s attention due to their heat transportation rate. Nanofluid comes in existence when we add a small quantity of nano-sized 10−9 − 10−7 particles to the base fluids. Low heat transportation fluids like fluorocarbons, glycol, deionized water, etc. have badly thermal conductivity and therefore deliberated necessary for heat transfer coefficient surrounded by heat transfer medium and surface. The nanoparticles are typically made up of metals (𝐴𝑙, 𝐶𝑢), nitrides (𝐴𝑙𝑁, 𝑆𝑖𝑁), carbides (𝑆𝑖𝑐), oxides (𝐴𝑙2 𝑂3 ), or nonmetals (carbon nanotubes, Graphite, etc.) and the base fluid (conductive fluid) is usually water or ethylene glycol. Also, it has been experimentally proved that rate of heat conduction of nanofluids is more than rate of heat conduction of the base fluids. The concept of nanofluid was initially proposed by Choi and Eastman [9] to indicate engineered colloids composed of nanoparticles dispersed in a base fluid. An MIT based comprehensive survey has been done by Buongiorno [10] for convective transportation in nanofluids by considering seven slip conditions that may produce a relative velocity within the base fluid and nanoparticles. Only two (Brownian motion and thermophoresis) out of these seven slip mechanisms were found to be important mechanisms. By adopting Buogiorno's model, Kuznetsov and Nield [11] explored the nanofluid boundary layer uniform convecting fluid flow. In recent years, MHD fluid flow has gained researchers attention due to its controllable heat transfer rate. Magnetohydrodynamics (MHD) effect also play and influential role in controlling the rate of cooling as well as segregation of molten metal’s from various non-metallic impurities. Magnetohydrodynamic (MHD) fluid flow has enormous utilization in manufacturing processes, even in the industrial areas as well. The terminology “Magnetohydrodynamic” is combination of three elementary terms magneto that stands for magnetic field, hydro that stands for fluid / liquid and dynamics that stands for evolution of particles. The existence of external magnetic field gives rise to Lorentz drag force which acts on the fluid, so potentially altering the characteristics of fluid flow especially velocity, temperature and concentration. Grouping of electromagnetism Maxwell’s equation and fluid mechanics Navier’s stokes equations therefore provides Magnetohydrodynamic (MHD) relation [12, 13]. Hayat et al. [14] studied the MHD fluid flow transportation over stretching surfaces. Later, the influence of viscous and Ohmic dissipation (i.e. joule heating) in nanofluid has been presented by Hussain et al. [15]. Vajravelu and Canon [16] studied the flow behavior of fluid towards a non-linear stretching sheet. Further, Matin et al. [17] analyzed the entropy effect in MHD nanofluid flow over stretching surface. Shawky et al. [18] studied the Williamson nanofluid flow in porous medium and he acknowledged that enhancement in non-Newtonian parameter escalates skin friction coefficient along with the rate of heat transfer. Basir et al. [19] examined the consequences of Peclet and Schmidt number in existence of partial slip towards a stretching surface. After that, rate of heat transfer along with partial slip condition was generalized by Pandey and Kumar [20]. Recently, Vinita and Poply [21] discussed MHD slip fluid flow of nanofluid in the existence of free stream velocity or outer velocity towards a stretching surface. Vinita et al. [22] studied MHD fluid flow with variable slip conditions over non-linear stretching surface. Furthermore, non-linearity effect towards the stretching surface under different physical circumstances has been examined by researchers in [6, 23-25]. Casson fluid is classified as a non-Newtonian fluid due to its rheological characteristics in relation to the shear stress–strain relationship. It behaves like an elastic solid at low shear strain and above a critical stress value; it behaves like a Newtonian fluid. A Casson fluid can best be described as a shear thinning liquid with infinite viscosity at zero shear rates, and zero viscosity at an infinite rate of shear. Some common examples of liquids that exhibit Casson fluid characteristics include tomato sauce, honey, soup, orange juice and human blood. In 1959, Casson fluid model was pioneered by Casson [26] for prognosis pigment-oil intermission flow. Furthermore, nanofluid boundary layer fluid flow over disparate geometries was premeditated by numerous researchers in [27-36]. Fluids behave as a solid, if the yield stress is more than applied shear stress. On the other hand, fluid behaves as a liquid for lesser yield stress. Kameshwaran et al. [37] investigated the dual solution in shrinking / stretching surface with Casson fluid flow to

Journal of Thermal Engineering, Research Article, Vol. 7, No. 2, Special Issue 13, pp. 1-17, February, 2021 determine Lorentz force effect on heat and flow transportation. Also, Khan et al. [38] studied consequence of viscous dissipation on MHD nanofluid flow past a stretching sheet. Afify [39] investigated influence of partial slip chemical reaction and viscous dissipation over Casson nanofluid by considering stretched sheet. Motivated by the above mentioned literature survey and widespread biological, industrial as well as engineering applications, it is of prime importance to explain the importance of mass and heat transportation of MHD flow past a non-linear stretching sheet with viscous and Ohmic dissipation effects by considering Navier velocity slip conditions. A precise review of already published research discloses that no such attempt has been made earlier even though the fruitfulness of thoughts and phenomenon explained in the present work can be expected to lead to the extremely productive interaction across disciplines. The mathematical difficulties and non-linearity appearing in the Casson nanofluid equations led us to use the numerical approach. The methodology adopted will be to solve the non-linear differential system through Runge-Kutta Fehlberg method. Behaviors of arising variables on concentration, temperature and velocity are graphically interpreted. The outcomes of current study reveal that non-linearity behavior of stretching sheet enhances the values of Sherwood number and skin friction coefficient while declination in value of Nusselt number is noticed. We give qualitative and quantitative comparisons with previously published work to show that our results have highly accurate. This study holds important industrial application, particularly in the field of extrusion where the fluid dispersed with particles is used to augment the strength and durability of the material.

Materials And Methods

In present analysis, 2-D incompressible fluid flow in MHD Casson nanofluid over non-linear stretching sheet has been considered. Non-linear behavior generates flow and sheet is stretched in both direction of x axis with stretching velocity 𝒖𝒘 = 𝒂𝒙𝑵 , where 𝒂, 𝒙 and 𝑵 denotes a constant, stretching surface coordinate and non-linear stretching parameter respectively. 𝑻𝒘 = 𝑻∞ + 𝑻𝟎 𝒙𝒎 at 𝒚 = 𝟎, where 𝑻𝟎 refers to the positive constant, 𝑻∞ refers to the ambient temperature attained and m refers to the physical parameter known as surface temperature parameter. Also, by introducing 𝒎 = 𝟎, we have a special case of constant surface temperature (CST). Figure 1 represents the physical model of the current study. The continuity, momentum, energy and concentration equations of the incompressible Casson nanofluid boundary layer flow are as follows (see Buongiorno [10], Mustafa and Khan [40], Usman et al. [41], Gireesha et al. [42], Seth et al. [43]) 𝜕𝑢 𝜕𝑢 + =0 (1) 𝜕𝑥 𝜕𝑦 𝜕𝑢 𝜕𝑢 1 𝜕 2 𝑢 𝜎𝐵2 +𝑣 = 𝜈 (1 + ) 2 − 𝑢 𝜕𝑥 𝜕𝑦 𝛽 𝜕𝑦 𝜌

Here horizontal and vertical velocities are represented by 𝑢 and 𝑣, respectively. Also 𝜈 denotes kinematic (𝜌𝑐)

viscosity, 𝜌𝑝 is the density of particle, 𝛼𝑚 = (𝜌 𝑚 is the thermal diffusivity, 𝑎 is a positive constant, 𝜏 = (𝜌𝑐)𝑝 defines 𝑐)𝑓

a proportion of heat capacities, 𝛽 denotes Casson fluid parameter, 𝐷𝑇 reflects thermophoretic diffusion coefficient, 𝐵

Journal of Thermal Engineering, Research Article, Vol. 7, No. 2, Special Issue 13, pp. 1-17, February, 2021 is the magnetic field intensity, 𝐷𝐵 denotes Brownian diffusion coefficient, 𝜎 represents electrical conductivity and 𝑢𝑠 refers to slip velocity that is considered to be proportional to wall stress described as: 𝜕𝑢 𝑢𝑠 = 𝑙 ( ) 𝜕𝑦 𝑦=0

here, 𝑙 refers to the slip length which is proportionality constant for slip velocity. The fundamental equations (1) to (4) with boundary conditions (5) to (6) are transformed using similarity variables [25]

Figure 1: Physical model and coordinate system Inserting equation (8) into equations (2) to (4), the governing equations (1)-(4) takes the form 1 2 2𝑁 2 (1 + ) 𝑓𝛿′′′ + 𝑓𝛿 𝑓𝛿′′ − ( ) 𝑀𝑓𝛿′ − ( ) 𝑓′ = 0 𝛽 𝑁+1 𝑁+1 𝛿

1 𝑁𝑡 ′′ 𝜙𝛿′′ + 𝑆𝑐𝑓𝛿 𝜙𝛿′ + 𝜃 =0 2 𝑁𝑏 𝛿 The relevant boundary conditions are reduced to

Journal of Thermal Engineering, Research Article, Vol. 7, No. 2, Special Issue 13, pp. 1-17, February, 2021 𝑓𝛿 (𝜉) = 0, 𝑓𝛿′ (𝜉) = 1 + 𝑑𝑓𝛿′′ (𝜉), 𝜃𝛿 (𝜉) = 1 𝑎𝑛𝑑 𝜙𝛿 (𝜉) = 1 𝑎𝑡 𝜉 = 0 𝑓𝛿′ (𝜉) → 0, 𝜙𝛿 (𝜉) → 0 𝑎𝑛𝑑 𝜃𝛿 (𝜉) → 0 𝑎𝑠 𝜉 → ∞

where prime denotes derivative with respect to 𝜉 and the key crucial parameters are defined by: 𝑀=

Here 𝑀 is the magnetic parameter, 𝑁𝑡 is the thermophoresis parameter, 𝑆𝑐 is the Schmidt number, 𝑃𝑟 is Prandtl number, 𝑑 is the velocity slip parameter, 𝐸𝑐 is the Eckert number and 𝑁𝑏 is the Brownian motion parameter. It is notes significantly that this boundary value problem condenses into the classical problem of mass, heat and flow transportation because of the stretching surface in a viscous fluid when 𝑁 = 1 and in equations (10) and (11); 𝑁𝑏, 𝑁𝑡 are zero (The boundary value problem is of no physical significance and condenses into the classical problem for becomes ill-posed). It is worth mentioning that equation (9) with the boundary conditions (12), (13); where 𝑁 = 0, are the classical Blasius flat-plate flow problem and a comprehensive numerical analysis of that problem has been conceded by the author of this work. For the linear stretching problem (i.e., when 𝑁 = 1), the exact solution for 𝑓𝛿 is 𝑓𝛿 (𝜉) = 1 − 𝑒 −𝜉 , which was first obtained by the great scientist Crane [1] and this exact solution is unique, while for the non-linear stretching problem (i.e., when 𝑁 ≠ 1), there is no exact solution. Also, the physical quantities of interest skin friction coefficient, local Nusselt number and local Sherwood number are respectively defined as: 𝐶𝑓𝑥 =

where 𝜏𝑤 , 𝑞𝑤 and 𝑞𝑚 are wall shear stress, local heat flux and local mass flux at the stretching surface serially given as: 3𝑁−1 𝑎(𝑁 + 1) 1 2 √ (1 + ) 𝑓𝛿′′ (0)

The system of partial differential equations (9)-(11) are highly non-linear, hence cannot be solved by using analytical methods. The variational Runge Kutta Fehlberg method has been implemented.

Numerical Procedure

The system of differential equations (9)-(11) are not solved analytically due to extremely non-linear behaviour. To solve this system of non-linear differential equations (9)-(11) together with boundary conditions (12)-(13), we adopt Runge Kutta Fehlberg approach (numerical procedure) and solution is investigated against the variation in Eckert

Journal of Thermal Engineering, Research Article, Vol. 7, No. 2, Special Issue 13, pp. 1-17, February, 2021 number 𝐸𝑐, Casson fluid parameter 𝛽, velocity slip parameter 𝑑, thermophoresis 𝑁𝑡, Brownian motion parameter 𝑁𝑏, Schmidt number 𝑆𝑐, non-linear stretching parameter 𝑁 and magnetic parameter 𝑀. The value of horizontal axis - 𝜉 is chosen in such a way that a velocity, temperature and concentration profiles asymptotically tends to the boundary condition. All the simulations are conceded with 𝜉𝑚𝑎𝑥 = 10. However, to illustrate the characterization of curve efficiently, much lower values of 𝜉 are used. The major advantage of shooting method over many numerical techniques is that it has fifth order truncation error. Also the process of solution computation is easier as compared to other numerical methods or techniques. For this purpose, the coupled differential equations (9)-(11) are rehabilitated into a set of seven initial value problems by launching the new set of dependent variables 𝑠𝑖 (i=1 to 7) as:

𝑁𝑡 2𝑚 1 1 𝑁𝑡 2 𝑃𝑟 [𝑠1 𝑠5 − ( ) 𝑠2 𝑠4 + 𝑁𝑏𝑠5 𝑠7 + 𝑁𝑡𝑠52 + (1 + ) 𝐸𝑐 𝑠32 ] − 𝑠1 𝑠7 𝑆𝑐 + Pr ( ) 𝑀𝐸𝑐𝑠22 𝑁𝑏 𝑁+1 𝛽 2 𝑁𝑏 𝑁+1

The subjected boundary conditions (12) and (13) are reduced to 𝑠1 = 0, 𝑠2 = 1 + 𝑑 𝑠3 , 𝑠6 = 1, 𝑠4 = 1 𝑎𝑡 𝜉 = 0 𝑎𝑛𝑑 𝑠2 = 𝜆, 𝑠6 = 0, 𝑠4 = 0 𝑎𝑠 𝜉 → ∞

To solve the above system of differential equations along with the boundary conditions, shooting technique is implemented which consists of the following steps [15]:

1. Choose the appropriate value for the limit 𝜉∞ .

2. Next important step is to select good initial approximations for 𝑠3 (0), 𝑠5 (0) and 𝑠7 (0). 3. Therefore the above system of differential equations (19)-(25) is converted into the initial value problem and then it is solved with the help of Runge-Kutta-Fehlberg method. 4. The computed solution will converge if absolute difference of given and computed values of 𝑠2 (∞), 𝑠4 (∞) and 𝑠6 (∞) is absolutely less than tolerance error i.e. 10−6 . 5. If these differences are greater than tolerance error, then guessed values of 𝑠3 (0), 𝑠5 (0) and 𝑠7 (0) are modified with the help of Newton's method. This procedure is repeated until the computed solution meets the convergence criteria as shown in Figure 2.

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Figure 2: Flow chart of numerical technique Validation of Numerical Scheme In current study, we use RKF method for finding the final outcomes and the adopting numerical process (i.e. shooting technique) is valid. We take constant vale of nanoparticle concentration and temperature at the boundary. For the purpose of verification of the code, we compare our results for local Nusselt number with those of Rana and Bhargava [25] and Cortell [6] for different values of Prandtl number 𝑃𝑟 and non-linear stretching parameter 𝑁 by taking the values of Brownian motion parameter 𝑁𝑏 and thermophoresis parameter 𝑁𝑡 as zero as shown via Table 1. A very good agreement in the results can be seen which strengthens the reliability of the present work (see Figures 3 and 4). Table 1: Comparison of results for local Nusselt number −𝜃𝛿′ (0) with 𝑁𝑡 = 𝑁𝑏 = 0 and different values of Prandtl number 𝑃𝑟 and non-linear stretching parameter 𝑁 𝑃𝑟

Rana and Bhargava [25] 0.6101 0.5955 0.5756 0.5660 1.5683 1.5512 1.5269 1.5144

Cortell [6] 0.6102 0.5952 0.5745 0.5644 1.6071 1.5867 1.5574 1.5423

Present results 0.6102 0.5952 0.5747 0.5647 1.6077 1.5867 1.5576 1.5431

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Figure 3: Bar graph showing comparison of local Nusselt number for non-linear stretching sheet when 𝑃𝑟 = 1.0

Figure 4: Bar graph showing comparison of local Nusselt number for non-linear stretching sheet when 𝑃𝑟 = 5.0

Results And Discussion

Present study finds numerical solution of differential equations (9) to (11) subjected to the boundary conditions (12) to (13) that are computed using RKF method by applying shooting technique. The main reason behind to solve the present problem are to determine the impact of prominent fluid parameters namely Eckert number 𝐸𝑐, Casson fluid parameter 𝛽, velocity slip parameter 𝑑, thermophoresis 𝑁𝑡, Brownian motion parameter 𝑁𝑏, Schmidt number 𝑆𝑐, non-linear stretching parameter 𝑁 and magnetic parameter 𝑀 on 𝑓𝛿′ (0), 𝜃𝛿 (0) and 𝜙𝛿 (0). Physically, the values of prominent physical parameters are taken as 𝛽 = 0.4, 𝑑 = 0.1, 𝑁𝑏 = 0.3, 𝐸𝑐 = 0.4, 𝑁𝑡 = 0.3, 𝑆𝑐 = 1.5, 𝑁 = 𝑀 = 1.0, 𝑚 = 1.0 and 𝑃𝑟 = 5.0. Table 2 demonstrate the impact of fluid ′′ ′ parameters 𝛽, 𝑑 𝑁 𝑏, 𝑁 𝑡, 𝐸𝑐, 𝑆𝑐, 𝑀 and 𝑁 on skin friction coefficient 𝑓𝛿 (0), local Nusselt number −𝜃𝛿 (0) and local Sherwood number −𝜙𝛿′ (0) by taking fixed entries of fluid parameters Prandtl number, 𝑃𝑟 as 5.0 and surface temperature parameter 𝑚 as 1.0 and an excellent convergence has been achieved in current problem.

Figure 6: Impact of velocity slip parameter 𝑑 on velocity profile 𝑓𝛿′ (𝜉) for 𝑚 = 1.0, 𝑃𝑟 = 5.0, 𝛽 = 0.4, 𝑁𝑡 = 𝑁𝑏 = 0.3, 𝐸𝑐 = 0.1, 𝑆𝑐 = 1.5, 𝑁 = 𝑀 = 1.0

Figure 5: Impact of Casson fluid parameter 𝛽 on velocity profile 𝑓𝛿′ (𝜉) for 𝑚 = 1.0, 𝑃𝑟 = 5.0, 𝑑 = 0.1, 𝑁𝑏 = 0.3, 𝑁𝑡 = 0.3, 𝐸𝑐 = 0.1, 𝑆𝑐 = 1.5, 𝑀 =

Conclusion

Present study reflects the heat, mass and flow transportation of Magnetohydrodynamic (MHD) Casson nanofluid towards a sheet which is stretched non-linearly. Final outcomes of the current study incorporates the significance of Ohmic and Viscous dissipation effects in addition to the MHD Casson fluid flow velocity profile, temperature distribution and nanoparticle volume fraction. Similarity conversion is used for transforming partial into ordinary differential equations. Key findings of current analysis are summarized as: 1. Skin friction coefficient elevates with increment in magnetic parameter 𝑀 due to produced Lorentz force that ultimately improves local Sherwood number along with Nusselt number for higher magnetic parameter 𝑀. 2. A declination in the value of Velocity profile is observed with variation in the values of Casson fluid parameter 𝛽, velocity slip parameter 𝑑 whereas rises for larger values of non-linear stretching parameter 𝑁. 3. Fluid temperature enhances for greater values of physical parameters Eckert number 𝐸𝑐, Brownian motion parameter 𝑁𝑏 and thermophoresis parameter 𝑁𝑡. On the other hand, it reduces for higher values of Casson fluid parameter 𝛽 and velocity slip parameter 𝑑. 4. An enhancement in the profile of nanoparticle concentration is noticed for greater values of thermophoresis parameter 𝑁𝑡, velocity slip parameter 𝑑, Casson fluid parameter 𝛽. Whereas, it declines for Brownian motion parameter 𝑁𝑏 and Schmidt number 𝑆𝑐. It is hoped that current analysis serves impetus for the scientists to develop technological applications over stretching surfaces. Results of the present study helps to control heat transfer rate in different manufacturing processes, biological systems to produce the quality product this analysis can be extended to shrinking surface as well.

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Nomenclature

Cartesian coordinates Magnetic field intensity Stream velocity Positive constant Surface temperature parameter Schmidt number Concentration Nanoparticle volume fraction Mass flux Prandtl number Nusselt number Thermophoresis parameter Ambient temperature attained Sheerwood number Vertical velocity Brownian diffusion coefficient Brownian motion parameter Non linear stretching parameter Temperature Stretching velocity Thermophoresis diffusion coefficient Slip velocity Slip length Temperature at the sheet Hass flux Ambient nanoparticle volume fraction Horizontal velocity Magnetic parameter

Kinematic viscosity Casson fluid parameter Electrical conductivity Similarity variable Density of base fluid Thermal diffusivity Ratio of heat capacities Non-dimensional temperature Non-dimensional nanoparticle concentration

Acknowledgement

First author (Vinita) would like to thank Mr. Madan Lal Makkar and Mrs. Neelam Makkar for their love, support and encouragement. Additionally, all authors appreciate the constructive and valuable comments of the reviewers, which improved the manuscript.

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