Abstract

This paper examined the three-dimensional steady thin film flow of tangent hyperbolic fluid with nonlinear mixed convection flow and entropy generation past a stretching surface under the influence of magnetic field. For the flow problem, the Cattaneo–Christov heat and mass diffusion model was employed to examine heat and mass transfer characteristics and impacts of the normally directed magnetic field. To transform nonlinear PDEs into ODEs, the variable transformation technique was used. The bvp4c algorithm was implemented to solve these ODEs. The behavior of every leading parameter on the velocities, temperature, concentration profile, entropy generation, and Bejan number was reported with tabular and figurative form. The results show that as the values of Br increase, the entropy generation enhances, but the Bejan number decreases. Moreover, as the values of B increase, the opposite characteristics are observed in entropy generation and Bejan number graphs. Furthermore, the skin friction coefficient number, local Nusselt number, and Sherwood number are graphically discussed for the active involved parameters. The best agreement is recorded when we compare this paper with the previous literature for various values of .

1. Introduction

In physical sciences, one of the essential factors of flow of fluid which has very common applications is entropy generation. In different fields of sciences and technologies, some researchers have done a lot of work on it. In case of thin film fluid flow, very little work is available in the literature. The entropy productions due to transmission of convective heating were reported by Bejan [1, 2] for the first time. Later on, Lin et al. [3] examined the compressible flow through a high pressure turbine for local entropy generation due to delayed detached eddy simulation. Further, Khan et al. [4] studied entropy generation of rotating disks due to bioconvection flow of nanofluid. Opanuga et al. [5] have also carried out a study on entropy generation analysis with the effect of ion-slip and hall current of couple stress fluid. Moreover, Bibi and Xu [6] have discussed entropy generation analysis of Jeffrey nanofluid under magnetic field effect on horizontal channel.

The study of entropy generation analysis nowadays is an active research area. Accordingly, Ji et al. [7] have presented performance evaluation of entropy generation due to forced convective heat transfer. Shit et al. [8] also investigated entropy generation performance under the effect of thermal radiation and unsteady flow over a flat plate. Gireesha et al. [9] have presented a study on entropy generation with inclined porous microchannel for Casson fluid flow with viscous and Joule heating. Muhammad and Sohail [10] have computed entropy generation under the impact of radiative heat transfer on rotating disks with Ree–Eyring fluid. Leonid and Martyushev [11] have analyzed the misconception about the production of entropies. A report on nonequilibrium thermodynamics with productions of entropy has been presented by Velasco et al. [12]. Gallavotti [13] has computed stationary states of nonequilibrium in a point of view. Additionally, Kurnia et al. [14] have studied the cooling of microchannel with heat transfer performance on entropy production, and Şahin [15] has numerically explored entropy generation in a square cavity filled with boron-water nanofluid. Moreover, Soltanipour et al. [16] have numerically examined ferrofluid forced convection with magnetic field and entropy generation in a curved pipe. Additionally, Riaz et al. [17] have investigated the features of entropy generation and applied magnetic field in an annular part of two micro nonconcentric pipes on Jeffrey fluid with the inner one being of rigid nature traveling for uniform speed, and Riaz et al. [18] have discussed the entropy generation and Williamson fluid on the asymmetric peristaltic propulsion of non-Newtonian fluid with convective boundary conditions.

The attention of most recent researchers is focused on the study of flow and heat transfer in a thin liquid film on a stretching surface with its numerous applications in industrial and engineering fields. In manufacturing processes of aerodynamic extrusion of plastic sheets, foodstuff processing, fibre coating, reactor fluidization, annealing, and thinning of copper wires in relation to fluid flow and heat transfer within a thin liquid film are very important. It is essential to maintain surface quality of the extubate in the process of extrusion of materials. The best appearance with strength, good translucency, and low impeding friction for producing smooth glossy surfaces for all coating processes is necessary for a thin film. The behavior of flow and heat transfer in thin film is determined by the quality of the end product in the extrusion process of the fluid flow and heat transfer characteristics on stretching sheet. Gul et al. [19] have analyzed thin film flow and entropy generation over a stretching sheet with variable fluid properties. Sarojamma et al. [20] and Shah et al. [21] have discussed the influence of magnetic field on thin film flow and the generation of entropy for unsteady flow of Newtonian and non-Newtonian fluid over stretching sheet.

The attention of scientists and engineers in recent years is attracted by the magnetohydrodynamic (MHD) phenomenon owing to its realistic application in MHD electric transformers, accelerators, and cooling of a metallic plate. Natural convection flow with electrically conducting fluid past different geometries with different fluids has been studied widely in most of the literature. Despite the fact that there are numerous studies on natural convection flow of electrically conducting fluid, there are just a few studies regarding natural convection flow of an electrically conducting tangent hyperbolic fluid. Consistent magnetic environment with steered square geometry under mixed convection thermal transfer and micropolar liquid material is discussed by Siddiqui et al. [22], and Jha et al. [23] have investigated magnetohydrodynamic flow in a vertical microchannel with hall effect under natural convection flow. Moreover, Khatera et al. [24] presented the impact of magnetic field diffusion using cylindrical coordinates system of Fokker–Planck equation which relates potential symmetry and invariant solution. Furthermore, Pal and Talukdar [25] have solved the problem of rotating porous channel with the effect of magnetic field, convective heat and mass transfers, hall current, and thermal radiation. Abiodun et al. [26] have checked MHD micropolar fluid with effect of hall current and analysis of entropy production. Liao et al. [27] have reported a numerical study on magnetohydrodynamics with planetary spheres. Basant et al. [28] have computed flow of magnetohydrodynamics with ion-slip current at vertical microchannel. Barik et al. [29], Yeh et al. [30], Usman et al. [31], Singh and Bhardwaj [32], and Kumar et al. [33] have numerically computed the effect of chemical reaction, free convection, and magnetic field on heat and mass transfer phenomenon past different geometry with different initial and boundary conditions for different kinds of fluids.

Mixed convection is the combined influence of both free and forced convection. Most technical applications of it have been discovered by some of the following researchers. Accordingly, Shayma and Litan [34] have analyzed incompressible fluid past a wedge with boundary layer flow mixed convection transient under magnetic field effects. Rashed et al. [35] have also investigated temperature distribution of a vertical porous annulus with mixed convection opposing flow. Arani et al. [36] and Mohamed et al. [37] also presented mixed convection, optimal distribution of heat transfer, and viscous dissipation effect of discrete heat sources. Alsabery et al. [38] have checked mixed convection flow and entropy generation for a heat source and rotating solid cylinder inside a wavy-walled enclosure. Wubshet and Temesgen [39] have reported Eyring–Powell nanofluid with ion-slip and hall effects under stretching sheet. Hashmi et al. [40] have computed mixed convection flow with homogeneous heterogeneous reactions of magnetized Maxwell fluid. Sajida et al. [41] have studied viscoelastic fluid under mixed convection flow of permeable parallel vertical plates. Basant et al. [42] have examined temperature dependent of vertical channel under viscosity and flow reversal of mixed convection flow. Ahmad et al. [43] have introduced Jeffrey fluid on mixed convection flow for magnetohydrodynamic effects with exponentially stretching surface. Khan et al. [44] have carried out a theoretical analysis for mixed convection flow of Maxwell fluid between two infinite isothermal stretching disks. Basant et al. [45] have analyzed mixed convection flow on hall and ion-slip effects with wall heating at vertical microchannel. Additionally, Jawali et al. [46] have investigated mixed convection flow with Soret effects under condition of Robin boundary conditions. Wubshet and Gosa [47] have presented inclined stretching cylinder with mixed convective flow under Cattaneo–Christov heat flux theory. Ashraf et al. [48] have solved three-dimensional mixed convection radiative flow in presence of convective condition and thermophoresis stretching sheet. Shah and Dennis [49] have solved mixed convection flow of vertical plate under oscillating of heat transfer analysis. Moreover, Khan et al. [50] have checked nonlinear mixed convection flow of tangent hyperbolic fluid and entropy generation optimization under activation energy stretching sheet. In many engineering and industrial processes, such as food mixing, multigrade oils, plasma, and composite materials, non-Newtonian fluids are found. For those applications, many researchers concentrated on non-Newtonian fluids. Hyperbolic tangent fluid is considered the best one of branches of the non-Newtonian fluid which is capable of describing the thinning shear effects. Recently, Kumar et al. [51] have reported the dusty flow of hyperbolic tangent fluid with magnetic field and thermal radiation over stretching sheet. Naseer et al. [52] have computed the hyperbolic tangent fluid with boundary layer flow of cylindrical stretching sheet. Khan et al. [53] have introduced double stratification tangent hyperbolic fluid flow under radiation and chemical reaction. Shahzad et al. [54] have developed MHD tangent hyperbolic fluid with Joule heating, viscous dissipation, and chemical reaction. Naseer et al. [55] have analyzed hyperbolic tangent fluid of cylinder under vertical stretching surface. Hayat et al. [56] have investigated Soret–Dufour effects on hyperbolic tangent fluid flow at stagnation point. Kumar et al. [57], Rehman et al. [58], and Hayat et al. [59] have studied tangent hyperbolic fluid under the effects of unsteady flow, variable thermal conductivity, Joule heating, and thermal radiation. Nadeem et al. [60] also reported magnetohydrodynamic peristaltic flow in a vertical asymmetric channel with heat transfer. Additionally, Akbar et al. [61] and Hussain et al. [62] have examined the effects of convective heating and nonlinear stretching parameters on the MHD boundary layer flow analysis of tangent hyperbolic fluid. Furthermore, behavior of incompressible magnetohydrodynamic flow tangent hyperbolic fluid with the effect of melting heat transfer in the stagnation point has been reported by Hayat et al. [63], and the analysis of tangent hyperbolic material in the presence of both nanoparticles past an exponential surface with the heat, mass, and motile microorganisms transfer rate in the magnetohydrodynamic convective flow has been performed by Shafiq et al. [64]. Moreover, Alaidrous et al. [65] and Ibrahim and Gizewu [66, 67] have studied tangent hyperbolic nanofluid past bidirectional extending surface under the application of some physical conditions such as thermal radiation, Joule heating, activation energy, and chemical reaction of high order by employing Cattaneo–Christov heat flux model. To our knowledge, no one has attempted to investigate the effect of three-dimensional thin film flow of tangent hyperbolic fluid with nonlinear mixed convection flow and entropy generation past stretching surface.

2. Mathematical Formulation

This paper deals with three-dimensional flow of hyperbolic tangent fluid thin film. The fluid satisfies the condition of incompressible, steady, laminar, and viscous flow due to entropy generation and mixed convection with applied magnetic field in the perpendicular direction. For the flow formulation, Cattaneo–Christov heat and mass diffusion model is employed for the examination of heat and mass transfer characteristics. and are established as z is normal to the surface in the Cartesian coordinate system. Magnitudes of velocities at are and where represents the contraction of the sheet and represents the stretching of the sheet. In this article, the description of temperature field and the concentration distribution are defined as and , where , , , and are demonstrated by the temperature at the surface of a sheet, the nanofluid concentration at the surface of a sheet, the constant reference temperature, and the constant reference concentration, respectively, as shown in Figure 1.

Figure 1 

Physical model of the problem.

Constitutive equation of non-Newtonian fluid which is assumed in this article is presented by [66] as follows:where is defined asfor

The equation is assumed when and ; by using this idea, (1) becomeswhere , , , , , , and are extra stress tensor, zero shear rate viscosity, infinity shear rate viscosity, time constant, dimensionless power-law index, strain rate tensor of second invariant, and shear-thinning characteristics which represent tangent hyperbolic fluid being considered, respectively.

Governing equations of this problem are defined as follows:

The equations of Cattaneo–Christov heat and mass model flux as used by [67] are

, ; ; ; ; ; and are the relaxation time parameter for the heat flux and mass flux, the heat flux, the temperature of the fluid, mass flux, concentration of the fluid, and thermal conductivity of the fluid, respectively.where and are defined as

The boundary condition announcement by [19] in case of the present flow model is given as follows:

Similarity transformations used by [19] are given as

The flow equations as a result of conservation principles such as momentums, energy, and concentration are the following equations, respectively.

Thickness of film of a flowing liquid in nondimensional form is by considering the unknown parameter to determine the boundary of the problem according to the next equation:where the governing parameters are

The skin friction coefficient , the local Nusselt number , and the local Sherwood number are physical quantities as used by [67], defined as follows:when

2.1. Entropy Generation Analysis

Thermal irreversibility of the system is measured by systems of entropy generation. Mathematically, the present model of the entropy generation rate per unit volume is given in [19]. This model is used to reduce entropy generation of physical parameters in any scientific and engineering applications by controlling the output.where representation of entropy generation due to the first term is thermal irreversibility, the second term is nanoparticle interdependence with base fluid, the third term is local mass transfer of the nanoparticle volume fraction, the fourth term is the viscous dissipation, and the fifth term is the magnetic field, respectively, in the above equation.

Dimensionless form of the entropy generation can be expressed as follows by using the similarity transformation.where the representations of temperature difference variable, concentration difference parameter, Brinkman number, Diffusion parameter, and entropy generation are given as , , , , and , respectively.

Bejan number for locality of quantifying the irreversibility of heat transfer process is now appropriate to be determined. Mathematically, the Bejan number in the boundary layer is defined by [32] according to the following equation:

3. Numerical Method

The bvp4c method is used to solve the converted ODEs of the problem, considering , , , , , , , , , and . The given matrix form isfor

The vector form of boundary conditions is given as follows:

The vector form of boundary conditions is given as follows:

Numerical integration of the bvp4c Matlab solver is used to solve (22).

4. Results and Discussion

The characteristics of velocity against are illustrated by the graph in Figure 2. Physically, an enlargement in the values of reduces the resistance to flow for the case of small viscosity shear-thinning fluid by enhancing the fluid velocity. By this fact and as can be seen from Figure 2, the values of extremely enlarge as the velocity of nanofluid is deceased. Figure 3 is plotted to demonstrate the behaviors of on the velocity of the fluid. As the value of is upgraded from 0.2 to 0.4, the nanofluid velocity is enhanced. Figures 4 and 5 are plotted to investigate the characteristic of mixed convection (or thermal buoyancy) parameter and nonlinear mixed convection due to temperature on velocity profile along x-axis, respectively. As we seen from the graphs, the influences of these parameters are similar.

Figure 2 

Characteristics of on velocity profile along x-direction.