# Entropy Generation and Regression Analysis of Magnetohydrodynamic Stagnation Point Flow of a Casson Fluid with Radiative and Dissipative Heat Transfer and Hall Effects

Authors: Prashu; Raj Nandkeolyar; Premful Kumar; Vivek Kumar Sinha; Vivek Sangwan
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

The phenomenon of heat transfer is prevalent in industries and has an extensive range of applications. However, mostly the discussion of heat transfer problems is limited to the study of the first law of thermodynamics, which deals with energy conservation. It is just restricted to the quantity of energy, not to its quality; i.e., there is no difference between the work (high-grade energy) and the heat (low-grade energy). A measurement of the degree of randomness of energy in a system is known as entropy. It is unavailable for doing useful work because work takes place only from ordered molecular motion. Even though many boundary layer models exist in the literature to investigate the flow and heat transfer of various fluids along a stretching surface, they have not yet been used at their maximum ability. The main motive of the current research is to discuss entropy generation or its minimization during heat transfer. This work presents an entropy generation analysis for the transient three-dimensional stagnation point flow of a hydromagnetic Casson fluid flowing over a stretching surface in the existence of Hall current, viscous dissipation, and nonlinear radiation. The physical configuration of the present work is described in terms of partial differential equations (PDEs) of nonlinear nature. Furthermore, these PDEs are converted into ordinary differential equations by using some relevant similarity transformations. An efficient numerical method named as the spectral quasilinearization method (SQLM) is used to solve this model. The expression of the Bejan number and volumetric entropy generation rate is also computed. A parametric analysis, including the essential physical parameters, is performed to examine the influences of distinct flow parameters on the velocity profile, temperature profile, Bejan number, entropy generation number, and the coefficients of skin friction and the Nusselt number. In order to further insight into the emerging physical quantities of engineering interest, multiple quadratic regression models are used to estimate the coefficients of skin friction and heat transfer.


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## Body

## 1. Introduction

The boundary layer concept derived its name from the great German professor Ludwig Prandtl at the beginning of the 1900s. Even though general fluid flow equations existed for many years, the solutions of these equations could not describe the effects of the flow field properly. To overcome this problem, Prandtl was the first who suggests dividing the whole fluid flow into two parts, namely boundary layer region and outside the boundary layer region. He observed a change in values of inertia and viscous force from an area that is very close to the surface to an area far from the surface. His excellent work was published in 1904 at Heidelberg, entitled “On the motion of a fluid of very small viscosity.” Thus, with the help of the boundary layer theory, the effects of the flow field can be analyzed properly. Then, inspired by Prandtl’s theory, numerous researchers made different inventions on it with new aspects. It was the Sakiadis [1] who initially examined the boundary layer theory of fluid flow towards a moving surface with uniform speed. He analyzed the behavior of both laminar and irregular flow in the boundary layer region. In recent decades, the theory of boundary layer flow of fluid along a sheet was extended by many researchers by considering distinct aspects and conditions such as stretching surface and shrinking surface. In the aspect of stretching surface, Crane [2] was the first who extended Sakiadis’s theory and introduced the boundary layer 2-D fluid flow towards a linearly stretching surface. The boundary layer flow of viscous, incompressible fluid towards a stretching surface is popular among the numerous researchers ascribable to its extensive applications in the industrial and technology fields such as metallurgical, petrochemical, and material processing. Erickson et al. [3] discussed the boundary layer flow of viscous incompressible fluid for heat and mass transfer along a plate with suction and blowing parameters. Dutta et al. [4] did an analysis of the temperature field for two-dimensional boundary layer fluid flow toward a stretching surface with uniform heat flux. Patil et al. [5] formulated a 2-D mixed convection flow model towards the semiinfinite power-law stretching surface and solved it numerically using a finite difference approach. Maleki et al. [6] analyzed the influence of heat source/sink parameters for the boundary layer two-dimensional flow of nanofluid along a moving plate under the consideration of viscous dissipation. Some literature regarding boundary layer flow can be seen through the references [7–14].Fluid has two crucial properties, stretching rate and cooling rate; both play a crucial role in a better outcome. During the manufacturing process, rapid changes in these two properties of fluid diminish the quality of the product. To control these fluid properties and to avoid solidification, some external magnetic field is required. Therefore, the demand for modern technology has raised the interest in magnetohydrodynamics (MHD). The phenomenon of MHD was originated at the beginning of the twentieth century. At present, the existence of MHD is possible due to the pioneering work of various researchers. The fundamentals of MHD were derived by Hartman [15]. After 1950, this phenomenon evolved expeditiously and attracted numerous researchers due to its various real-life applications. The famous scientist Alfven [16] contributed greatly to developing the concept of magnetohydrodynamics in the area of plasma physics. For this, he was awarded the Nobel Prize in 1970. Initially, the flow of hydromagnetic fluid towards a stretching surface was examined by Pavlov [17]. Kumaran et al. [18] investigated the flow model of hydromagnetic fluid along a quadratically stretching surface and solved the governing model for the exact solution. Mahanthesh et al. [19] analyzed the 3-D MHD nanofluid flow towards a bidirectional nonlinear stretching surface in the existence of radiation and dissipation effects. He obtained numerical solutions by using an efficient technique, the shooting method. The combined impacts of heat source/sink and thermal radiation effects on 3-D MHD Jeffrey fluid flow, flowing over a stretching surface, were investigated by Gireesha et al. [20].While studying the flow of hydromagnetic fluid, the concept of the Hall effect takes an important place as it arises due to the consideration of some external magnetic field. It was the great American physicist Edwin Hall [21] who introduced the concept of the Hall effect in 1879. Even though the concept of Hall current was introduced in 1879, it took several years to get established properly. The principle of the Hall effect describes the charging behavior when it is subject to electric and magnetic fields and is widely used in modern technology equipment such as magnetometers, wheel speed sensors, positional sensors, switches, and MHD power generators. Also, it is used to cool nuclear reactors, large metallic plates, and MHD pumps in the industry. Thus, these applications have intensified the interest of researchers in the study of Hall current. Gupta [22] discussed this phenomenon on the flow of viscous electrically conducting incompressible fluid flowing over a permeable surface in the consideration of a magnetic field effect. He concluded that for the small magnetic Reynolds number, under the influence of Hall current, the pattern of the flowing fluid is just similar to the nonconducting fluid flow, flowing over a sheet in a rotating frame. The combined effect of Hall current and heat source/sink on laminar hydromagnetic fluid flow is discussed by Saleem and Aziz [23]. Some relevant studies regarding the Hall effect can be seen through the references [24–28].All the studies mentioned above are time-independent (steady state), but to examine the flow configuration more realistically, the study of time-dependent (unsteady state) fluid flow is necessary. All the flows which vary at a particular point with time come under the category of unsteady flows. Initially, the boundary layer theory was established only for time-independent laminar flows. Later, it was realized that, all the fluid flow problems, whether man-made or natural, depend on time.. Wang [29] used the particular type of similarity transformation to change a set of PDEs into the set of ODEs. Elbashbeshy and Bazid [30] examined the transient flow of the hydromagnetic fluid and concluded that there is a rise in the thickness of momentum and thermal boundary layer with an increment in the unsteadiness parameter. The discussion on the boundary layer problem of unsteady fluid flow flowing along a vertical stretching surface was done by Ishak et al. [31]. Aziz [32] formulated a boundary value model to analyze the transient flow of the fluid and solved it for similarity solutions. Pal [33] examined the Hall and radiation effects on the transient flow of electromagnetic fluid along a porous sheet. Do et al. [34] discussed the transient Darcy–Forchheimer nanofluid flow along the stretching surface under the consideration of Brownian and thermophoresis effects. He solved the governing model using the spectral relaxation method. Recently, Prashu et al. [35] considered the transient behavior of the hydromagnetic flow of a couple-stress dusty fluid in the consideration of dissipative heat transfer effects. They solved the developed mathematical model using a spectral method-based successive linearization technique.Since fluids have multiple properties, it is essential to examine the fluid flow problems by formulating models with distinct combinations. In research, the aspect of non-Newtonian fluids is prominent because of its wide range of real-life applications. The fluid that does not follow Newton’s law of viscosity comes under the category of non-Newtonian fluids. Paint, condensed milk, chocolate flows, ink, Mayonnaise, sauces, lubricant, and jelly all are examples of non-Newtonian fluids. To avoid the complexities of such fluids, these are categorized into different flow configurations such as micropolar, Casson, Maxwell, and power law. Among all of these models, Casson fluid model comes under the category of the essential models that specify yield stress characteristics. It was Casson who introduced Casson fluid model in 1995. The said model tells the fact of the interaction of solid and liquid states. It acts as solid if yield stresses are dominant on shear stress; otherwise, it starts flowing. Chocolate flows, jelly, and blood flow are all examples of Casson fluid flow. This phenomenon attracts many researchers because of its extensive applications in distinct fields such as biophysics, industries, engineering, rheology, and cancer therapy. The authors of [36] are the first who investigated the Casson fluid model within the two coaxial cylinders. The phenomena of Casson fluid took many years to establish properly. Patel [37] investigated the flow of Casson fluid flowing along an oscillating sheet in the existence of heat source/sink, Hall, and radiation effects. He solved the governing formulated model using Laplace transform technique. Hamid et al. [38] have formulated a model for the 2-D flow of Casson fluid towards a stretching surface considering the effect of thermal radiation and magnetic field. They have solved the governing model for both the transient and nontransient flows.All the above-mentioned studies of heat transfer are just restricted to the first law of thermodynamics, which associates with the quantity of energy, not with its quality. In this law, heat and work both are the same. However, as the concept of heat transfer is widespread in all fields such as engineering, industries, and technical equipment, it demands an understanding of the difference between heat and work. Heat relates to that molecule that is not in order when energy flows in a system or from one system to another; some disorganization occurs, which diminishes the quality of the working system, and this process is known as entropy. Besides, work relates to the molecule flowing in well order, leading to raising the working system’s quality. Despite numerous flow configurations that have been discussed above to examine the flow and heat transfer of distinct fluids along various surfaces, it is still required to study the entropy generation and its minimization during the heat transfer process. Bejan [39, 40] was the first who introduced the concept of entropy generation in thermal engineering. Later, numerous researchers have contributed to examining the entropy generation within various flow configurations having different properties. Rashidi et al. [41] performed an entropy generation analysis for investigating stagnation point flow towards a permeable surface under the heat source/sink impacts. Khan et al. [42] performed an entropy generation analysis for the boundary layer flow of viscous fluid passing a stretching surface, including the frictional and Joule heating terms. To study the entropy generation, Aziz and Afify [43] formulated a flow configuration for MHD Casson fluid flowing in the existence of a Hall current. Hassan [44] investigated boundary layer MHD fluid flow of couple stress fluid passing through a permeable channel. Alsaedi et al. [45] have formulated a mixed convection flow model for Eyring–Powell nanofluid and have done an analysis of entropy generation and minimization on the governing flow configuration. The influence of the nondarcy medium on the boundary layer flow of viscous incompressible fluid which contains micropolar water-based tio2 nanomaterial has been observed by Zaib et al. [46] along with mixed convection entropy generation aspects.In the present work, we discuss an entropy generation analysis for a transient three-dimensional hydromagnetic stagnation point flow of Casson fluid along a stretching sheet in the existence of Hall current, viscous dissipation, and nonlinear radiation. The physical configuration of the present work is presented as a set of nonlinear PDEs. Furthermore, some relevant similarity transformations are utilized to convert the PDEs into the set of ODEs. An efficient numerical method, namely SQLM, is used for solving the model. The expression of the Bejan number and volumetric entropy generation rate is also computed. A parametric analysis, including the essential physical parameters, is performed to examine the influences of distinct flow parameters on the velocity profile, temperature profile, Bejan number, entropy generation number, and the coefficients of skin friction and the Nusselt number. In order to further insight into the emerging physical quantities of engineering interest, multiple quadratic regression models are used to estimate the coefficients of skin friction and heat transfer. The present work provides many applications in the lubrication phenomenon, solar power collectors, heating process, rotating reactors, cooling of devices, thermal engineering, solar heat exchangers, and extrusion processes.


## 2. Mathematical Formulation

The following facts are assumed to formulate the model of the present work:(i)
The electrically conducting, incompressible, viscous, Casson fluid flowing over a stretching sheet near a stagnation point is considered.(ii)
The rheological equation expressing the stress components for the isotropic and incompressible flow of the Casson fluid is given by(1)τij=2μB+py2πeij,π>πc,2μB+py2πceij,π<πc,where eij presents the i,jth component of the deformation rate, μB indicates the plastic dynamic viscosity of the non-Newtonian fluid, py is the yield stress of the fluid, π=eijeij denotes the product of the component of the deformation rate with itself, and πc signifies the critical value of this product. Besides, β=μB2πc/py stands for the Casson parameter. It should be noted that, as β⟶∞, non-Newtonian behavior of the fluid disappears and the fluid performs similar to a Newtonian fluid.(iii)
The boundary layer flow is considered to be transient, hydromagnetic, and three-dimensional.(iv)
The sheet is assumed to be placed in thexz plane. The sheet surface is stretched with velocity u=uwx,t=bx/1−γt along the x axis, whereas the free-stream velocity along x axis is ue=ax/1−γt.(v)
An external magnetic fieldBt=B01−γt−1/2 is applied in a direction normal to the sheet surface. Also, the induced magnetic field is considered as negligible compared to the applied one by assuming that the fluid has a considerably small magnetic Reynolds number. These facts are represented in Figure 1 depicting the physical sketch of the problem.(vi)
The effect of Hall current is considered in the fluid flow whereas the heat transfer characteristics are being affected by the thermal radiation and viscous dissipation effects.Figure 1
Physical sketch of the problem.Taking into account the above assumptions, the fluid flow and heat transfer problem are modeled as the following set of partial differential equations (24):(2)∂u∂x+∂v∂y=0,(3)∂u∂t+u∂u∂x+v∂u∂y=−1ρ∂P∂x+ν1+1β∂2u∂y2−σB2tρ1+m2u+mw,(4)∂w∂t+u∂w∂x+v∂w∂y=−1ρ∂P∂z+ν1+1β∂2w∂y2+σB2tρ1+m2mu−w,(5)∂T∂t+u∂T∂x+v∂T∂y=αm∂2T∂y2−1ρcP∂qr∂y+νcp1+1β∂u∂y2+∂w∂y2,where u, v, and w denote the velocity components towards the x, y, and z directions, t is time, ν=μ/ρ directs to kinematic viscosity, m is Hall current parameter, μ is dynamic viscosity, ρ is density, σ is electrical conductivity, and β is Casson fluid parameter. Since the inviscid flow velocity is u=ue=ax/1−γt, where a and γ are constants, the pressure gradient terms in (3) and (4) are eliminated by applying the boundary layer approximation as follows:(6)−1ρ∂P∂x=∂ue∂t+ue∂ue∂x+σB2tρ1+m2ue,−1ρ∂P∂z=−mσB2tρ1+m2ue.In (5), T is used for the temperature within the boundary layer, αm=k/ρcp denotes the fluid’s thermal diffusivity, and cp is specific heat capacitance at constant pressure. Also, the radiative heat flux qr for thermal radiation is simplified as follows by using Rosseland approximation [25, 47]:(7)qr=−4σ∗3k∗∂T4∂y=−16σ∗3k∗T3∂T∂y,where σ∗ denotes the Stefan–Boltzmann constant, and k∗ denotes the coefficient of Rosseland’s mean absorption. Therefore, the governing temperature equation is written as follows:(8)∂T∂t+u∂T∂x+v∂T∂y+w∂T∂z=αm∂2T∂y2+16σ∗3k∗ρcP∂∂yT3∂T∂y+νcp1+1β∂u∂y2+∂w∂y2.The appropriate boundary conditions are(9)Aty=0:u=uw=bx1−γt,v=0,w=0,T=Twx,t=T0ax2ν1−γt2+T∞;Asy⟶∞:u⟶ue=ax1−γt,w⟶0,T⟶T∞,where T0 is reference temperature. We now apply some relevant similarity transformations as follows:(10)u=ax1−γtf′η^,v=−aν1−γtfη^,w=ax1−γtgη^,(11)θη^=T−T∞Tw−T∞,η^=yaν1−γt.These transformations are chosen in such a way so that equation (2) is automatically satisfied, and the remaining equations (3), (4), and (8) are converted into(12)f‴1+1β+ff″−f′2−Af′−Aη^2f″−M1+m2f′+mg+A+M1+m2+1=0,(13)g″1+1β−f′g+fg′−Ag−Aη^2g′+M1+m2−m+mf′−g=0,(14)1+43Nr1+θθw−13θ″+fθ′Pr+4Nr1+θw−1θ2θ′2θw−1+EcPr1+1βf″2+g′2−2Prf′θ−2APrθ−PrAη^2θ′=0.The boundary conditions (10) and (11) convert to(15)Atη^=0⇒f′η^=ba=s,fη^=0,gη^=0,θη^=1;Asη^⟶∞⇒f′η^⟶1,gη^⟶0,θη^⟶0,where A=γ/a is the unsteadiness parameter. Ec=ue2/cpTw−T∞ and Pr=ν/αm are for Eckert and Prandtl number. θw=Tw/T∞ and s=b/a represent as temperature and stretching ratio parameter. M=σB02/ρa stands for magnetic field parameter, and Nr=kk∗/4σ∗T∞3 denotes the radiation parameter.The friction coefficientsCfx, Cfz towards the x and z direction, and the local Nusselt number Nux are written as follows:(16)Cfx=τwxρue2,Cfz=τwzρue2,Nux=xqwkTw−T∞,where(17)τwx=μ1+1β∂u∂yy=0,τwz=μ1+1β∂w∂yy=0,qw=−k∂T∂y+qry=0.Therefore, the governing values ofCfx, Cfz, and Nux are(18)CfxRex1/2=1+1βf″0,CfzRex1/2=1+1βg′0,Rex−1/2Nux=−1+43Nr1+θw−1θ03θ′0,where Rex=xue/ν denotes the local Reynolds number.


## 3. Entropy Generation Analysis

The local volumetric rate of entropy generationSg for an incompressible viscous Casson fluid under the consideration of Hall current, magnetic field, viscous dissipation, and nonlinear radiation along the stretching sheet is defined as follows:(19)Sg=kT∞2Ty2+Tx2+16σ∗T33kk∗Ty2+Tx2+μT∞1+1βuy2+wy2+1T∞σB21+m2u2+w2.There are three sources, which are the reasons of entropy generation (EG) in our formulated model. In the above equation, the first term concerns the EG due to heat transfer; the second one refers to the EG cause of the fluid friction, and the last one deals with the EG cause of the magnetic field and Hall current. We can calculate the dimensionless numberNs for the rate of entropy generation by dividing the Sg with the rate of characteristic generation Sg0, which is defined as follows:(20)Sg0=kTw−T∞2L2T∞2,where L is the characteristic length. So the entropy generation number Ns=Sg/Sg0 is written as(21)Ns=ReL1+43Nr1+Πθ3θ′2+4θ2χ2ReL+ReLBrΠ21+1βf″2+g′2+ReLBrΠ2M1+m2f′2+g2,where ReL=uLL/ν denotes the Reynolds number depends on characteristic length L, χ=x/L is the nondimensional surface length, Br=μue2/kT∞ is the Brinkman number, and Π=θw−1 is the dimensionless temperature difference.Also, the dimensionless number of entropy generationNs which is defined by equation (21) can be described in the form as(22)Ns=NH+Nf+Nm=NH+NF,where NH,Nm,Nf are the sources of entropy generation concern to the heat transfer, magnetic field, and fluid friction, respectively. So we can define an irreversibility distribution ratio namely as ϕ as follows:(23)ϕ=NFNH=entropygenerationbecauseoffluidfrictionandmagneticfieldentropygenerationbecauseofheattransfer,if the value of ϕ∈01, the heat transfer source will have a dominant impact in the entropy generation process, while when ϕ>1 irreversibility effects due to viscous dissipation and magnetic field are dominant. Also, for ϕ=1, both NH,NF have equal contributions.Also, another essential irreversibility distribution parameter, namely Bejan numberBe, can be written as follows:(24)Be=NHNs=11+ϕ.Clearly, Bejan number lies between the [0, 1]. WhenBe is more than 0.5, it results that the contribution of the heat transfer source in entropy generation is dominant. Besides, if the contribution of flow friction and magnetic field sources are dominant in the entropy generation, then for sure, Be<0.5. Also, both will have an equal contribution to Be=0.5.


## 4. Numerical Solution

The solution of governing nonlinear ODEs (14)–(16) is computed by performing a numerical technique namely the spectral quasilinearization method (SQLM). In this technique, firstly, the nonlinear ODEs are converted into linearized differential equations by performing linear Taylor series approximation, and the governing linearized iteration scheme is written as follows:(25)a^113fr+1‴+a^112fr+1″+a^111fr+1′+a^110fr+1+a^120gr+1=Oˇ1,(26)a^222gr+1″+a^221gr+1′+a^220gr+1+a^211fr+1′+a^210fr+1=Oˇ2,(27)a^332θr+1″+a^331θr+1′+a^330θr+1+a^312fr+1″+a^311fr+1′+a^310fr+1+a^321gr+1′=Oˇ3.where(28)a^113=1+1β;a^112=fr−Aη^2;a^111=−2fr′−A−M1+m2;a^110=fr″;a^120=−Mm1+m2;Oˇ1=−A−1−M1+m2+frfr″−fr′2;a^222=1+1β;a^221=fr−Aη^2,a^220=−fr′−A−M1+m2;a^211=−gr+Mm1+m2,a^210=gr′;Oˇ2=−grfr′+gr′fr+Mm1+m2;a^332=43Nr1+θw−1θr3+1;a^331=−APrη^2+Prfr+8θw−1Nr1+θw−1θr2θr′;a^330=4θw−1Nr1+θw−1θr2θr″+8θw−12Nr1+θw−1θrθr′2−2APr−2Prfr′;a^312=21+1βEcPrfr″;a^311=−2Prθr;a^310=Prθr′;a^321=21+1βEcPrgr,′Oˇ3=1βNrβEcNrPrfr″2+βEcNrPrgr′2+βNrPrfrθr′+4βθw−13θr3θr″−2βθrNrPrfr′−2θw−14θw−1θr′2+θr″+4βθwθr′2+4βθw−12θ23θw−1θr′2+2θr″−4βθr′2+EcNrPrfr″2+EcNrPrgr′2.According to the numerical technique, the boundary conditions can be written as follows:(29)Atη^=0:fr+1′η^=s;fr+1η^=0;gr+1η^=0;θr+1η^=1,Asη^⟶∞:fr+1′η^⟶1;gr+1η^⟶0;θr+1η^⟶0.There is a need for an initial guess to begin the iterative scheme, which is taken as(30)f0=1−se−η^+s−1+η^;h0=0;θ0=e−η^.For finding the solution, the governing linear equations (25)–(27) afterwards dealt with using Chebyshev pseudospectral collocation method. Since the present model is having the infinity boundary conditions, therefore to deal this condition, we truncate it as 0,L^. Afterwards, an transformation η^=L^ξ^+1/2 is used to convert the domain 0,L^ to −1,1 and so that a mesh can be made by using Gauss-Lobatto points. The values of the functions’ derivatives, which are not known in the linearized equations, are approximated at the nodes with the Chebyshev differentiation matrix. The final governing model in the form of system of equations is written as follows:(31)Eˇ11Eˇ12Eˇ13Eˇ21Eˇ22Eˇ23Eˇ31Eˇ32Eˇ33fr+1gr+1θr+1=Oˇ1Oˇ2Oˇ3,where each Eˇij is a square matrix of size Nˇ+1×Nˇ+1. fr+1,gr+1,θr+1,Oˇ1,Oˇ2, and Oˇ3, are each of size Nˇ+1×1.


## 5. Error Analysis and Discussion of Results

An entropy generation analysis for the transient three-dimensional hydromagnetic stagnation point flow of a Casson fluid along a stretching surface under the consideration of Hall current, viscous dissipation, and nonlinear thermal radiation is carried out. The equations (12)–(14) are coupled as well as highly nonlinear in nature. Due to this, it is required to validate the outcomes in order to trust the obtained numerical solutions. The solutions which the SQLM calculates are verified by performing the residual analysis of the equations (25)–(27). The governing residuals of the primary velocity, secondary velocity, and temperature distribution are shown in Figure 2 using 180 collocation points and 20 iterations. It resembles from Figure 2 that the values of governing residuals are <10−5 in just about 20 iterations. Thus, the approximate solutions derived using the SQLM are trustworthy and usable. Also, a comparison of the coefficients of skin friction in x direction, i.e., 1+1/βf″0 obtained for setting β⟶∞,m=0,A=0,s=0, and M=0 is made with the values obtained by Guram and Smith [48]. The value of 1+1/βf″0 is 1.2326 for Guram and Smith [48] and 1.23258829 for present research. From this comparison, it is clearly visible that our results are in excellent with the results of Guram and Smith [48].Figure 2
Error inf′η^,gη^,θη^.
(a)(b)(c)Behavior of distinct physical quantities, such as primary and secondary velocities, temperature, the dimensionless entropy generation number, Bejan number, and the skin friction and heat transfer coefficients, is analyzed for varying values of the pertinent flow parameters, and the effects are shown using figures and tables. For performing the numerical computations, the default values of the parameters are considered to beA=1.5,θw=1.5,β=0.4,Nr=1,Pr=15,Ec=1.5,m=0.5,M=4,s=2χ=0.5,ReL=0.3, and Br=3.The influence ofm and M on the profiles of primary and secondary velocities and the temperature is shown graphically in Figure 3. It is found that on increasing the magnetic parameter M, there is a decrease in f′η^ profile. An enhancement in the value of the magnetic parameter corresponds to an increase in the force resisting the flow in the primary direction; however, the influence of the Hall current parameter m results in an acceleration of the primary fluid velocity f′η^. The magnetic field and Hall current both have the tendency to accelerate the secondary flow field, except for the region away from the sheet, where the strength of the induced Lorentz force weakens, and the secondary flow profiles show different behavior. Temperature profile θη^ rises as the M increases. m results in reducing the fluid temperature. It is concluded from Figure 3 that the momentum boundary layer for both directions gets thicker with the rising influence of the Hall current. The resistive Lorentz force tends to reduce the boundary layer thickness in the primary flow direction. The thermal boundary layer gets thinner with increasing Hall current and gets thicker with Lorentz force.Figure 3
Influence ofM and m on (a) f′η^, (b) gη^, and (c) θη^ when A=1.5,θw=1.5,β=0.4,Nr=1,Pr=15,Ec=1.5,s=2χ=0.5,ReL=0.3,Br=3.
(a)(b)(c)The effects ofA and β on the profiles of primary velocity, secondary velocity, and temperature are shown graphically in Figure 4. It is found that the increase in both A and β causes retardation in the primary flow-field f′η^. Closer to the surface, the profile of gη^ increases for the larger Casson parameter β, whereas away from the surface, it changes its characteristics, and the secondary velocity decreases with rising values of Casson parameter β. At the same time, the secondary velocity profile behaves oppositely for the unsteadiness parameter A. Profiles of θη^ fall with the rising values of β and A. From Figure 5, it is visible that for the larger values of stretching ratio parameter s, the profiles of f′,g and θ are rising. It is noticed that by enhancing the stretching ratio parameter s>1, the relative velocity of the surface is increased, which is the main reason for the acceleration in the flow field and enhancement in the temperature of the fluid.Figure 4
Influence ofβ and A on (a) f′η^, (b) gη^, and (c) θη^ when θw=1.5,Nr=1,Pr=15,Ec=1.5,m=0.5,M=4,s=2χ=0.5,ReL=0.3,Br=3.
(a)(b)(c)Figure 5
Influence ofs on (a) f′η^, (b) gη^, and (c) θη^ when A=1.5,θw=1.5,β=0.4,Nr=1,Pr=15,Ec=1.5,m=0.5,M=4,χ=0.5,ReL=0.3,Br=3.
(a)(b)(c)Figure6 represents the profiles of the temperature field for several values of Nr and Pr. Since Nr is reciprocal to the thermal radiation influence, Pr is the ratio of the viscous diffusivity to thermal diffusivity, indicating that the increase in the value of Nr means a fall in the thermal radiation and an increment in Pr implies a fall in the thermal diffusivity. This figure shows that there is a fall in the fluid temperature with rising values of the Nr and Pr, meaning that the radiative heat transfer and thermal diffusivity tend to increase the fluid temperature within the boundary layer. Figure 7 represents the distribution of the temperature field for varying values of Ec. The parameter Ec measures the influence of viscous dissipation in the heat transfer process, and it implies that an increment in viscous dissipation causes a rise in the fluid temperature.Figure 6
Influence ofNr and Pr on θη^ when A=1.5,θw=1.5,β=0.4,Ec=1.5,m=0.5,M=4,s=2χ=0.5,ReL=0.3,Br=3.Figure 7
Influence ofEc on θη^ when A=1.5,θw=1.5,β=0.4,Nr=1,Pr=15,m=0.5,M=4,s=2χ=0.5,ReL=0.3,Br=3.To study the influence of pertinent nondimensional parameters onCfx, Cfz, and Nux, values of these quantities are shown in Table 1. It is clearly visible that Cfx decreases with the increase in β and m, whereas it gets increased s, A, and M. The skin-friction coefficient Cfz decreases with rising A and β while it increases with increasing s, M, and m. This shows that the wall shear-stress along the primary direction enhances with the stretching of the velocity of the sheet, unsteadiness, and the strength of the magnetic field, and it decreases with the Casson parameter and Hall current. The wall shear-stress in the secondary flow direction is directly proportional to the stretching velocity, magnetic field, and Hall current, whereas it is inversely proportional to the unsteadiness and Casson parameter. The heat transfer coefficient, i.e., the Nusselt number Nux, enhances with rising values of θw, Pr, and Nr. The other parameters Ec, s, A, β, M, and m have reducing influence on the Nusselt number. Therefore, the heat transfer coefficient rises with the rising temperature of the sheet and viscous diffusion, whereas it is reversely influenced by thermal diffusivity, viscous dissipation, unsteadiness, Casson parameter, magnetic field, and the Hall current.Table 1
Table for the numeric values of skin-friction coefficients and Nusselt number.
θwPrEcsAβNrMm−CfxRex1/2CfzRex1/2Rex−1/2Nux1.5151.521.50.4140.55.265194020.56298252−1.628419881.25.265194020.56298252−2.762496911.85.265194020.562982520.06573688105.265194070.562982520.12859466155.265194020.56298252−1.62841988205.265194000.56298252−3.324180370.25.265194010.5629825218.584220831.55.265194020.56298252−1.6284198835.265193990.56298252−26.968476741.52.548572320.2881714914.3580937125.265194020.56298252−1.628419882.58.142155930.82582006−32.818317480.14.917564400.60554672−7.987164291.55.265194020.56298252−1.6284198835.621293810.523664852.997107560.19.334191330.99806133−6.433562550.45.265194020.56298252−1.628419880.84.221546880.451390220.318250500.55.265194040.562982521.2102425315.265194020.56298252−1.628419881.55.265194000.56298252−2.5278543224.680817150.322016992.7611355245.265194020.56298252−1.6284198865.794953720.75914345−5.880554290.15.484433030.13314107−3.004635890.55.265194020.56298252−1.6284198814.878712230.767360870.92385836The behavior ofNS and Be for distinct values of m and M is visible in Figure 8. The entropy generation NS rises with increasing M. The increase in entropy generation with a rising magnetic field reports a deceleration of the fluid velocity in the x direction. The entropy generation number decreases with increasing values of m. Also, the values of Be get decreased with an increasing value of M. Physically, when we increase M, magnetic field sources’ contribution becomes dominant compared to the heat transfer sources in the entropy generation process. The Be values increase for rising values of m. Figures 9 and 10 depict that both NS and Be decrease for increasing values of β, A, and Nr while both the quantities behave contrarily for the rising values of θw.Figure 8
Effect ofm and M on (a) NS, (b) Be, when A=1.5,θw=1.5,β=0.4,Nr=1,Pr=15,Ec=1.5,s=2χ=0.5,ReL=0.3,Br=3.
(a)(b)Figure 9
Effect ofβ and A on (a) NS, (b) Be, when θw=1.5,Nr=1,Pr=15,Ec=1.5,m=0.5,M=4,s=2χ=0.5,ReL=0.3,Br=3.
(a)(b)Figure 10
Effect ofNr and θw on (a) NS, (b) Be, when A=1.5,β=0.4,Pr=15,Ec=1.5,m=0.5,M=4,s=2χ=0.5,ReL=0.3,Br=3.
(a)(b)The impact ofPr and χ on NS and Be is depicted in Figure 11. Closer to the surface NS enhances for rising values of Pr. However, away from the surface, it starts diminishing. The reason behind this is that the temperature profiles decrease for large values of Pr, which leads to a sharp fall in the entropy generation of the flow system. It is evident from Figure 11 that the Bejan number decreases for both Pr and χ. The profiles of NS and Be, for distinct values of ReL and Br, are shown in Figure 12. Raising the Reynolds number also rises entropy generation. The reason behind this is that the larger Reynolds number flows have more disturbances in the flow field. Also, it is noticed that an increment in Br causes more entropy in the flow field. The Bejan number Be decreases for rising values of NS and Be.Figure 11
Effect ofPr and χ on (a) NS, (b) Be, when A=1.5,θw=1.5,β=0.4,Nr=1,Ec=1.5,m=0.5,M=4,s=2ReL=0.3,Br=3.
(a)(b)Figure 12
Effect ofReL and Br on (a) NS, (b) Be, when A=1.5,θw=1.5,β=0.4,Nr=1,Pr=15,Ec=1.5,m=0.5,M=4,s=2χ=0.5.
(a)(b)


## 6. Regression Analysis

Regression analysis is used in many fields including industries, prediction, and forecasting. Keeping this in mind, a quadratic regression model is formulated for the statistical analysis of skin friction coefficient and local Nusselt number. These results have potential applications in many industrial processes such as electrical power generation, nuclear energy plants, melt spinning techniques for cooling liquid, astrophysical flows, space vehicles, geothermal extractions, and solar systems. For this,CfxRex1/2 is estimated for the fixed values of s, and a set of 100 values of M and β is arbitrarily chosen from 1.5,6.5 and 0.05,0.9, respectively. Likewise, to evaluate CfzRex1/2 for fixed values of θw, a set of 100 values of A and m is picked arbitrarily from 0,4 and 0.05,1.1, respectively. Furthermore, Rex−1/2Nux is figured for fixed values of Nr, and a set of 100 values of Pr and m is randomly picked from 9,21 and 0.05,1.1, respectively. The approximate quadratic regression models of CfxRex1/2, CfzRex1/2, and Rex−1/2Nux are given as follows:(32)Cfxest=Cfx+a1M+a2β+a3M2+a4β2+a5Mβ,(33)Cfzest=Cfz+b1A+b2m+b3A2+b4m2+b5Am,(34)Nuest=Nu+c1Pr+c2m+c3Pr2+c4m2+c5Prm,where a1, a2, a3, a4, a5 and b1, b2, b3, b4, b5, and c1, c2, c3, c4, c5 are the coefficients corresponding to the parameters, which are given in equations (32)–(34).Thep values of independent parameters determine their significance. The Nusselt number and skin friction coefficient are significant for the parameters with p values less than equal to 0.05.The values of the parameter coefficientsM,β,M2,β2,Mβ, standard error, and p values are displayed in Table 2. All physical parameters are significant because their p values are less than 0.05.Table 2
Regression values and error bounds of the multiple quadratic regression model approximated forCfx.
MultipleR0.984424R-square0.96909AdjustedR-square0.967446F589.4164ModelsCoefficientsStandard errorp-valueconstant33.084486.2947159.18×10−7M−25.92758.6738390.00357β111.664245.987290.017079M27.129952.9664680.018201β2195.54482.040310.019159Mβ−71.763131.269460.023958Table3 lists the standard error, p values, and coefficient values for the parameters A,m,A2,m2,Am. The means of the associated physical parameters are significant when all p values are less than 0.05. It can be seen here that only m and m2 are significant.Table 3
Regression values and error bounds of the multiple quadratic regression model approximated forCfz.
MultipleR0.999911R-square0.999822AdjustedR-square0.999812F105578.1ModelsCoefficientsStandard errorp-valueconstant−0.00440.0010570.009334A−0.005220.0067360.440245m2.0323870.0287362.6×10−83A20.0020050.0073160.784692m2−0.959380.105531.57×10−14mA−0.03460.0554910.534487Similarly, Table4 shows standard error, p values, and the parameter coefficients pr,m,pr2,m2,prm. Since only pr and m2 physical parameter’s p value is less than 0.05, they are considered significant.Table 4
Regression values and error bounds of the multiple quadratic regression model approximated forNu.
MultipleR0.998137R-square0.996278AdjustedR-square0.99608F5032.329ModelsCoefficientsStandard errorp-valueconstant2.3161540.8547480.008003pr−0.613170.1605970.000241m0.9533891.8575620.608983pr2−0.005990.007420.421407m24.846510.9399991.39×10−6prm0.1199120.1674060.475585


## 7. Conclusions

An entropy generation analysis for the transient three-dimensional hydromagnetic stagnation point flow of a Casson fluid is performed successfully along a stretching surface in the presence of Hall current, viscous dissipation, and nonlinear radiation. The main findings of the study are as follows:(1)
Velocity distribution along thex direction is diminished by the magnetic effect. Besides, it is enhanced by the Hall effect. However, the magnetic field and Hall current both tend to accelerate the velocity distribution in the z direction, except for the region away from the sheet, where the strength of the induced Lorentz force weakens, and the secondary flow profiles show different behavior. The fluid temperature increases due to the magnetic effect and gets reduced under the Hall effect.(2)
The coefficients of skin friction inx and z directions decrease with increased values of the Casson fluid parameter. In comparison, it has a reducing influence on the Nusselt number. Unsteadiness parameter has the tendency to diminish the skin friction coefficient in z direction and the Nusselt number and to increase the skin-friction coefficient in x direction.(3)
Entropy generation is increased by the presence of a magnetic effect. However, the Bejan number diminishes with an increment in the magnetic effect since the increase in the influence of the magnetic field causes the contribution of magnetic field sources to become dominant compared to the heat transfer sources in the entropy generation process. The increments in the values of Casson, radiation, and unsteadiness parameters decrease the Bejan number and entropy generation.(4)
Performing quadratic regression analysis, it is found that all the parameters which have been considered for skin friction inx direction are significant. The parameters m and m2 are significant for skin friction along z direction, and the parameters pr and m2 are significant for the Nusselt number.



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