Thermal Analysis of a Casson Boundary Layer Flow over a Penetrable Stretching Porous Wedge

Tis work aims to analyze the Casson thermal boundary layer fow over an expanding wedge in a porous medium with convective boundary conditions and ohmic heating. Moreover, the efects of porosity and viscous dissipation are studied in detail and included in the analysis.Teimportanceofthisstudyisduetoitsapplicationsinbiomedicalengineeringwheretheanalysisofbehaviorofnon-Newtonian bloodfowinarteriesandveinsisdesired.Withinthecontextofbloodfow,itisalsoapplicabletomanyotherfelds,forinstance,radiative therapy,MHDgenerators,soilmachines,melt-spinning,andinsulationprocesses.TemodeledproblemisasetofPDEs,whichis nondimensionalizedtoderiveanonlinearboundaryvalueproblem(BVP).TeobtainedBVPissolvedusingtheshootingtechnique, endowedwiththeorderfourRunge-KuttaandNewtonmethods.Teimpactofdiferentparametersonthemomentumandtemperature feldsisinvestigatedalongwithtwoimportantparametersofphysicalsignifcance,i.e.,theNusseltnumberandthesurfacedragforce. Resultsarevalidated,andanexcellentagreementisseenfortheparametersofinterestusingMATLABbuilt-infunctionbvp4c.A signifcantfndingisthatbyincreasingtheCassonliquidparameter,thevelocitydecreasesasthewedgeexpandsquickerthanthefree streamvelocityat R � 2. However, the velocity increases for the case when R � 0 . 1. A decrease in the Darcy number increases the temperature profle. Furthermore, the convective parameter accelerated the heat transmission rate, and a rise in the Prandtl number thickens the thermal boundary layer. Te fndings of this investigation contribute to problems in fuid dynamics and heat transfer that involve studying the behavior of a non-Newtonian fuid with Casson rheological properties near a solid porous wedge surface.


Introduction
Mixed convection across an impermeable or permeable wedge is used in many manufacturing processes, including extruding molten polymers, fabricating plastic sheets for solar energy, and storing thermal energy.According to Shah et al. [1], mixed convection is a fuid fow phenomenon characterized by the simultaneous efect of forced convection, which is caused by external forces such as a pump or a fan, and natural convection, which is caused by buoyant forces caused by temperature gradients within the fuid.Systems such as technical equipment and natural phenomena such as the movement of temperature stratifed masses of air and water on Earth exhibit mixed convection fow [2,3].It has uses in heat exchangers [4], thermal design, cooling electronic components, and other felds.It is also pertinent to heat transfer and fuid fow analysis.Karwe and Jaluria [5,6] explained that the investigation is necessary to check the quality of resulting products and cooling rates.Ishak et al. [2] investigated the mixed convection magnetohydrodynamics boundary layer across a vertically expanding sheet while the wall temperature remained unchanged.Te reader is referred to [1,7,8] for a good overview on the mixed convection MHD viscoelastic fuid fow analysis.
Some recent studies have also considered the mixed convective fow in analyzing the porous fow structures.In this connection, thermal analysis in a higher grade Forchheimer porous nanostructure bounded between nonisothermal plates is investigated by Saleem et al. [9].Tey utilized a fnite diference approach in their analysis.Te transient problem of mixed convection over a vertically moving wedge was examined by Ravindran et al. [10].Numerical results show that the buoyant force enhances the skin friction and Nusselt number.Te thermal properties of magnetohydrodynamics of Falkner-Skan Casson fow sideways a moving wedge are analyzed by Ullah et al. [11].Tey used fnite diference and quasilinearization techniques to derive the problem's solution.It is detected that an augmentation in the values of the Eckert number has a negligible impact on force convection fow.Tey discovered that the existence of the suction or injection changes the rates of skin friction, concentration, and heat transfer.Unsteady fow with mixed convection dusty fow across a vertical wedge was investigated by Hossain et al. [3].
Many of the above-described studies addressed the fow convection over a vertical porous wedge.It is worth mentioning that there is limited research about convection fow in the presence of suction/injection, viscous and ohmic dissipation upon a permeable/impermeable wedge.Te numerical solution of thermal radiation infuence on magnetohydrodynamics with forced convective fow adjacent to a wedge with a heat source/sink [12] was computed by Chamkha et al. [13] employing the implicit fnite difference technique.Anwar Hossain et al. [14] have explored the problem of MHD fow via a wedge with variable surface temperature.Additionally, perturbation solutions for different values of τ dimensions of time are discovered.Su et al. [15] investigated the MHD mixed convective fow of heat transfer across a permeable stretched wedge using numerical and analytic (DTM) methods.Also, the authors demonstrated that as the domain becomes unbounded, the DTM-BF solutions diverge.Te infuence of chemical reaction on the mixed convection magnetohydrodynamics and heat transfer across a porous wedge was investigated by Kandasamy et al. [16].Tis research concludes that in a mixed convective regime, there is an increase in the hydrodynamic layer.In contrast, the concentration and temperature boundary layers decrease by lowering the values of the suction parameter.Te issue of MHD fow through a porous nonisothermal wedge was examined by Prasad et al. [17].Tey observed that among all variables, the mixed convection parameter substantially impacts the surface drag force, thermal rate, normal velocity of the wall, and temperature profle of the MHD fow through a porous nonisothermal wedge.Transient Falkner-Skan liquid fow of Carreau microliquid via a still/moving wedge was demonstrated by Khan et al. [18].
Due to the numerous applications in radiative therapy, metal casting technology, and ceramic engineering, several studies [4,19] have explored the infuence of radiative efect at the boundary layer with the transfer of heat characteristics.In biomedical engineering, within the context of blood fow [20,21], the Casson parameter has been considered in analyzing the behavior of non-Newtonian fow in arteries [22][23][24].For a detailed review of the use and applications of Casson nanofuid fow, the reader is also referred to the work of Nayak et al. [25] who have investigated the Casson nanofuid fow subjected to biomedicine applications and chemical reactions [26][27][28].For some other important applications related to heat transfer enhancement, the reader is referred to the work of [29] and references therein.Ullah et al. [30] have investigated the Casson liquid fow across a nonstationary wedge using the efect of thermophoresis and Brownian difusion.Te transportation of heat for homogeneous/heterogeneous Casson liquid in a porous medium was studied by Bilal et al. [31].Te velocity, temperature, and concentration of solutions for homogeneous and heterogeneous reactions are fundamental limitations of the basic subsidized fow parameters.Several researchers investigated the impact of heat radiation on different fow problems [32][33][34] using fnite element strategies.
Viscous dissipation can signifcantly afect the temperature distribution and overall energy balance of the system.Similarly, porous media have a large contact surface with fuids, which can signifcantly increase the heat transfer efect.Te porous medium changes the fow feld conditions, and the conduction heat transfer coefcient is usually higher than that of the fuid studied.Te main aim of this study is to extend the work of Hussain et al. [35] by considering the following additional physical efects: (i) Te Casson thermal boundary layer fow rheology is assumed (ii) Te efects of porosity and viscous dissipation are taken into account In this respect, two important parameters of physical interest arises are the viscous dissipation parameter E c and the porosity parameter D a .Te specifc goals and objectives of the study are to investigate the following research questions: (i) How does a porous wedge fow's thermal reaction to Casson fow rheology behave?(ii) How does the porosity afect the velocity and thermal profles in the presence of Casson rheology?(iii) How does the viscous dissipation afect the hydrodynamic profles of the present rheology?(iv) How do the viscous dissipation and porosity of the Casson fuid afect the skin-friction coefcient and Nusselt number?
Te next section presents the modeling aspects of the modeled problem which is frst transformed to a boundary value problem in ordinary diferential equations.Te problem is then solved numerically by the shooting method, and obtained results are discussed.

Mathematical Modeling
Tis problem considers an electrically conducting thermal radiative 2D Casson fuid fow with mixed convection on a moving wedge as shown in Figure 1.Te conservation equations for mass, momentum, and energy for viscous incompressible fuid in the presence of the viscous dissipation, Rosseland radiation, and porosity constitute the modeling description of the considered fow and are described respectively as [36] ∇ and ( where in equation ( 2), the symbol S represents the stress tensor of the considered fuid, which is represented by Te rheological equation for the incompressible fow of the Casson fuid is given by [37] where e ij are the components of the strain tensor defned as In equation (5), π is the product of the component of the deformation rate with itself, π � e ij e ij , π c is a critical value of this product based on the non-Newtonian model, and p y is the yield stress of the fuid.Te Cartesian coordinates' system is denoted by (x, y) from which x-direction is along the wedge and y-direction is normal.Te velocity of the wedge is represented by U w , and the velocity that is distant from the wedge is U, which depends on x, i.e., U � U 0 x m .Te stretching wedge has suction and injection velocity, denoted by v w .During the heating of the wedge, a hot fuid with temperature T f produces a high heat transfer coefcient h f , where T ∞ is the temperature of the fuid away from the wedge.Te uniform magnetic feld is denoted by B, acting along y-direction and equivalent to B � B o x m− 1/2 .Te wedge angle is denoted by Ω � βπ.

Dimensional Equations.
Connected fow equations with boundary conditions are (see for reference [15]) Te boundary conditions are described as follows: For the physical description of these boundary conditions, the reader is referred to the work of Su et al. [15].Here, ], g, β o , σ, ρ, c p , and α denote the kinematic viscosity, gravity feld, coefcient of thermal expansion, conductivity of fuid due to electric current, fuid density, heat capacity, and thermal conductivity, respectively.Te term zq r /zy here q r is radiative heat fux and is equal to − 4σ o /3k * zT 4 /zy, where σ o and k * are known as the Stefan-Boltzman parameter and mean absorption coefcient, respectively.β � μ B ��� 2π c  /p y is the parameter of Casson fuid.

Similarity Transformations.
Here, in this subsection, the similar similarity transformations are introduced.Similar transformations are employed [15] to simplify the set of governing equations of a complex physical problem by converting them dimensionless and making them similar in structure.
Tese are particularly useful for analyzing the fow and heat transfer phenomenon scales with changing parameters like Reynolds and Nusselt numbers.Tere are a lot of situations when one needs to introduce nonsimilar transformations [38] to convert this physical system into a nondimensional form.Such situations arise when nonstandard boundary conditions [39][40][41] are introduced in the mathematical description of the fow problems.8)-( 13) are nondimensionalized using transformations in Section 2.2.Te resulting equations take the form:

Quantities of Interest
3.1.Surface Drag Force.Te dimensional form of the skin friction is as follows: where μ β represents the viscosity of the liquid and p y is the liquid yield stress.Te nondimensional form of surface drag force is given as

Local Nusselt Number.
Te local Nusselt parameter is described as where q w is the conductive heat fux from the surface of the wedge.

Solution Method through Shooting and bvp4c
Te above nonlinear ODEs are now solved through the shooting method.Tis numerical technique is used for fnding the solution to a boundary value problem.Using the following derivatives, the boundary value problem is transformed into a frst-order initial value problem (see Table 1).
Using the above notations, equations ( 15) and ( 16) are transformed into the following ffteen ODEs: where s and q are the missing initial conditions.Choose the missing initial conditions that satisfy the above boundary conditions.Te RK-4 method is applied to solve the frst-order initial value problem.Newton's numerical technique is used to refne the starting values.Tis iteration is repeated until the values meet the required accuracy, achieved by getting sufciently close to the given boundary conditions.Te stopping criteria for Newton's method are given as follows: Here, ϵ � 10 − 8 .Te solution of the above model boundary value problem can also be calculated through the built-in MATLAB function bvp4c.Tis built-in function is suitable for solving higher-order boundary value problems in fuid mechanics.Te criteria behind the bvp4c function are the three-stage Lobatto IIIa formula, implemented by the fnite diference algorithm [42].Tis formula uses collocation, and the collocation polynomial ofers a C 1 -continuous solution that is consistently fourthorder accurate across the integration interval of the problem.
Te residual of the continuous solution serves as the foundation for mesh selection and error control in the model problem.Te integration interval is split into smaller intervals using the collocation technique and a mesh of points.Te solver can arrive at a numerical solution by resolving a global system of algebraic equations caused by the boundary conditions and the collocation conditions placed on each subinterval.Te solver then determines how much each subinterval's numerical solution error is.Te solver adjusts the mesh and restarts the procedure if the solution does not meet the tolerance conditions.In addition to an initial approximation of the solution at the mesh points, one must supply the initial mesh points.

Analysis of Results
Tis research is focused on analyzing the following aspects: (1) Te steady nonlinear conducting radiative Casson fow through a penetrable wedge is examined with ohmic heating, thermal behavior, suction/injection, and convective BC.
Journal of Mathematics (5) Termal boundary layer is analyzed through the numerical results by varying the Eckert number E c , Biot number B i , Darcy number D a , Prandtl number P r , and radiative heat transfer parameter N r .

Results and Discussion
Tis section describes the outcomes of the model presented above through the discussed shooting technique.Table 2 shows the values of the heat rate parameter |θ ′ (0)| and drag force parameter f ″ (0) for various values of the wedge angle β and velocity ratio parameter R. When R is less than or more than 1, surface drag force increases as the angle β increases.It is also seen that by an increase in β, the temperature gradient coefcient of Nusselt number |θ ′ (0)| decreases.Te results acquired by the shooting approach are verifed by BVP4C as well.Here, the values of diferent parameters are fxed as δ � 1, m � 0.1, λ � 0.8, B i � 5, M � 0, Nr � 1, Pr � 1, Ec � 0.5, C � 1, (Da) − 1 � 0.01, and α 1 � 0.01.Te numerical efects of drag force and Nusselt number for various values of the velocity ratio parameter R and the magnetic parameter M are shown in Table 3. Te values of the Nusselt coefcient θ ′ (0), surface drag force coefcient f ″ (0), and magnetic coefcient M rise when R is less than 1.When R is more than 1, the skin-friction coefcient and |θ ′ (0)| both drop as the magnetic coefcient M values increase.Te outcomes obtained using the shooting approach are likewise met by bvp4c.

Te Velocity Profles. Te impact of the Casson parameter δ on the velocity profle
{ } is seen in Figure 2(a).Increases in δ steepen the velocity profle f ′ (η) when R is less than one, whereas increases in δ deplete the velocity profle f ′ (η) as R � 2. For the case R < 1, Figure 2(b) shows an inverse relationship between the velocity profle and the magnetic parameter M; however, for R < 1, the relationship is the reverse.Te Lorentz force that the magnetic feld produces is what causes this phenomenon.For both situations of R, Figure 2(c) shows an increase in velocity f ′ (η) through an increase in the wedge's β angle.Te velocity boundary layer thickens with an increase in the velocity ratio (R), as seen in Figure 3(a).Te velocity boundary layer thickens with an increase in the velocity ratio (R), as seen in Figure 3. Te efects of radiation parameter Nr and Prandtl number Pr on the velocity profle are shown by generating Figures 3(b) and 3(c) from the numerical results.Te fgures show that for both situations of R, there is a fall in the velocity profle due to cumulative values of the Prandtl number and decreasing values of the radiation parameter.Tree key implications are shown in Figure 4. To demonstrate how the Biot number afects the velocity profle, Figure 4(a) is drawn.Te ratio of the body's internal conductive resistance to its surface-measured external convective resistance is known as the convection parameter Bi.It is observed that temperature is improved when internal conductive resistance is increased rather than exterior convective resistance and thus raises the velocity profle f′(η).Figure 4(b) illustrates how Eckert number Ec has a major infuence.Tis fgure shows that for R < 1, f ′ (η) is a decreasing function of Ec; however, the behavior is reversed when R � 2. As can be seen in Figure 4(c), the depth of the momentum barrier layer decreases as the Darcy parameter Da decreases.

Te Temperature Profle.
Te infuence of the Casson coefcient δ on the temperature profle θ(η), at R � 0.1 and R � 1.5, is shown in Figures 5(a) and 6.Te temperature profle decreases in all scenarios when δ increases.Figure 5(b) illustrates how changing the magnetic parameter M afects the momentum boundary layer's behavior.Te graph indicates that θ(η) decreases for rising values of M for R � 0.3 and exhibits opposite behavior at R � 2. A direct relationship between the wedge angle β and the temperature profle is seen in Figure 5(c).Tree important efects on thermal profles are depicted in Figure 7. Here, Figure 6(a) demonstrates how the thickness of the thermal boundary layer decreases as the radiation parameter Nr decreases.Te thermal profle θ(η) decreases as the Prandtl number accumulates, as seen in Figure 6(b).Te temperature profle θ(η) decreases as the velocity ratio parameter R increases, as      decreases as Pr increases.Te Nusselt number is an increasing function of Pr, as Figure 7(c) illustrates, but for a constant value of Pr, it decreases with decreasing values of the Casson parameter δ.Furthermore, Figure 7(d) shows that skin friction rises with the increasing values of δ for a given value of Pr.

Conclusions
Tis paper considers magnetohydrodynamics thermal fow and boundary layer analysis using Casson fuid over an extending wedge by considering porosity, ohmic heating, convective boundary layer conditions, and viscous dissipation.Te mathematical model of the problem is a system of nonlinear PDEs.Tis system is transformed into a set of ODEs with suitable transformations.Te obtained boundary value problem is numerically solved by using the shooting method.Tables and graphs give a detailed picture of the diferent thermodynamic profles.
Te following observations are of notable signifcance: (i) As Casson parameter δ increases when R � 0.3, f ′ (η) increases and increase in δ when R � 2, reduces the depth of momentum boundary layer.
(iii) At R � 0.3 and R � 2, the momentum boundary layer decreases by an increase in Da − 1 .
(iv) As the velocity ratio parameter R increases, f ′ (η) also increases.Te fndings of this investigation contribute to problems in fuid dynamics and heat transfer that involve studying the behavior of a non-Newtonian fuid with Casson rheological properties near a solid porous wedge surface.Such problems are common in various engineering and scientifc applications, including chemical engineering, biomedical engineering, and materials processing.In biomedical engineering, within the context of blood fow, the Casson parameter considered in this analysis describes the behavior of non-Newtonian blood fow in arteries and veins.In the pharmaceutical industry, the Casson parameter characterizes the rheological properties of pharmaceutical suspensions and emulsions, thus playing a vital role in the formulation and manufacturing of drugs.In polymer suspensions, the Casson parameter is vital in extrusion and injection molding processes.Te fndings of this investigation and the data obtained through numerical calculations delineate the variational changes in the physical parameters of importance, i.e., the skin-friction coefcient and Nusselt number.Terefore, it is recommended to reinvestigate some related physical problems in the feld of applications as mentioned earlier based on the fndings of the present investigation.

Figure 1 :
Figure 1: Geometry of the problem.

( 2 )
Te efects of porosity and viscous dissipation are considered.(3) Drag caused by the friction of a fuid against the surface and heat transfer due to pure conduction is examined by calculating the numerical values of the skin-friction coefcient and Nusselt number by varying the values of wedge angle β and the magnetic parameter M. (4) Tickness of the momentum boundary layer is examined for magnetic parameter M, wedge range β, Casson parameter δ, Prandtl number P r , and radiative heat transfer parameter N r .

Figure 7 :
Figure 7: Efects of Biot number B i and Casson parameter δ on drag force and Nusselt number.(a) Efects of B i on Nusselt number.(b) Efects of B i on surface drag force.(c) Efects of δ on Nusselt number.(d) Efects of δ on surface drag force.

Table 1 :
Denotation of unknown variables.

Table 2 :
Results of (Re x ) 1/2 C f and (Re x ) − 1/2 Nu x for values of β.
seen in Figure6(c).Te temperature's relationship with the Eckert number is seen in Figure8(a).When Ec rises, the temperature profle decreases.Te efect of convective number Bi is seen in Figure8(b).Increasing Bi levels increases the body's internal conductive resistance.Te behavior of temperature θ(η) is steepened.Figure8(c)

Table 3 :
Results of (Re x ) 1/2 C f and (Re x ) − 1/2 Nu x for values of M.