Numerical and Computer Simulations of Cross-Flow in the Streamwise Direction through a Moving Surface Comprising the Significant Impacts of Viscous Dissipation and Magnetic Fields: Stability Analysis and Dual Solutions

Department of Mathematics, Huzhou University, Huzhou 313000, China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, China Department of Mathematics and Social Sciences, Sukkur IBA University, Sukkur 65200, Sindh, Pakistan Department of Natural Sciences, 0e Begum Nusrat Bhutto Women University, Sukkur 65170, Pakistan Department of Mathematical Sciences, Federal Urdu University of Arts Science & Technology, Gulshan-e-Iqbal, Karachi-75300, Pakistan


Introduction
e investigations of CRFS started after the pioneering research by Blasius [1] and Prandtl [2] on the laminar flow from a flat surface through a miniature viscosity. Prandtl [3] seemed to be a primary researcher to give the result for the regular depress gradient flow through a yawed infinite cylinder. e study through cross-flow or secondary flow is significant in many engineering applications such as a flow of wind phenomena, mechanical, aerospace, and rotating disk. Jones [4] discussed the vital results involving the problem of CRF, where he scrutinized the influence of sweepback on the boundary layer flow (BOUNLF). e three-dimensional flow past flat surface, as well as curved surfaces, was explored by Mager [5]. Bhattacharyya and Pop [6] examined the dissipation impact on forced convective flow in the way of CRF. Weidman [7] found the outcomes of the flow over an exponentially stretching surface involving the power law at which CRFS is shaped via the activity of the transverse wall shearing. Haq et al. [8] scrutinized the CRF in the direction of streamwise (STW) from a heated moving surface comprising the influence of viscous dissipation (VISD). Itu et al. [9] utilized the finite element technique to improve the rigidity of circular composite surfaces via radial ribs. Khan et al. [10] considered the combined impacts of a chemical reaction (CHEMR) and activation energy (ACTIE) comprising nanoparticles through a CRF and STW direction with nonlinear radiation. ey reported the dual nature of solutions. Recently, Nisar et al. [11] explored the nonlinear radiation impact on MHD flow of titanium and aluminum alloy particles in the CRF and STW flow direction. e investigations of the properties of magnetic assets also named magnetohydrodynamics (MHD) flow on electrically conducting liquids conferred its importance in industrial and engineering fields such as heat insulation, geothermal systems, boilers, saltwater, nuclear process, energy storage, and biological transportation. Abbas [12] obtained the equation of frequency for radial vibrations through a poroelastic cylinder in a porous lid by utilizing Biot's theory for wave propagation. Groza and Pop [13] explored the fractional problems of linear multipoint as well as two-point boundary value and obtained the numerical solution by implementing the Haar wavelet technique. Niculita et al. [14] used the finite element technique by utilizing the skin panel for the adaptive wing. Ali [15] found the dual outcomes of magnetoviscous flow through a nonlinear permeable shrinking surface. Mabood et al. [16] analyzed the influence of CHEMR on magnetorotated liquid from a vertical surface entrenched in a permeable medium with a heat source. Kumar et al. [17] scrutinized the frictional heating impact on magneto flow (MAGF) comprising Ferro liquid in the existence of radiation. Bhatti and Rashidi [18] explored the entropy analysis on magnetonanoliquid through a stretching sheet. Abbas and Marin [19] established the generalized thermoelasticity through a pulsed laser and obtained the analytic solution. e stimulus of thermal and exponential space-dependent heat sources on MHD nanoliquid through a rotating disk was observed by Makinde et al. [20]. Riaz et al. [21] considered the impact of bioheat on the 3D flow of non-Newtonian fluid in a peristaltic motion through a rectangular cross-section. Mabood Shateyi [22] deliberated the time-dependent MAGF with heat and mass transport embedded in a porous stretching sheet with radiation. Recently, Ahmad et al. [23] investigated the magnetic field on time-dependent squeezing flow between two horizontal surfaces involving non-Newtonian liquid. e fluctuating destruction of the velocity gradients through the viscous stresses is signified as viscous dissipation.
is fractional irreversible process is frequently referred to as the conversion of kinetic energy into internal energy of the liquid (heating up the liquid owing to viscosity because dissipation is elevated in the regions through great gradients). e concept of VISD was first considered by Brinkman [24]. He utilized the viscous dissipation (VISD) concept in the capillary flows. Gebhart [25] scrutinized the impact of VISD in free convective liquid flows. e stimulus of VISD on free convective flow in the non-Darcy regime was explored by Murthy and Singh [26], and they explored the ten percent decrement in three split regions of fluids. Tunc and Bayazitoglu [27] utilized the technique of integral transform to solve the problem containing viscous dissipation along with the transfer of heat in microtubes. e influences of thermal radiation (THERMR) and VISD on time-dependent magneto flow through a heated infinite vertical surface were inspected by Cookey et al. [28]. Partha et al. [29] discovered the VISD influence on mixed convective flow through an exponential stretched sheet. Ibrahim et al. [30] studied the combined effects of THERMR and VISD on time-dependent flow involving micropolar fluid. e BOUNLF and the transfer rate of heat involving Sisko liquid from an expanded or stretched cylinder through erratic thermal conductivity and VISD were inspected by Malik et al. [31]. Recently, Hussain et al. [32] discussed the impact of VISD on MAGF of a non-Newtonian liquid over a nonlinear heated stretched surface.
As discussed earlier, the problem containing combined effects of the magnetic field and viscous dissipation is not yet explored. erefore, we are examining the viscous dissipation effect along with the magnetic field in the cross-flow and streamwise directions with the characteristic of heat transport. Another important contribution regarding the problem is the complex dual nature of solutions and the stability analysis which of most researchers missed out. is assessment added an innovative methodology for the researchers, engineers, and scientists to find out the key features of the transfer rate of heat in the STW direction of CRF. e subsequent model is numerically handled using bvp4c software. e effects of the important parameters are contested with the assistance of graphs. e problems involving cross-flow taken here are exceptional in that the transverse motion is believed to be fully developed. In several engineering conditions, the cross-flowing stream may be dependent strongly on the orientation and position of a second leading edge.

Problem Formulation
e current proposed model is measured using the approach of streamwise (STW) and cross-flow (CRF) directions within the boundary layer. e Cartesian coordinates and their corresponding velocities are symbolically denoted by (x, y, z) and (u, v, w), respectively. Based on the secondary flow phenomena, to deal with the three-dimensional (cross-flow and streamwise directions) flow over a heated surface under the variable applied magnetic field, . ey accomplish a motion along with an unchanged velocity −λU 1 into and out of the origin located at a position x equal zero, where the coordinate x is run in the path of a flat surface, while the dimensionless moving parameter and the unchanged velocity are denoted by λ and U 1 , respectively. Also, T w and T ∞ are the uniform wall and ambient temperature of the liquid, whereas the constant and ambient or free-stream concentration of the liquid are symbolically signified by C w and C ∞ , respectively. e problem is schematically captured in Figure 1. Also, it is presumed that the CRF has a range that is extended widely in the spanwise direction. So, the components of the momentum, energy, and concentration equation did not depend upon the z coordinate. e exercising of the aforementioned conventions along with the Boussinesq approximation and the BOUNDL scaling the leading PDEs is [6,8,11] with the corresponding boundary conditions Here, the component velocities in the respective directions of x−, y−, and z−axes are signified by u(x, y), v(x, y), and w(x, y), respectively, while the subscript such as x and y of the velocity components is called the partial derivatives. e other symbols involved in equations (1)-(6) are abbreviated as the dynamic viscosity μ f , density ρ f , temperature of the liquid T, thermal diffusivity α f , effective heat capacity (ρc p ) f , concentration C, Brownian diffusion coefficient D B , and thermal conductivity k f . For simplicity, the technique for solving the problem here by introducing the similarity transformations is as follows: us, equation (1) is true identically, while equations (2)-(5) are worked out by the similarity variables. e governing dimensional form of ordinary differential equations become as follows: e converted boundary conditions are

Mathematical Problems in Engineering
In the aforementioned equations, the dimensionless sundry parameters are mathematically expressed as follows: are, respectively, called the magnetic parameter, Prandtl number, Lewis number, and Eckert numbers.

Skin Friction.
e coefficient of skin friction over the STW (C fx ) in the x-direction and CRF (C fz ) in the z-direction is written as

Nusselt Number.
e local rate of heat transfer in dimensionless form is written as

Sherwood Number. e local Sherwood number in dimensionless form is written as
where the correlation of the Reynolds number is

Stability Analysis
Within this section, the linear stability analysis of the solutions achieved is carried out to test their stability. Because of such purpose, we cited the work concluded through the study of Merkin [33]. Considering the time-dependent form of equations (2)-(5) along with BC (6), we may write subject to the boundary conditions By letting the fresh time-dependent similarity variable τ, the old transformations (7) may be in mathematical form as follows: where τ � U 1 t/2x and η � y ������� U 1 /2xυ f . Using (21), equations (16)- (19) along with the appropriate BC (20) reduce to the simplest form as follows: 1 Pr along with the corresponding BCs e key physical dimensional constraints involved in the above equations are the same and are defined earlier. e stability analysis of the free time-dependent solution f(η, 0) � f 0 (η) and g(η, 0) � g 0 (η) for the momentum equation in both directions and further θ(η, 0) � θ 0 (η) and ϕ(η, 0) � ϕ 0 (η) signify the corresponding energy and concentration equations. Let where β highlights the eigenvalues. Using (27) into (22)-(26), we may achieve the following linearized eigenvalue problem: Θ″ + Pr f 0 Θ′ + Fθ 0 ′ + Ec 1 2f 0 ′ F″ 2 + Ec 2 2g 0 ′ G′ 2 + βΘ � 0, along with the appropriate BCs e eigenvalue outcome of the linearized-type problem provides the class of eigenvalues. Because of the approach of linear stability analysis, the initial deceleration of disturbance, which is a physically reliable outcome (stable), is observed because of the positive coarse eigenvalue, while the initial development of disturbance is occurred owing to the negative coarse values which provide the unstable outcome. Furthermore, it is easy to note all of the positive smallest eigenvalues which indicates that the results achieved are stable and physically consistent. erefore, in the current problem, equations (28)-(32) are solved for the eigenvalue β with the new BC F′′ � 1 by relaxing the condition that F ′ ⟶ 0 as η ⟶ ∞.

Results and Discussion
In this current section, the numerical results of the streamwise and cross-flow problem governed by the set of nonlinear partial differential equations (PDEs) are presented. To solve the nonlinear PDEs (2)-(5), the similarity variables are first introduced to transform the problem into a system of highly nonlinear ODEs (8)- (11). ese ODEs are very difficult to solve exactly; therefore, it can be tackled numerically by using the scheme of finite difference for boundary value problem based on the three-stage Lobatto IIIA. For the computations, the fixed values assigned to the parameters are S � Le � 0.5, λ � −3.5, M � 0.1, Ec 1 � 0.1, Ec 2 � 0.2, and Pr � 6.2. e dual or multiple solutions of the problem under consideration are obtained for the specific range of the stretching and shrinking constraints. In our results, the upper branch solution and the lower branch solution is depicted graphically with the solid and dotted lines, respectively.

Effects of Magnetic Parameter.
e influence of the magnetic parameter M is captured in the STW and CRF directions in Figures 2(a) and 2(b), respectively. e

Mathematical Problems in Engineering
solutions obtained for f ′ (η)and g(η) satisfy the initial and boundary conditions. e velocity profiles in the STW and CRF directions decrease in the first branch solution and increase in the second branch solution. Physically, this is happening due to a larger amount of M which can slow down the liquid motion; this occurs due to retarding force known as Lorentz force. Figures 3(a) and 3(b) explain that the skin friction in both directions increases owing to the constraint M in the first branch solution, while the changed behavior is observed in the second branch solution. On the other hand, the rate of heat and mass transfer decreases in the first branch solution and enhances the second branch solution as portrayed in Figures 4(a) and 4(b), respectively. e huge magnetic limit develops the force called drag which is also entitled as Lorentz force.
is force controls as mediators that resist the fluid flow and craft progress in the local transfer rate of heat and mass along with the local coefficient of skin coefficient.

Influence of Suction Parameter.
In the porous medium, the mass suction/injection parameter is important and is defined with the expression S � −v 0 ������� 2x/] f U 1 . In case of suction, (S > 0) and for injection, (S < 0). If (S < 0), the dual nature of solutions cannot be obtained. However, the dual solutions exist in the case of mass suction. In Figures 5(a) and 5(b), the variation of the suction parameter S on both the velocity profiles in the STW direction in terms of f ′ (η) and the CRF in terms of g(η) is displayed. ese results suggest that the velocity of liquid augments due to increasing values of S in the upper branch solution which in turn reduces the velocity BOUNLF thickness, while there is a decline in the lower branch solution in the STW and CRF directions. Generally, because of S, the heated liquid is shoved toward a surface of the wall where the forces can retard the liquid owing to lofty control of the viscosity. e skin friction coefficients in both directions are displayed in Figures 6(a) and 6(b). ese figures show that the

Effects of Eckert Numbers.
e Eckert parameters appear in the energy equation due to the viscous dissipation term.
e Eckert numbers have a direct impact on the rate of heat transfer. As the Eckert numbers upsurge, more heat is transferred from the surface due to which the thermal boundary layer increases. is impacts of E c1 and E c2 on temperature θ(η) is displayed in Figure 8(a). However, the increases in Eckert numbers show a decrease in Nusselt number (Figure 8(b)). is is predictable since the implementation of the Eckert number proposes the ratio of kinetic energy and enthalpy.
us, an augment in E c1 and E c2 suggests that the heat dissipated is stored in the liquid via fractional heating that enhances the liquid temperature and consequently declines the heat transfer rate.

Effects of Lewis Number.
Lewis number L e appears in the concentration equation and is a ratio of the kinematic viscosity and the Brownian diffusion coefficient. e influence of L e on concentration ϕ(η) is demonstrated in Figure 9(a). e graphical result shows that the concentration profile decreases for both solutions. is implies that the heat will disperse more quickly than concentration. e concentration boundary layer becomes steeper as L e enhances. Moreover, the increase in L e shows an increase in the Sherwood number (Figure 9(b)).    Figures 10(a)-10(d), the variations of stretching/shrinking parameters on the velocities f ′ (η) and g(η) in both directions are displayed. e effect of the stretching parameter (Figures 10(a) and 10(b)) shows that the lower branch solution decreases while the upper branch solution increases. On the other hand, the altered behavior is noted for the shrinking parameter, as illustrated in Figures 10(c) and 10(d). It is also examined from these figures that, in the phenomenon of the lower branch solution, initially, the velocity profiles are negative, and then as the value of η progresses, it starts to augment and then becomes positive. Physically, this trend happens due to the contrary directions of shrinking and stretching velocities.

Conclusions
In the current investigation, the impact of VISD on MAGF near a stagnation point in the directions of STW and CRF through a moving surface is scrutinized. e leading PDEs are converted into nonlinear ODEs via suitable variables. ese converted ODEs are worked out by utilizing bvp4c software based on 3-stage Lobatto IIIA. e influences of pertinent constraints on the flow fields such as concentration, velocity, and temperature are illustrated graphically and numerically scrutinized. e core finding of this exploration can be precised as follows: Finally, this paper can be expanded by taking a timedependent flow or mixed convective flow. Also, the non-Newtonian fluid or hybrid nanofluid may be added due to numerous applications.

Abbreviations
B 0 : Magnetic field intensity C: Concentration C ∞ : Ambient fluid concentration C w : Wall concentration C fx and C fz : Skin friction coefficients D B : Brownian diffusion coefficient Ec 1 and Ec 2 : Eckert numbers f and g: Dimensionless velocities k f : ermal conductivity L e : Lewis number M: Magnetic parameter Nu x : Nusselt number Pr: Prandtl number R d : Radiation parameter Re x : Local Reynolds number S: Suction Sh x : Sherwood number T: Temperature T ∞ : Free-stream temperature T w : Temperature of wall U 1 : Constant velocity w 0 : Transverse velocity (u, v, w): Velocity components (x, y, z): Cartesian coordinates α f : ermal diffusivity β: Eigenvalues λ: Moving parameter μ f : Dynamic viscosity ρ f : Density (ρc p ) f : Effective heat capacity θ: Dimensionless temperature υ f : Kinematic viscosity ψ: Stream function τ: Time-dependent similarity variable ': Derivative w. r. t. η.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.