Preprint Title Stability analysis of heat transfer in nanomaterial flow of boundary layer towards a shrinking surface: Hybrid nanofluid versus nanofluid

Many boundary value problems (BVPs) have dual solutions in some cases containing one stable solution (upper branch) while other unstable (lower branch). In this paper, MHD flow and heat transfer past a shrinking sheet is studied for three distinct fluids: 2 3 / Al O ZnO  kerosene hybrid nanofluid, 2 3 / Al O kerosene nanofluid, and / ZnO kerosene nanofluid. The partial differential equations (PDEs) are turned into ordinary differential equations (ODEs) using an appropriate transformation and then dual solutions are obtained analytically by employing the Least Square method (LSM). Moreover, stability analysis is implemented on the time-dependent case by calculating the least eigenvalues using Matlab routine bvp4c. It is noticed that negative eigenvalue is related to unstable solution i.e., it provides initial progress of disturbance and positive eigenvalue is related to stable solution i.e., the disturbance in solution decline initially. The impacts of various parameters, skin friction coefficient, and local Nusselt number for dual solutions are presented graphically. It is also noted that the results obtained for hybrid nanofluids are better than ordinary nanofluids.

, ss Solid-nanoparticles for 23 Al O and ZnO , respectively

Introduction
The remarkable analysis in the flow of boundary layer past a stretching/shrinking sheet has been completed by many specialists as it has vast uses in industries and engineering. Some common examples are packaging of products, manufacturing of polymers, glass blowing, drawing of wires and aerospace coatings, etc. The real behavior of the surface depends on the rate of stretching/shrinking and cooling (exchange of heat) during the process of stretching/shrinking. Miklavcic and Wang [1] initiated the flow of fluid caused by a shrinking sheet and presented 3 numerical, exact, and close form solutions and they obtained dual solutions for the case of the shrinking sheet. The analysis of the flow of stagnation point past a shrinking sheet was studied by Wang [2]. He found dual solutions by taking some range of velocity ratio parameters. Ishak et al. [3] made an extension on the work of Wang [2] by taking micropolar fluid and finding multiple solutions. The solution in the analytical form for the flow of boundary layer caused by a shrinking sheet was given by Fang and Zhang [4]. They studied the close form solutions with special parameters. The MHD flow of fluid near the region of stagnation point towards a shrinking sheet was discussed by Lok et al. [5]. The study of flow problems through a permeable shrinking surface in the region of stagnation point having dual solutions was presented by Bhattacharyya and Layek [6] and they evaluated numerical solution by examining the impacts of suction/blowing and radiation. The slip effects on the flow of stagnation point caused by shrinking sheet were presented by Bhattacharyya et al. [7]. They also established dual solutions by employing shooting techniques to solve self-similar equations and they also noticed the increment in the range of dual solutions with slip parameter. Few other researchers [8][9][10] also found the dual solutions for shrinking sheets by using several physical effects.
In the 19th century, Maxwell [11] studied the influence of fluid's thermal conductivity by taking various substances with improved conductivity. Latterly, to improve the fluid conductivity the idea of nanofluid as a new type of heat transfer fluid was given by Choi and Eastman [12].
They studied that nanofluid is formed when the nanoparticles are suspended in a base fluid. With the development in the process of nanotechnology-based transfer of heat, nanofluid is defined as a colloidal suspension of nanomaterials (1-100 nm) in a base fluid. By adding the nanoparticles like carbon material, metal oxides, and metals in base fluids, the outgrowth thermal conductivity of fluids increases in conventional heat transfer fluids and limits the ability of cooling. Nanofluids have vast applications in the manufacturing and engineering industries like fuel generators, cooling of electronics, and engine [13]. Therefore, nanofluids impact a great effort and interest to the researcher. Few other researchers have extended a great analysis of nanofluids [14][15][16].
In recent times, a new type of nanofluid called the hybrid nanofluid is introduced which is formed by suspension of assorted nanoparticles in the base fluid. Hybrid nanofluid enhances properties of heat transfer as well as provides extra substantial thermal, physical, and rheological features. This fluid captivated many researchers to extend the given problem for the transfer of heat. In this regard, an experimental analysis is conducted by Suresh et al. [17] [19]. Their findings reported that the optimal rate of heat transfer for hybrid nanofluid is gained by selecting different nanoparticles. Due to this regard, many investigators studied the hybrid nanofluids by taking physical assumptions over a stretching/shrinking sheet-like, Khashi'ie et al. [20], Zainal et al. [21], and Waini et al. [22].
In the last two or more decades, researchers started to apply a stability analysis on problems having multiple solutions. Some of them are stable and others are unstable. Due to the existence of dual solutions, stability analysis in fluid dynamics problems plays an important role and is roughly linked with some numerical errors. In this era, Merkin [23] was the first who perform For the flow of boundary layer past a shrinking sheet with nonlinear differential equations, analytical solutions have a significant role. But in the case of strictly nonlinear coupled equations, it is very challenging work to find an analytical solution. In this regard, various analytical methods 5 have been employed to find estimated solutions to such nonlinear differential equations. The analytical outcomes of very weak nonlinear BVP were found by Nayfeh [33]. He used perturbation methods for this purpose but for a certain range of parameters, this method is not so efficient. The homotopy perturbation method was applied by Khan et al. [34] for the influence of thermal conductivity on the transfer of heat inside a hollow sphere with heat generation. Some other analytical techniques have also been employed for solving the nonlinear problems like linearization methods [35], Lindstedt-Poincare method [36], differential transformation method [37], and optimal homotopy perturbation method [38]. Moreover, some other simple and more accurate analytical methods are available for solving differential equations namely: weighted residual methods which consist of the Least square method, method of Moments, Collocation method, and Galerkin method. The least-square method was initiated by Bouaziz and Aziz [39].
They employed this method for predicting the longitudinal fin performance and found this method more simple and accurate than others. Furthermore, the detailed study on the Least square method was completed by Hatami and Ganji [40][41][42] and they employed this method on different problems of fluid mechanics.
The main focus of the present paper is to use an analytical method namely: Least square method to find the dual solutions of MHD flow and heat transfer past a shrinking sheet for three distinct fluids. Also to check the reliable solution using stability analysis and evaluate the corresponding eigenvalues for both solutions. B is also taken in the normal 6 direction to the surface. Since it is an earlier assumption that the magnetic Reynolds number is too low, which causes the higher magnetic diffusion than magnetic advection, due to which induced magnetic field is ignored.

Mathematical model
Under all assumptions stated above, the governing Navier-Stokes and energy equations for the case of steady flow with radiation are written as: Al O 3970 The radiative heat flux is denoted by r q and is given by Rosseland approximation [43]: The variation of temperature ( 4 T ) is taken as Taylor's expansion, then ignoring the terms of higher-order and after that, the expansion of ( 4 T ) about T  gives 4 Eq. (3) then becomes 8 Now the similarity transformations are introduced in the following forms: here prime specifies differentiation of function with respect to variable  . The dimensionless form of governing equations are presented as follows: where the magnetic number M , Prandtl number r P and radiation parameter R are given as 23 01 And the constant parameters 1 2 3 4 , , , A A A A and 5 A are: BCs: Here, a b   represents the velocity ratio parameter.
The physical terms of attention are coefficient of skin friction f C and local Nusselt number x Nu and both of them are given below: here Re u x   indicates the local Reynolds number.

Stability analysis
Stability analysis has been performed by many researchers like Awaludin et al. [28], Hamid et al. [29], and Waini et al. [30] Now for a dimensionless form of the above equations, new transformations with time-variable  are used as Using the above transformation (16) in Eqs. (14) and (15) 1 1 0 Following Awaludin et al. [28], Hamid et al. [29] and Waini et al. [30], the unknown functions are: .
Here,  is the unidentified eigenvalue parameter, 0 () f  and 0 ()  are steady solutions of a problem given in Eqs.
with BCs:

Least square method
The Weighted Residual technique such as the Least square method is an approximation technique that gives the most useable procedure that applies to nonlinear dynamical models. The central idea of this method is to obtain an estimated solution of the differential equation.
Consider a differential equation subjected to the boundary conditions jj B F g  .

11
In order to discover an estimated solution to the given problem, consider a linear combination set (linearly independent) of a basis functions. That is, Here 0 F is selected in a manner that the boundary conditions are satisfied, exactly if possible. j  are the linearly independent functions, also called trial functions, and are supposed to be known.
The coefficients j c are unknowns and can be obtained by solving a system of equations. When substituted Eq. (29) in Eq. (27), it will not satisfy the equation. Hence an error or residual R , which is a continuous function of spatial coordinates, exists and is written as In one spatial coordinate, the approximating functions may be the trigonometric functions or the polynomials of the form The notion in the least square method is to make the error (residual) equal to zero over the entire domain (say) X in an average sense.
Where weight functions and the unknown coefficients j c are exactly equal. We take the sum of the squares of residuals rather than the sum of residuals. So, this sum is minimized and given as Now to get the minimum of the given function, the derivative of Eq. (33)

Solutions
The domain of the problem under consideration is [ Using Eq. (29), trial functions satisfying the boundary conditions given in Eq. (39) take the form:   ) ), Using Eq. (36), the weight functions are obtained: 1  1  1  1  2  2  1  2  3  4  5  6  1  2  3  4  5  6 , , , , , Substitution of the weights along with the residuals in Eq. (32) gives a system of six nonlinear equations in six unknowns 16 () cc  . By applying Newton's method [34]       and 4b display the graph of a velocity profile () f'  for the first and second solution, respectively whereas Fig. 5a represents the first and Fig.5b represents the second solution for the temperature profile ()  . It is determined from Fig. 4(a,b) Fig. 6 that the velocity profile increases  while Fig.  7 depicts that the temperature profile decreases when  is increased. It is concluded from these figures that the behavior of  is the same for all three types of fluid.

Conclusion
The BVP presented in Eqs. enhances with the enhancement of radiation parameter R for 1 st solution but declines initially for 2 nd solution and then after a certain value, it begins to increase.