Local Non-Similar Solutions for Boundary Layer Flow over a Nonlinear Stretching Surface with Uniform Lateral Mass Flux: Utilization of Third Level of Truncation

: The present study aims to examine the effects of uniform lateral mass ﬂux on the boundary layer ﬂow induced by a non-linearly stretching surface. For uniform mass ﬂux, the boundary layer ﬂow does not conform to a similarity solution. The problem may be resolved by the similarity solution only when the transverse velocity at the boundary of the porous stretching surface is of the form v w ∼ x p − 12 . In other words, the ﬂow becomes non-similar; to date, this has not been reported in the literature. That is why, in the current study, the local-similarity approximation up to the third level of truncation is utilized to solve the problem. The pseudo-similarity variable, stream function and transformed streamwise coordinate are deﬁned such that the continuity equation is identically satisﬁed, and the momentum equation reduces to a non-similar dimensionless boundary layer equation. We derived the non-similar equations of the ﬁrst, second and third levels of truncations and compared the numerical results obtained from different levels of truncations. In order to ﬁnd numerical solutions to these equations, the built-in MATLAB routine, known as bvp4c, is used. Further, all non-similar terms that appear in the momentum equations are retained without any approximations. The approximations are introduced only in the subsidiary equations and relative boundary conditions. For the case of suction, the rate of increase in the numerical values of skin friction coefﬁcient obtained from the ﬁrst level of truncation with increasing velocity index parameter is found to be underestimated, while overestimation is found in the case of injection. The numerical results that were obtained from the third level of truncations are plotted against the embedding physical parameters and are then discussed.


Introduction
The phenomenon of suction or blowing is of considerable practical significance in many applications and has been used effectively in transpiration cooling, energizing the inner boundary layer in an adverse pressure gradient, increasing the lift on aerofoils, preventing corrosion or scaling, and cooling of surfaces exposed to high-temperature flows. In drag reduction, suction is also used in chemical processes to remove reactants, while injection is applied to add reactants, to control boundary layers, etc. [1][2][3][4]. Boundary layer flows over a stretching surface have many engineering and industrial applications. Crane [5] was the first to investigate the flow of a boundary layer over a stretching surface. He opened a new direction for researchers, which have started working on the boundary layer flows induced by stretching surfaces to benefit industries involved in manufacturing processes like continuous casting, polymer extrusion, wire drawing, hot rolling, glass blowing, paper production, annealing and drawing of plastic films, fiber spinning, etc. [6][7][8][9][10][11][12]. The first attempt to analyze the effect of suction or injection (blowing) on the boundary layer flow over a linearly stretching surface was made by Gupta and Gupta [13]. They reported a similarity solution in the presence of constant mass suction/injection and observed that an increase in suction causes a gradual thinning of the boundary layer, whereas blowing has the opposite effect. Chen and Char [14] also investigated a linearly stretching surface subject to suction or injection. A few recent investigations on the flow over permeable (porous) linearly stretched sheets have been performed by [15][16][17][18][19][20][21][22]. Bank [23] firstly examined the flow over a nonlinear stretching (power law velocity variation of the form U 0 x m , here m is the velocity exponent parameter and U 0 is dimensional constant having dimension (length/time)(length) −m ) and computed the similarity solutions numerically. Research by Vajravelu [24] focused on the investigation of fluid motion in conjunction with heat transfer while maintaining a constant temperature at the wall. Ali [25] extended the analysis reported in [13] from linear to non-linear stretching flow, and that in [14] by considering the porous stretching sheets instead of impermeable ones. They also analyzed the impact of suction or blowing on the boundary layer thickness and skin-friction coefficient. Ali [25] proposed that the suction/injection velocity V w = νU 0 x 1−p in which (V w , U 0 , x, p, ν) respectively represent the vertical velocity component at the boundary, the dimensional constant, the Cartesian coordinate along the stretching boundary, the velocity index parameter and the kinematic viscosity, must be a function of the distance from the leading edge to permit similarity solutions. Hence, for the constant wall suction/blowing velocity, a similarity solution does not exist for the flow over a non-linearly stretching surface [26,27]. In some prior studies, the number of independent variables was not reduced after performing the proposed transformations, and a space variable x appeared in the mass flux parameter. Researchers merged this space variable in the mass suction/injection parameter and treated this as constant velocity and the corresponding equations as locally self-similar, which actually corresponds to non-similar situations. The elementary flow quantities in non-similar boundary layer flow changes in a stream-wise direction. In summary, a more realistic scenario for the flow over a nonlinear porous stretching sheet would use uniform surface mass transfer [28]; this corresponds to non-similar boundary layer flow.
The Navier Stokes equations are a set of coupled and nonlinear partial differential equations that govern fluid motion by employing the basic conservation laws of mass, momentum, and energy. The nature of these equations (parabolic, hyperbolic, and elliptic), as well as their nonlinearity and complexity, strongly depend on the flow configuration. Exact solutions of these equations are very rare. Numerous analytical and numerical techniques have been developed to solve fluid flow problems in different configurations [29][30][31][32][33][34][35][36][37][38]. Notably, the techniques developed to solve local non-similar boundary layer problems are mentioned in [39][40][41][42][43][44][45][46][47][48][49][50]. We employed the local non-similar method initially introduced by Sparrow et al. [39] and derived the equations up to the third level of truncation to compute the solutions of non-similar boundary layer equations. The built-in MATLAB routine bvp4c [51,52] was used to numerically solve the boundary value problem in a fourth-order method. The slope linear regression approach was utilized to compare the numerical results obtained from the local non-similarity method and local similarity approach. Moreover, the relative error was also computed to compare the solutions with different levels of truncations.

Description of Mathematical Modeling
This comprehensive work incorporates the local non-similar flow induced by a nonlinear stretching surface with uniform lateral mass flux. This non-linear stretching surface has a velocity u s = u 0 x p where u 0 > 1 is a constant having dimension T −1 L 1−p , x represents the streamwise coordinate, and p > 1 is a dimensionless number which is called the velocity index parameter. The flow configuration, coordinate system, and boundary conditions are presented in Figure 1.
The following set of boundary conditions has been applied [24]:  The basic equations that govern the flow in the current circumstance are as follows [24] ∂u ∂x The following set of boundary conditions has been applied [24]: Introducing the following variables: Equation (1) satisfies identically, whereas Equation (2) and the relevant boundary conditions provide: Mathematics 2022, 10, 4159

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As seen in the preceding system, the number of independent variables does not reduce after performing the proposed transformations, and space variable x has appeared in the physical parameter (i.e. the mass flux parameter), as defined below: Consequently, the elementary flow quantities change in a streamwise direction. Because of this, the governing equations can be turned into dimensionless partial differential equations using non-similarity transformations. For this, we must select a non-similarity variable that correlates to this mass flux parameter.
The transformations defined in (8) satisfy Equation (1) identically, whereas Equations (2) and (3) become: We can neglect the right-hand side of Equation (9) and the last term on the right-hand side in Equation (10b) for the first truncation level [39]. By doing this, Equations (9) and (10a-c) can be re-written as Equations (11) and (12) represent the local similarity model. In this model, the variable ξ is treated as a parameter. The above local similarity model can be reproduced by assuming that f is a function of χ only and treats ξ as a parameter.
We now introduce the following variable to set up the second level of truncation.

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Substituting Equation (13) into Equations (9) and (10) yields: With one new dependent variable introduced into the problem, it is necessary to create one new equation with the corresponding boundary conditions. Therefore, Equations (14) and (15) are differentiated with respect to ξ We can neglect the right-hand side of Equation (16) and the last term on the right hand in Equation (17b) for the second level of truncation [39]. Consequently, the set of equations for the second level of truncations along with boundary conditions are as follows: with boundary conditions We now introduce the following variable to set up the third level of truncation.
Mathematics 2022, 10, 4159 6 of 14 Substituting Equation (22) into Equations (16) and (17) yields: Since we have introduced a new dependent variable into the problem, it is necessary to create one further equation with the corresponding boundary conditions. Therefore, Equations (23) and (24) are differentiated with respect to ξ: We can neglect the right-hand side of Equation (25) and the last term on the right hand in Equation (26b) for the third level of truncation [39]. Consequently, the set of equations for third level of truncations along with boundary conditions are as follows: With boundary conditions After the utilization of Equation (8), the skin friction coefficient C f ∂u ∂y y=0 takes the following form:

Results and Discussion
The uniform lateral mass flux causes no similarity in the momentum boundary layer if the flow is driven by a nonlinear porous stretching surface. Therefore, the local nonsimilarity approach is utilized to solve Equations (9) and (10). The one, two, and threeequation model are developed with sufficient detail and are solved numerically by application of bvp4c solver for −6.4 ≤ ξ ≤ 6.4. The numerical values of −C f x √ Re x as obtained from the different level of truncation by taking wide range of withdrawal and injection parameters are respectively presented in Tables 1 and 2. Strengthening the suction parameter enhances the skin friction coefficient. Table 2 reveals that −C f x √ Re x decrease with increasing strength of injection. Further, the first level of truncation overestimates skin friction coefficient for injection, as shown in Table 2. In addition, the relative error between the first and second truncation is high as compared to second and third levels. Tables 3 and 4 Tables 3 and 4. It is clearly seen from the values of S l p in Table 3 that the rate of increase in the numerical values of −C f x √ Re x obtained from the first level of truncation is low, compared to those of the second and third levels of truncation. The S l p values in Table 4 show that the rate of increase in numerical values of −C f x √ Re x obtained from the first level of truncation is high, compared to those of the second and third levels of truncation. In other words, the local similarity solution (solution obtained from the first level of truncation) against various velocity index parameter values underestimates fluid suction and overestimates for fluid injection. The relative percentage errors between the first and second levels of truncation and between the second and third levels of truncation are shown in Tables 3 and 4. The error in the numerical values obtained from local similarity method is high for fluid injection, as compared to the fluid suction. This fact can be seen from the relative percentage error in the third column of Tables 3 and 4. Further, it is clearly seen that the relative percentage error is high between the first and second levels of truncation compared to the second and third levels. In other words, the results obtained from local similarity approach is less accurate than the local non-similar approach. This fact can be explained by the fact that the non-similar terms are neglected in the first level of truncation and related boundary conditions (Equations (11) and 12(b)); therefore, it would be expected that the first level solutions would be less accurate than those using the second and third levels of truncation. Deriving the second level of truncation, the non-similar terms are neglected in the auxiliary equation and boundary conditions, but these non-similar terms are retained in the momentum equation and related boundary conditions; this is why the solution obtained from the second level of truncation is more accurate than that from the first level of truncation. Similarly, for the three-equation model, the truncation is removed twice from the system of equations and relevant boundary conditions, and therefore, the results are more accurate than those from the two-model equation.
for withdrawal (v w < 0) of fluid obtained from different levels of truncation by assuming that p = 2.
for injection (v w > 0) of fluid obtained from different levels of truncation by assuming that p = 2.
for different values of velocity index parameter (p) from different levels of truncation, assuming that ξ = 3.0.
for different values of velocity index parameter (p) from different levels of truncation, assuming that ξ = −3.0. p

First Level of Truncation (Local Similarity Solution)
Relative Error

Second Level of Truncation (Local Non-Similarity Solution)
Relative Error   Figure 2 shows the behavior of the velocity profile against various suction parameter values. As shown in the plot, the local non-similar velocity boundary layer thickness is reduced as the suction parameter increases. Physically, if there is suction, the fluid under ambient conditions will be drawn closer to the stretching surface, which will result in a reduction in the thickness of the momentum boundary layer (MBL). Figure 3 demonstrates the variation of the velocity profile with increasing injection parameter values. The plot clearly shows the increase in the velocity profile as the injection parameter increases. Additionally, as the values of the injection parameter go up, so does the MBL thickness. Physically, the fluid under ambient conditions will be pulled away from the stretching surface as the injection parameter is increased, resulting in an increase in the thickness of the momentum boundary layer (MBL). Figure

Concluding Remarks
The momentum boundary layer, which does not permit similarity solutions, is the subject of this article. One, two, and three-equation models were developed and solved numerically using the fourth-order boundary value solver (the MATLAB inbuilt boundary value solver called bvp4c). The key outcomes are as follows: • The uniform lateral mass flux causes the non-similarity in the boundary layer flow over nonlinear stretching.

•
The problem allows the similarity solution only when the transverse velocity at the boundary of the porous stretching surface is of the form The skin friction coefficient increases with enhancing the withdrawal parameter and decreases for the injection parameter.

•
With an increasing injection, the one-equation model (local similarity method) overestimates the skin friction coefficient.

•
For the increasing value of the velocity index parameter, the one-model approach, also known as the local similarity approach, underestimates the skin friction coefficient in the presence of suction.

Concluding Remarks
The momentum boundary layer, which does not permit similarity solutions, is the subject of this article. One, two, and three-equation models were developed and solved numerically using the fourth-order boundary value solver (the MATLAB inbuilt boundary value solver called bvp4c). The key outcomes are as follows: The uniform lateral mass flux causes the non-similarity in the boundary layer flow over nonlinear stretching.

•
The problem allows the similarity solution only when the transverse velocity at the boundary of the porous stretching surface is of the form v w ∼ x p−1

. •
The skin friction coefficient increases with enhancing the withdrawal parameter and decreases for the injection parameter.

•
With an increasing injection, the one-equation model (local similarity method) overestimates the skin friction coefficient.

•
For the increasing value of the velocity index parameter, the one-model approach, also known as the local similarity approach, underestimates the skin friction coefficient in the presence of suction. • For increasing velocity index values, the one-model overestimates the skin friction coefficient in the presence of injection.

•
The thickness of the non-similar boundary layer reduces with increasing suction and velocity index parameters.

•
The fluid inside the non-similar momentum boundary layer accelerates with increasing injection parameters. • Future research may focus on thermal and second law analyses of fluid flow over a nonlinearly stretching surface with uniform lateral mass flux.