Flow of magnetohydrodynamic viscous fluid by curved configuration with non-linear boundary driven velocity

This paper discusses the flow of magnetohydrodynamic (MHD) viscous fluid driven by non-linear stretching curved surface. The relevant system of the flow configuration is considered using orthogonal curvilinear geometry in presence of radially varying magnetic field. The governing partial differential equations of the flow problem are reduced into boundary layer regime which are then transformed using similarity variables into ordinary differential equations. The resulting equations of the curved case are not amenable to analytic solutions due to non-linearity of the system. Thus a computational approach through Runge-Kutta fourth order together with the shooting technique is adopted for the numerical solution. The existing solution for flat surface is recovered in validating the present models. The novelty of this work contains the correctness and analytical solution of the existing models over flat surface. The results are interesting and can be useful in polymer dynamics.


Introduction
The linear and non-linear stretching of sheets have been broadly discussed in the literature because of its wide applications in the chemical and polymer industry where the stretching phenomena are used to improve the quality of the finished product. To avoid repetition, the relevant background literature and recent studies with applications for stretching cases have been presented in the articles [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. In this present study, the subsequent discussion is confined to some relevant and fundamental studies for non-linear stretching. These papers serve as a spotlight to the current findings, which consequently extends to curved structures. Takhar et al. [20] investigated the flow over nonlinear quadratic stretching of the flat plate over decades ago in which they showed that the fluid velocity increases in comparison to linear stretching. Kumaran et al. [21] explained that the velocity due to non-linear, quadratic, of the stretching sheet superposes the linear case. Kumar and Sanjayanand [22] showed in consequences with linear mass flux for viscoelastic fluid flow in the boundary layer region, the choice of quadratic stretching velocity parallel to the boundary sheet. The effect of suction and blowing on impermeable surface was addressed by Raptis and Perdikis [23]. In this study, an inverse solution is proposed that transformed the governing equations into three self-similar partial differential equations. Kelson [24] made an observation that the solution procedures given in [20,23] lack authenticity. Meaning two components of the momentum equations are addressed while finding the solution and ignored the third equation resulting in a nonphysical solution. He presented an analytical solution for the problem discussed by Raptis and Perdikis but the models remain incomplete in comparison to the current analysis. Cortell [25,26] revisited the problem using the same incomplete similarity variable and presented an alternate analytical approach to get a solution of non-linear stretching for the impermeable surface in non-Newtonian viscous-elastic MHD fluid with heat generation. The objective of this paper can be highlighted as follows: first, the discussion ensued by Kelson has been settled by considering the current transformation variables that generate complete models in generalized curvilinear coordinates. A straightforward procedure is adopted and an analytical solution is provided using the homotopy analysis method and satisfies all the three equations and boundary conditions. The non-linear part of quadratic stretching from the curved surface has been undertaken which was hitherto not known to date. All the plus points of the plane non-linear stretching are far more important for curved non-linear stretching from mathematical and practical points of view. Also, linear and nonlinear stretching strength are examined and are important coefficients that have potential influences on the flow field. To invoke further applications in industrial and engineering processes, electrically conducting fluid with a magnetic field is also considered in the realm of MHD. The magnetic field is considered to be variable rather than constant to satisfy Maxwell equations. To our understanding, mentioning some background literature on curved surfaces will be another spectrum to this demand. The study of viscous fluid over the curved stretching surface was initiated by Sajid et al. [27]. They discovered that the contribution of pressure inside the boundary layer cannot be neglected when compared to the stretching of plane surfaces. This investigation was further extended to include radiative heat transfer and magnetic fields by Abbas et al. [28]. The flow of a viscous fluid along a power law curved stretching sheet is given by Sanni et al. [29]. The heat and mass transfer analysis of Jeffrey fluid flow in the presence of homogeneous and heterogeneous reactions was reported by Imtiaz et al [30]. Rosca and Pop [31] analysed the flow on curved shrinking/stretching permeable surface and discussed the stability analysis. Hayat et al. [32] documented the numerical study of the flow over a non-linear curved stretching surface with heat and mass transfer. Moreover, recent and relevant papers can be found in Refs. [33][34][35][36][37][38][39][40][41]. In view of the above articles, we realized that non-linear quadratic stretching of hydromagnetic viscous fluid over the curved surface has not been discussed as of now; therefore, we tend to incorporate two new features of non-linearity and curvature to the stretching sheet problem; whereas the addition of a magnetic field is a bonus. Precisely, the viscous flow of an electrically conducting fluid over a quadratic stretching curved surface is examined to study the combined effects of linear and non-linear stretching and radius of curvature. This form of stretching surface redefines the polynomial power law as investigated for m = 1 and 2. The novelty of this paper can be described as follows. Curved linear stretching exists, but quadratic curved stretching was not available in the literature. It is observed that the non-linear stretching enhances the fluid flow, which has great significance regarding its applications in the polymer industry. The strength of the non-linear stretching velocity is found to matter a lot. This point has been investigated for the first time showing that the strength of the non-linear part, on the flow field, is significantly higher than the linear part of the stretching velocity. The boundary layer equations governing curved structure are a 2D problem represented by the three scalar components of the momentum equations. The present geometry gives a 3D view of the existing physical description with no flow kinetics being considered in the z-direction. The Lorentz force due to the applied magnetic field is a resistive force restraining the mutual velocity between the medium and the magnetic field; it is shown that an additional resistive force due to non-linear stretching is also there to be reckoned with. The results are found worthwhile in controlling fluid flow rate in medical apparatus made with prestressed bladder for dispensing fluids.

Flow description
Consider 2D boundary layer flow of an incompressible viscous fluid by a curved surface. The surface is driven by non-linear velocity u(s) = αs + βs 2 in the s− direction.
The build-up of the flow in r− direction determines the boundary layer thickness. The fluid is conducted under the application of a variable magnetic field acting in the transverse direction to the fluid motion. It is given by [39,40] B(r) = λB 0 e r (1) where λ = 1 R+r is the curvature, R the radius of curvature, B 0 denotes the strength of applied magnetic field parameter where induced magnetic field is neglected for small magnetic Reynolds number and e r is the unit vector in the radial direction. If u, v are x− and r− components of velocity V, then the flow kinetics Figure 1.
The current density and Lorentz force F = J × B can be expressed in the absence of electrical field (E = 0) as where σ and B 0 are the conductivity of the fluid and the strength of applied magnetic field. The continuity and momentum governing equations for 2-D fluid flow [27] are modified with the variable magnetic fields to satisfy The boundary conditions satisfied by the sheet are [20] u w = αs + βs 2 , v w = 0 at r = 0, In Equations (4)−(8), pis the pressure, α(1/t) and β(1/Lt) are the stretching strengths for α, β ε [0,1]. ν is the kinematic viscosity of the fluid, ρ the density of the fluid andLthe characteristic length. Through appropriate scaling, the following non-dimensional variables are defined.
After applying Equation (9) in Equations. (4)−(6), the continuity expression is satisfied identically and momentum boundary layer equations are v∂ r u + Rλu∂ s u + λuv Introducing the following dimensionless variables due to the geometry of the physical problem η = r α + βs ν 1 2 , u = αsf (η) + βs 2 h (η), where f (η) and h(η) represent the stream functions and the primes denote the differentials. Using Equations (12) and (13) in the above boundary layer equations, we get subject to the relevant conditions which take the form.
in which M = σ B 0 2 α 2 /μ denotes the Hartmann number. Total pressure P(η) inside the boundary layer region can be expressed as It is worth noting that the above differential equations subject to By comparing Equations (23) and (24) with the existing problems [20][21][22][23], the terms f (η) and h (η) are missing as a result of incorrect similarity variable Equations (23) and (24) can be reduced to After using Equation (26) and Equation (27), we get where ω = √ 1 + M 2 . Now, interest here is to solve (28) with (25) for function h(η).

Analytic solution
Consider homotopy analysis method (HAM) in which the convergence region is given as −c ≤ h ≤ c. The series approximations for the unknown is expressed in the form

First-order solution
The zeroth-order deformation through auxiliary function H h = e −η , linear operator L h , and initial guess h 0 (η) are given as follows: such that Equation (28) satisfies the property in the form where C j (j = 1 − 4) are arbitrary constants, h h represents the auxiliary parameter, and H h the convergence control parameter. Homotopy parameter is bounded in the form 0 ≤ q ≤ 1 and the required equations with the boundary conditions can now be written as In this analysis, the embedding auxiliary parameter provides us the liberty to select, choose and control the convergence region of this solution. The acceptable range 0 ≤ h h ≤ 1 is taken for simplicity thereby the solution h(η) is obtained when h h becomes unity.

Non-linear operator
The non-linear operator for Equation (28) is expressed as

mth-order deformation
The deformation of mth-order can be written as With the above analysis, Equation (28) and Equation (25) give Thus the velocity component becomes (40) and the corresponding vertical component velocity can be recovered by inserting Equations (26) and (40) This gives the complete solution of MHD viscous flow past a quadratic stretching of flat surface. In the subsequent analysis, our focus is to address non-linear (quadratic) in curved coordinates. After eliminating the pressure term, Equations (19)-(21) yield Equations (42)-(44) are self-similar. The physical quantity which is the frictional drag coefficient C f is expressed as where Using Equations. (12) and (46) in Equation (45), one can get

Computational methodology
In this section, our focus is to present physical and plausible solutions of the three momentum equations in response to the curved structure by employing numerical approach. In the existing solution of the problem involved flat surface, the third momentum equation has been ignored without any physical justification (see Refs. [20][21][22][23]). This omission was first pointed out by Kelson [24]. He mentioned that their choice of stream function and solutions are not able to satisfy the required conditions at the surface. Therefore, such expressions do not represent the physical solutions of the system. Here we take up all three equations simultaneously in the subsequent analysis. Substituting Equation (42) Equations (42) and (49) are higher non-linear coupled partial differential equations and are only amenable to numerical solution. The initial expression for the above higher-order system is converted into an initial value problem using (f , f , f , f , h, h , h , h , h v ) T = (z 1 , z 2 , z 3 , z 4 , z 5 , z 6 , z 7 , z 8 , z 9 ) T The procedure of our numerical technique of the above system of equations including the boundary conditions gives the following expression: where λ is now in dimensionless form as (= 1/ξ + η). At this point, the shooting technique is employed with Runge-Kutta (RK) algorithm in MATLAB. The missing initial conditions remain the challenge and are determined using Newton's method until the boundary conditions in equations (16) and (17) are satisfied. An extra condition is specified as z 8 = 0 (h (∞) = 0). The initial guesses are chosen in the following form: Using the Taylor expansion in Equation (52) about η = ∞, we obtain In which subsequent conditions (f , h , h , h v ) are expanded in the form (53). After lengthy steps, the system handling η = ∞ in Jacobian matrix to enhance the convergence region while using Newton's algorithm is given asX whereX = (w 1 , w 2 , w 3 , w 4 , w 5 ) T and the iterations define below and ensuring that the condition z = 1 remains unchanged all through the loops to a fixed tolerance value less  than 10 −8 . The obtained numerical results are analysed under the effect of various characterizing parameters such as linear and non-linear stretching velocity factors "α" and "β", dimensionless radius of curvature "ξ ", and magnetic parameter "M" (Hartmann number). The values of the surface drag force (skin friction coefficients) are presented in tabular form for different curvature parameters. The results are compared with the existing literature in the limit β = 0. It is clearly shown in Table  1 that our results are in good agreement with results obtained by Abbas [28], Sanni et al. [29] and Rosca and Pop [31] and our numerical values are presented for different magnetic parameter M.

Results and discussion
In this section, the flow kinetics due to curved stretching of boundary surface is seen for velocity field u(η), the pressure distribution P(η) and the surface e s C f , in particular, the effects of  Figure 3(B) where linear term prevails. The influence of dimensionless radius of curvature on flow field is illustrated in Figure 3(C). It shows that the velocity and momentum boundary layer are found to decrease slightly with an increase in dimensionless radius of curvature. This observation can be seen as potential consequences of small curvature (large radius of curvature) in decreasing the centrifugal force and hence reduces the velocity. This observation has been emphasized in the introduction that the curvature plays an important role in improving the flow rate for the curved structures through the generation of secondary flow. Thus the effect of radius of curvature is significant along the surface due to centrifugal force that keeps the flow along the curved path. The effects of applied magnetic field on the flow field are presented in Figure 3(D) and (E). It can be seen from Figure 3(D) that for linear stretching (β = 0), the velocity decreases against the magnetic field. It is due to the opposing nature of the Lorentz force. Remarkably, a further decrease in the velocity field occurs by the imposition of non-linear stretching velocity through Figure  3(E). Thus an important value of magnetic field can be further accentuated through the consideration of non-linear velocity in controlling the flow. In fact, this observation can be seen as an additional potent result of the MHD effect in the hydromagnetic regimes. The effect of magnetic parameter, 0.3 ≤ M ≤ 1 on pressure is given in Figures 4(A and B). The graphs show that pressure decreases with the magnetic parameter due to the application of opposing Lorentz force both in the absence and presence of a non-linear part of stretching velocity. However, the non-linear part further decreases the pressure which is in agreement with the observation made for the velocity as well. In other words, the decreasing pressure decays the flow velocity. Figures  4(C, D, and E) show the effect of physical parameters α, β and ξ on P(η). Fixing either α or β at 0.3 and varying the other parameter, we observed from Figures 4(C and D) that the pressure rises with higher α or β. However, the increase is more prominent for the influence of the nonlinear stretching strength β in comparison with the linear strength α. This observation infers that the strength of the non-linear part of the stretching velocity has a significant role than the linear part. Figure 4(E) exhibits the effect of dimensionless radius of curvature ξ on the pressure in the boundary layer region. This explains the reduction of pressure to zero (agrees with the boundary layer flow conditions) as we move towards the flat surface. Finally, we make a comparative study of the stretching velocities at hand in the curved structure. We remember that for the linear stretching, the velocity and the boundary layer decrease, whereas both of these quantities increase for non-linear stretching. It is further observed that of the two strengths of linear and nonlinear parts, the non-linear part has a dominating role. These conclusions provide further insight to improve the quality of finished products in the polymer industry, to enhance and control the flow generated by the stretching of sheets specifically for the curved sheets. These results are well substantiated in Table 1 by comparing the numerical results with the published articles (see the upper part). Whereas the lower section of Table  1 presents the impact of magnetic field on skin friction coefficient. It is noticed that surface drag force enhances for increasing magnetic parameter Macross the table as we move towards the curved surface. On the other hand, it decreases for the higher dimensionless radius of curvature parameter ξ when the surface becomes flat. It shows substantial support for the observation made in Figure 3(C).

Conclusion
This study examines the steady MHD viscous flow by non-linear quadratic stretching surface. The problem is characterized by physically important parameters like the radius of curvature, magnetic field and stretching strength. The fifth-order coupled boundary layer equations are solved numerically by implementing shooting techniques with Runge-Kutta (RK) fourthorder algorithm. The significant results of these findings can be highlighted as follows: • Impact of the radius of curvature ξ has a positive effect on the velocity u(η) and pressure p(η) • Effect of magnetic parameter M decreases the velocity, the momentum boundary layer and the pressure • It is also noticed that the pressure reduces further due to higher resistive force not only because of the magnetic parameter but also with non-linear stretching strength β. • The effects of varying non-linear terms "α" and "β" on the velocity and pressure inside the boundary layer are examined intuitively. • An analytical solution of non-linear, quadratic stretching velocity for the plane surface is provided with appropriate similarity transformation variables.

Disclosure statement
No potential conflict of interest was reported by the author(s).