Analytical Solutions for Unsteady Thin Film Flow with Internal Heating and Radiation

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Introduction
A boundary layer forms whenever a fuid fows over a body.Tis concept on the formation of the hydrodynamic and thermal boundary layers is being used in several engineering processes, such as cooling of the metallic plates, coating, continuous casting, reactor fuidization, extrusion of polymers and metals, and drawing of polymer sheets.Advancements in various industrial processes have been made based on the characteristics of unsteady fuid fow and the rate of heat transfer in a thin liquid flm on stretching sheets.Hence, considering the importance of unsteady thin flm fow in several engineering applications, researchers have actively investigated momentum and heat transfer during the past several decades.Wang [1] was the frst to study the thin flm hydrodynamics on a stretching surface of an unsteady fow.By employing similarity transformations, he reduced the Navier-Stokes equations into a system of ordinary diferential equations (ODEs) and solved them numerically.Te reason for such a reduction in the fow equations that possess three independent variables and the application of numerical procedures to the reduced equations that contain only one independent variable is the nonlinearity of both.Andersson et al. [2] extended the work of Wang by conducting a heat transfer analysis on a liquid unsteady flm fow over stretching surfaces and employing a shooting subroutine [3] for the solution of ODEs.Dandapat et al. [4] traced the efects of thermocapillarity on the dynamics of unsteady fuid fow in a liquid flm on a stretching surface.Wang [5] obtained analytical solutions for the fow dynamics considered in [2] by employing the homotopy analysis method (HAM) [6].In [7], Dandapat et al. considered changes in fow, temperature distribution, and skin friction coefcient as a consequence of temperature-dependent variations in viscosity, thermal conductivity, and thermocapillarity.Liu and Andersson [8] considered a more generalized form of temperature variation and investigated the efects of the Prandtl number and nondimensional unsteadiness parameter on the temperature distribution under various special circumstances.Mahmoud and Megahed [9] conducted research on unsteady magnetohydrodynamic (MHD) non-Newtonian fuid fow in a transverse magnetic feld.Tey imposed a few more physical constraints, namely variable viscosity and thermal conductivity, in studying the dynamics of fow and heat.Tey concluded that the velocity and temperature are inversely and directly proportional, respectively, to the magnetic parameter.A similar study on viscous dissipation was conducted by Aziz and Hashim [10], who concluded that the temperature is directly proportional to the Eckert number.
In [11][12][13], the authors considered internal heat generation, thermal radiation, and variable heat fux separately to analyze their impacts on the velocity feld and temperature distribution.In [14], Liu and Megahed studied the combined efects of internal heat generation, thermal radiation, and variable heat fux on the velocity feld, temperature distribution, and skin friction coefcient in liquid flm fow over an unsteady stretching sheet.Idrees et al. [15] used a similarity transformation to reduce the Navier-Stokes equations of the MHD fuid fow with variable physical properties into ODEs.Using these reduced equations, they investigated the efects of variable viscosity and thermal conductivity on unsteady MHD fuid fow with heat transfer using the HAM and shooting method.Te authors found good agreement between the results.Rehman et al. [16] investigated the dynamics of unsteady fow and the temperature of a nanofuid on a stretching sheet.Te efects of various parameters on velocity and temperature were investigated using an optimal HAM [17].
Te dependent and independent variables of diferential equations can be reduced if some Lie point symmetries are obtainable for them.Moreover, for diferential equations, an order reduction can be performed, conservation laws can be constructed, and nonlinear equations can be linearized with the obtained Lie symmetries [18,19].Tese Lie point symmetries are the invertible changes in the dependent and independent variables, which, when applied to the corresponding diferential equations, leave them invariant with each obtained Lie point symmetry, invariants, which are the main components in the construction of similarity transformations, exist.Such transformations are employed to map partial diferential equations (PDEs) of the fow to ODEs [20][21][22][23][24][25][26].Similar studies based on the Lie symmetry method were conducted in [27][28][29], where two successive invertible mappings of the fow model were achieved, and an analytic solution procedure was applied to present the velocity and temperature profles.
In this study, we developed seven Lie symmetry generators for time-dependent thin flm fuid fow with internal heating and thermal radiation.Tese Lie symmetry generators were obtained using MAPLE.When investigating the invariance of the associated boundary conditions of the fow model, we observed that single symmetries do not allow the stretching sheet velocity and temperature to be functions of time and space variables.Terefore, we used linear combinations of the obtained symmetries because they are also symmetries of the fow PDEs.We employed six linear combinations (two symmetries for each combination) to determine the Lie similarity transformations.Using these invertible maps, we reduced the fow PDEs into a system of ODEs, along with the conversion of the corresponding boundary conditions.In such mappings, the Lie similarity transformations reduce two independent variables and one dependent variable of the fow PDEs.Using the HAM, we constructed analytic solutions of the obtained ODEs.Te velocity and temperature profles are shown graphically with variations in the unsteadiness parameter S, Prandtl number Pr, radiative heat fux R, and internal heating parameter B. We obtained multiple solutions here because we are providing more than one Lie similarity transformation that is diferent from those that already exist.Tese transformations reduce the fow equations into multiple types of ODE systems.To the best of our knowledge, these solutions have not yet been generated using the Lie symmetry method for this type of fow.
Te remainder of this paper is organized as follows.Section 2 describes the formulation of the fuid fow problem.Section 3 presents the derivation of the Lie point symmetries, corresponding invariants, and similarity transformations, along with the reduction of the considered fuid fow problem.Section 4 describes the analytical solutions of the ODE systems derived through double reductions.Section 5 presents the results and a discussion of the obtained solutions.Finally, Section 6 summarizes the conclusions drawn from this study.

Governing Equation
Consider a thin flm of a thermally radiative Newtonian fuid with an internal source of heat generation on a horizontal elastic sheet emerging from a narrow opening at the origin.Figure 1 depicts this process.Te unsteady stretching of the elastic sheet along the x-axis at y � 0 with velocity U(x, t) causes fuid motion within the flm.Te velocity and temperature felds of the thin liquid flm are governed [14] by the following time-dependent boundary layer equations for mass, momentum, and energy conservation: where u and v are the velocities in the x and y directions, respectively, and the subscripts t, x, and y denote the partial derivatives with respect to these variables.Further, μ, ρ, κ, C p , T, and q r represent the dynamic viscosity, density, thermal conductivity, specifc heat at constant pressure, temperature of the fuid, and radiative heat fux, respectively.Te numerical value of Q may represent the internal heat generation or absorption per unit volume.
In the radiative heat fux q r , T 4 � 4T 3 0 T − 3T 4 0 and  U � x/t.B is the heat generation parameter (when positive) or heat absorption parameter (when negative), depending on the temperature, σ * is the Stefan-Boltzmann constant, and κ * is the mean absorption coefcient.Te following boundary conditions are considered in the fow model: ( Te fow diagram is shown in Figure 1.

Construction of Similarity Transformations
In this study, the Lie point symmetries are derived using the "PDEtools" package of the MAPLE software.A detailed algebraic procedure for deriving the Lie point symmetries of PDEs can be found in [27,28,30,31].Tese symmetry generators and associated Lie transformations leave the diferential equations and corresponding boundary conditions invariant.Te system of PDEs (1) and associated conditions (3) possess the following seven Lie point symmetries: Tese Lie point symmetries are used to determine the invariants corresponding to each of them or their linear combinations.We used linear combinations to construct the transformations.Te complete procedure for constructing the Lie similarity transformations can be found in [27,28,31].Te Lie similarity transformations and their corresponding ODEs for system (1) are presented in Table 1.
Te boundary conditions for the ODE system cases 1-6 in Table 1 are where the prime symbols in all systems in Table 1 and the above conditions represent the derivative with respect to the similarity variable η; R � (16σ * T 3 0 /3k * κ) and B represent the radiative heat fux and internal heating parameter, respectively; Pr � μC p /κ is the Prandtl number; Υ is the dimensionless flm thickness, which is equal to the square of β; and S is the dimensionless unsteadiness parameter, which is equal to the ratio of a to b.

Journal of Mathematics
Bold letters represent the diferential operators. 4 Journal of Mathematics

Solution Approach
In this section, all the systems of ODEs in Table 1 are analytically solved using the HAM.Te frst step is the derivation of HAM initial functions f 0 (η) and θ 0 (η), which are given in [5] and expressed as for cases 1-6 in Table 1.Te second step is to construct the m th -order deformation equations and integrate them to determine the solution built on the initial functions given above.Te detailed procedure for constructing the m th -order deformation equations can be found in [5].For example, the ODEs for case 1 of Table 1 are written as where h f and h θ are nonzero auxiliary parameters that must have identical signs for a convergent solution and H f (η) and H θ (η) are auxiliary functions that are normally set as 1.Moreover, L f � z 3 /zη 3 and L θ � z 2 /zη 2 are linear operators and Te frst equation is the same for all systems in Table 1; therefore, R f,m (η) is the same, which is expressed as

Case
R θ,m (η) R θ,m (η) for all the second equations in Table 1 are given in Table 2. Te boundary conditions based on [5] are as follows: Te m th -order approximation of f(η), θ(η), and c are expressed as Te accuracy of these approximations can be improved by increasing the order of the HAM.Te m th -order approximation of a function is the sum of all the approximate values of that function, that is, f 0 + f 1 + f 2 + . . .+ f m .Te same method can be employed for the analytical solutions of all cases listed in Table 1.

Results and Discussion
We solved the systems of nonlinear ODEs listed in Table 1 subject to boundary conditions (10) by applying a 10 th -order HAM.Tese solutions were constructed for diferent values of the unsteadiness parameter, Prandtl number, radiative heat fux, and internal heating Table 3: Variation of dimensionless flm thickness and reduced skin friction coefcient with respect to S for cases 1-6.Te spans of their ranges depend on the values of the other variables.Hence, the curves for multiple values of the unsteadiness parameter, Prandtl number, radiative heat fux, and internal heating parameter were obtained.
Te h f curves for the frst equations of cases 1-6 are shown in Figure 2, in which a range for the parameter h f that generates a solution for the fow equation given in all cases in Table 1 is provided.However, the unsteadiness parameter should be greater than 2.0.
A valid h f curve implies that the HAM can provide a solution under the considered conditions, and a valid h f curve exists when the unsteadiness parameter is greater than 2. It is evident from Figure 3 that f ′ (η) is directly proportional to the unsteadiness parameter S for cases 1-6.Te literature indicates that in viscous fuids, the fow boundary layer thickness increases when the fuid velocity decreases.Moreover, Figure 2 and Table 3 depict the increase in dimensionless flm thickness β and reduced skin friction coefcient − f ″ (0) with an increase in the unsteadiness parameter.However, the rate of increase in the flm thickness slows down when the unsteadiness parameter S > 3 and this phenomenon holds true for all cases.

Variation in Temperature with Unsteadiness Parameter S.
Te impact of the variation in the unsteadiness parameter S on the temperature profle is illustrated in Figure 4 for all systems (cases 1-6).Te nondimensional temperature parameter θ(η) is directly proportional to the unsteadiness parameter for cases 2, 3, and 6.In contrast, an inverse relationship exists for cases 1, 4, and 5.However, the change for cases 1, 2, and 5 is small compared to that of the remaining cases in Figure 4. We drew the temperature profles considering various Prandtl numbers, namely, Pr � 0.32, 0.64, and 1.28, as displayed in Figure 5.It has been shown in the literature for this type of fow that an inverse relationship exists between θ(η) and Pr, which is evident for cases 1, 2, 4, and 5.However, the temperature profles for cases 3 and 6 exhibit opposite trends.

Variation in Temperature with
Internal Heating Parameter B. Figure 6 illustrates how the internal heat generation afects the temperature transfer rates.When the internal heating parameter increases, the temperature also increases, except in case 3, in which the system is not afected by this parameter.It can be clearly seen from Table 4 that this phenomenon occurs because of the nonexistence of the heat generation parameter in the system in case 3.  Te efect of varying the radiative heat fux on the systems in cases 1, 2, and 5 is negligible, but it can be seen in Table 4.In case 4, θ(η) increases with R while for cases 3 and 6, an increase in R causes a decrease in θ(η), which is evident in Figure 7.

Conclusion
Tis study obtained lie point symmetries for a model representing the unsteady fow and heat transfer on a stretching surface with internal heating and radiation.Seven Lie point symmetries were obtained for the fow equations, and by employing the linear combinations of the Lie symmetries, the invariants were extracted.Tere were 21 such combinations, and of these, only seven provided the desired formats for the stretching sheet velocity, temperature, and thickness, which are suitable for double reduction through the Lie symmetry procedure.Using the linear combinations, we obtained invariants that were combined to provide similarity transformations.Subsequently, seven diferent sets of similarity transformations that enable the reduction of the fow and heat equations to ODEs were presented.Te analysis shows that six diferent types of reduced equations exist through the Lie similarity transformations, corresponding to the fow model, which, to the best of our knowledge, have not yet been presented or studied.Furthermore, homotopy analytical solutions were derived for the classes of ODEs presented by reducing the fow of PDEs into ODEs.Tese solutions are graphically illustrated and tabulated to reveal the fow and heat transfer profles.Among all the systems of reduced equations, only a few followed the same trends for the fow and heat transfer in response to the physical parameters and numbers when compared to the results presented in Reference [14].Unsteadiness causes fuctuations in the fow, and therefore, it acts as a resistive force against the fuid fow, thereby further resulting in energy loss.Tis is evident from the presented temperature profles with variations in unsteadiness S, whereas a few of the obtained systems do not exhibit this behavior.Te θ(η) profles corresponding to the systems in the considered cases were also constructed by considering the variations in the momentum difusivity to thermal difusivity ratio Pr.With the dominance of the momentum difusivity in this ratio, we observed that temperature gradients developed gradually, whereas for a few systems, this 10 Journal of Mathematics phenomenon developed rapidly.Te radiative heat fux R determines the rate of heat transfer, helps in making accurate predictions regarding the heat transfer in a thin flm, and possesses this type of behavior.Te response of heat transfer toward this parameter is dependent on the fow temperature.Hence, owing to the diferent temperature conditions that are imposed through the Lie similarity transformations presented in this study on the systems under consideration, we obtained diferent responses.In other words, at some instances, the systems were emissive, whereas at other instances they were absorbent.Tus, the internal heating parameter efect can be predicted in terms of the increment in the fuid temperature, and for all reduced systems of ODEs corresponding to the considered fow equations, the heat transfer rate was found to be directly proportional to the internal heating parameter B. However, there is a mixed pattern with respect to these parameters and numbers in the temperature profles.Note that the temperature profles follow the same patterns for all systems of reduced equations with variations in the internal heating parameter, except for one; hence, there is no change in this case.Furthermore, this study provided multiple analytical solutions for equations describing unsteady fow with heat transfer in a thin flm on a stretching sheet under the infuence of internal heating and radiation.Te Lie procedure reveals multiple solutions, thereby enabling the optimization of fow dynamics with heat transfer in a physical setup.In the future, these procedures can be further exploited to determine the fow and heat profles by introducing arbitrary constants as coefcients of the linear combinations of Lie symmetries.Some of these constants can be obtained from the resulting similarity transformations and systems of ODEs.Te presence of these constants enables control of the rate of unsteady fow and heat in thin flms over stretching sheets according to the requirements.

Table 1 :
Similarity transformations and transformed equations.

Table 2 :
m th -order deformation equations for the 2 nd ODEs in cases 1-6.