Abstract
This study determines Lie point symmetries for differential equations that mathematically express a time-dependent thin film fluid flow with internal heating and thermal radiation to construct invariants. These invariants are used in the derivation of similarity transformations for reducing the flow equations into systems of equations that possess only one independent variable. The homotopy analysis method is employed to analytically solve the reduced system of equations. The new similarity transformations and the corresponding analytical solutions comprehensively consider flow dynamics and heat transfer under multiple physical conditions. These solutions are presented graphically to demonstrate the effects of variations in the radiative heat flux with internal heating on the flow dynamics and heat transfer properties. Moreover, the variations in fluid dynamics are described graphically using the obtained analytical homotopy solution under different values of the unsteadiness parameter and Prandtl number.
1. Introduction
A boundary layer forms whenever a fluid flows over a body. This 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 fluidization, extrusion of polymers and metals, and drawing of polymer sheets. Advancements in various industrial processes have been made based on the characteristics of unsteady fluid flow and the rate of heat transfer in a thin liquid film on stretching sheets. Hence, considering the importance of unsteady thin film flow in several engineering applications, researchers have actively investigated momentum and heat transfer during the past several decades. Wang [1] was the first to study the thin film hydrodynamics on a stretching surface of an unsteady flow. By employing similarity transformations, he reduced the Navier–Stokes equations into a system of ordinary differential equations (ODEs) and solved them numerically. The reason for such a reduction in the flow 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 film flow over stretching surfaces and employing a shooting subroutine [3] for the solution of ODEs. Dandapat et al. [4] traced the effects of thermocapillarity on the dynamics of unsteady fluid flow in a liquid film on a stretching surface. Wang [5] obtained analytical solutions for the flow dynamics considered in [2] by employing the homotopy analysis method (HAM) [6]. In [7], Dandapat et al. considered changes in flow, temperature distribution, and skin friction coefficient 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 effects 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 fluid flow in a transverse magnetic field. They imposed a few more physical constraints, namely variable viscosity and thermal conductivity, in studying the dynamics of flow and heat. They 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 [11–13], the authors considered internal heat generation, thermal radiation, and variable heat flux separately to analyze their impacts on the velocity field and temperature distribution. In [14], Liu and Megahed studied the combined effects of internal heat generation, thermal radiation, and variable heat flux on the velocity field, temperature distribution, and skin friction coefficient in liquid film flow over an unsteady stretching sheet. Idrees et al. [15] used a similarity transformation to reduce the Navier–Stokes equations of the MHD fluid flow with variable physical properties into ODEs. Using these reduced equations, they investigated the effects of variable viscosity and thermal conductivity on unsteady MHD fluid flow with heat transfer using the HAM and shooting method. The authors found good agreement between the results. Rehman et al. [16] investigated the dynamics of unsteady flow and the temperature of a nanofluid on a stretching sheet. The effects of various parameters on velocity and temperature were investigated using an optimal HAM [17].
The dependent and independent variables of differential equations can be reduced if some Lie point symmetries are obtainable for them. Moreover, for differential 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]. These Lie point symmetries are the invertible changes in the dependent and independent variables, which, when applied to the corresponding differential 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 differential equations (PDEs) of the flow to ODEs [20–26]. Similar studies based on the Lie symmetry method were conducted in [27–29], where two successive invertible mappings of the flow model were achieved, and an analytic solution procedure was applied to present the velocity and temperature profiles.
In this study, we developed seven Lie symmetry generators for time-dependent thin film fluid flow with internal heating and thermal radiation. These Lie symmetry generators were obtained using MAPLE. When investigating the invariance of the associated boundary conditions of the flow model, we observed that single symmetries do not allow the stretching sheet velocity and temperature to be functions of time and space variables. Therefore, we used linear combinations of the obtained symmetries because they are also symmetries of the flow PDEs. We employed six linear combinations (two symmetries for each combination) to determine the Lie similarity transformations. Using these invertible maps, we reduced the flow 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 flow PDEs. Using the HAM, we constructed analytic solutions of the obtained ODEs. The velocity and temperature profiles are shown graphically with variations in the unsteadiness parameter , Prandtl number , radiative heat flux , and internal heating parameter . We obtained multiple solutions here because we are providing more than one Lie similarity transformation that is different from those that already exist. These transformations reduce the flow 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 flow.
The remainder of this paper is organized as follows. Section 2 describes the formulation of the fluid flow problem. Section 3 presents the derivation of the Lie point symmetries, corresponding invariants, and similarity transformations, along with the reduction of the considered fluid flow 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.
2. Governing Equation
Consider a thin film of a thermally radiative Newtonian fluid 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. The unsteady stretching of the elastic sheet along the -axis at with velocity causes fluid motion within the film. The velocity and temperature fields of the thin liquid film are governed [14] by the following time-dependent boundary layer equations for mass, momentum, and energy conservation:where u and are the velocities in the x and y directions, respectively, and the subscripts and denote the partial derivatives with respect to these variables. Further, represent the dynamic viscosity, density, thermal conductivity, specific heat at constant pressure, temperature of the fluid, and radiative heat flux, respectively. The numerical value of may represent the internal heat generation or absorption per unit volume.
In the radiative heat flux , and . is the heat generation parameter (when positive) or heat absorption parameter (when negative), depending on the temperature, is the Stefan–Boltzmann constant, and is the mean absorption coefficient. The following boundary conditions are considered in the flow model:
The flow diagram is shown in Figure 1.
3. 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]. These symmetry generators and associated Lie transformations leave the differential equations and corresponding boundary conditions invariant. The system of PDEs (1) and associated conditions (3) possess the following seven Lie point symmetries:
These 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. The complete procedure for constructing the Lie similarity transformations can be found in [27, 28, 31]. The Lie similarity transformations and their corresponding ODEs for system (1) are presented in Table 1.
The boundary conditions for the ODE system cases 1–6 in Table 1 arewhere the prime symbols in all systems in Table 1 and the above conditions represent the derivative with respect to the similarity variable ; and represent the radiative heat flux and internal heating parameter, respectively; is the Prandtl number; is the dimensionless film thickness, which is equal to the square of ; and is the dimensionless unsteadiness parameter, which is equal to the ratio of to .
4. Solution Approach
In this section, all the systems of ODEs in Table 1 are analytically solved using the HAM. The first step is the derivation of HAM initial functions and , which are given in [5] and expressed asfor cases 1–6 in Table 1. The second step is to construct the -order deformation equations and integrate them to determine the solution built on the initial functions given above. The detailed procedure for constructing the -order deformation equations can be found in [5]. For example, the ODEs for case 1 of Table 1 are written aswhere and are nonzero auxiliary parameters that must have identical signs for a convergent solution and and are auxiliary functions that are normally set as 1. Moreover, and are linear operators and
The first equation is the same for all systems in Table 1; therefore, is the same, which is expressed as
for all the second equations in Table 1 are given in Table 2. The boundary conditions based on [5] are as follows:
The -order approximation of and are expressed as
The accuracy of these approximations can be improved by increasing the order of the HAM. The -order approximation of a function is the sum of all the approximate values of that function, that is, . The same method can be employed for the analytical solutions of all cases listed in Table 1.
5. Results and Discussion
We solved the systems of nonlinear ODEs listed in Table 1 subject to boundary conditions (10) by applying a -order HAM. These solutions were constructed for different values of the unsteadiness parameter, Prandtl number, radiative heat flux, and internal heating parameter. All derived solutions are illustrated graphically and tabulated. The plots of the velocity and temperature profiles require appropriate values of auxiliary parameters and .
The 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 flux, and internal heating parameter were obtained.
The curves for the first equations of cases 1–6 are shown in Figure 2, in which a range for the parameter that generates a solution for the flow equation given in all cases in Table 1 is provided. However, the unsteadiness parameter should be greater than .
A valid curve implies that the HAM can provide a solution under the considered conditions, and a valid curve exists when the unsteadiness parameter is greater than . It is evident from Figure 3 that is directly proportional to the unsteadiness parameter S for cases 1–6. The literature indicates that in viscous fluids, the flow boundary layer thickness increases when the fluid velocity decreases. Moreover, Figure 2 and Table 3 depict the increase in dimensionless film thickness and reduced skin friction coefficient with an increase in the unsteadiness parameter. However, the rate of increase in the film thickness slows down when the unsteadiness parameter and this phenomenon holds true for all cases.
5.1. Variation in Temperature with Unsteadiness Parameter
The impact of the variation in the unsteadiness parameter on the temperature profile is illustrated in Figure 4 for all systems (cases 1–6). The nondimensional temperature parameter 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.
5.2. Variation in Temperature with Prandtl Number
We drew the temperature profiles considering various Prandtl numbers, namely, , as displayed in Figure 5. It has been shown in the literature for this type of flow that an inverse relationship exists between and , which is evident for cases 1, 2, 4, and 5. However, the temperature profiles for cases 3 and 6 exhibit opposite trends.
5.3. Variation in Temperature with Internal Heating Parameter
Figure 6 illustrates how the internal heat generation affects the temperature transfer rates. When the internal heating parameter increases, the temperature also increases, except in case 3, in which the system is not affected 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.
5.4. Variation in Temperature with Radiative Heat Flux
The effect of varying the radiative heat flux on the systems in cases 1, 2, and 5 is negligible, but it can be seen in Table 4. In case 4, increases with while for cases 3 and 6, an increase in causes a decrease in which is evident in Figure 7.
6. Conclusion
This study obtained lie point symmetries for a model representing the unsteady flow and heat transfer on a stretching surface with internal heating and radiation. Seven Lie point symmetries were obtained for the flow equations, and by employing the linear combinations of the Lie symmetries, the invariants were extracted. There were 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 different sets of similarity transformations that enable the reduction of the flow and heat equations to ODEs were presented. The analysis shows that six different types of reduced equations exist through the Lie similarity transformations, corresponding to the flow 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 flow of PDEs into ODEs. These solutions are graphically illustrated and tabulated to reveal the flow and heat transfer profiles. Among all the systems of reduced equations, only a few followed the same trends for the flow and heat transfer in response to the physical parameters and numbers when compared to the results presented in Reference [14]. Unsteadiness causes fluctuations in the flow, and therefore, it acts as a resistive force against the fluid flow, thereby further resulting in energy loss. This is evident from the presented temperature profiles with variations in unsteadiness , whereas a few of the obtained systems do not exhibit this behavior. The profiles corresponding to the systems in the considered cases were also constructed by considering the variations in the momentum diffusivity to thermal diffusivity ratio . With the dominance of the momentum diffusivity in this ratio, we observed that temperature gradients developed gradually, whereas for a few systems, this phenomenon developed rapidly. The radiative heat flux determines the rate of heat transfer, helps in making accurate predictions regarding the heat transfer in a thin film, and possesses this type of behavior. The response of heat transfer toward this parameter is dependent on the flow temperature. Hence, owing to the different temperature conditions that are imposed through the Lie similarity transformations presented in this study on the systems under consideration, we obtained different responses. In other words, at some instances, the systems were emissive, whereas at other instances they were absorbent. Thus, the internal heating parameter effect can be predicted in terms of the increment in the fluid temperature, and for all reduced systems of ODEs corresponding to the considered flow equations, the heat transfer rate was found to be directly proportional to the internal heating parameter . However, there is a mixed pattern with respect to these parameters and numbers in the temperature profiles. Note that the temperature profiles 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 flow with heat transfer in a thin film on a stretching sheet under the influence of internal heating and radiation. The Lie procedure reveals multiple solutions, thereby enabling the optimization of flow dynamics with heat transfer in a physical setup. In the future, these procedures can be further exploited to determine the flow and heat profiles by introducing arbitrary constants as coefficients of the linear combinations of Lie symmetries. Some of these constants can be obtained from the resulting similarity transformations and systems of ODEs. The presence of these constants enables control of the rate of unsteady flow and heat in thin films over stretching sheets according to the requirements.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2022R1C1C2003637) (to K.S.K.) RS-2023-00210403 (J.H.B) and Pusan National University Research Grant, 2022 (J.H.B).