Influence of temperature-dependent properties on a gravity-driven thin film along inclined plate

Abstract A numerical investigation into the effect of temperature-dependent fluid properties on the thin film flow along an inclined heated plate is presented. The equations governing the coupled flow and heat transfer are formulated based on couple stress non-Newtonian model. Solutions of the coupled nonlinear differential equations are tackled numerically by using the combination of shooting method and the Fehlberg-Runge-Kutta method. Findings are presented graphically and discussed precisely.


Introduction
The importance of thin uid ow along an inclined plate in many physical and engineering environment has accounted for the increasing number of studies in recent times. These studies are necessary to provide better understanding and improvement in the theory and applications of many real-life scenarios that are based on thin lms. Some of the examples includes photography, heat exchangers, lubrications, electroplating of surfaces, and synovial movements. In view of this, Kay et al. [1] presented the thin lm ow analysis over heated inclined plane using environmental model approach. Alharbi and Naire [2] introduced the R-adaptive mesh method for solv-ing spreading thin uid by taking the surface tension at the free surface into consideration. In a study by Shuaib et al. [3] a numerical study of a thin uid ow with a leakage was presented along an inclined surface. Yu and Cheng [4] presented experimental ndings of the thin uid ow behavior on the inclined plate. Du et al. [5] performed an experiment on the gravity driven thin lm of water under constantly changing inclination along an ellipsoidal heated plate. More interesting results can be found in ref. [6][7][8][9][10] and references cited within.
Motivated by several ndings of many thermal and mechanical engineering applications, the present article is devoted to temperature dependent properties of thin lm ow. Since ndings [11][12][13][14][15][16] documented in the literature have shown over the last few years that formulations based on constant uid properties are assumptions that are only valid in cases when the heat ow has been neglected or in small but nite thermal ranges. In the context of thin lm ow, several works have been documented in the literature on thermal instability with or without temperaturedependent properties. Closely related to this work is the brilliant e ort by Makinde [16,17], in which thin lm along heated plate with inclination and the thermal instabilities associated with the gravity-driven ow was accurately determined based on aspect ratio approximation of the governing equations. However, apart from the dynamic viscosity, other uid properties are very sensitive to temperature changes as con rmed in [11][12][13][14][15][16]. As a result, the speci c objective of this article is to investigate the e ect of variation of thermal conductivity, uid temperature and internal heat generation on the ow and thermal structure of the thin gravity-driven ow which to the best of our understanding has not been addressed in spite of its various application in the thermal and lubrication community. The present work is coupled and nonlinear, therefore, the exact solution of the problem is not possible. The numerical solution of the problem has been obtained by using the combination of shooting method to convert the boundary value problem to its equivalent initial value problem and solved using the fourth order Runge-Kutta integration scheme. The result revealed the non-existence of solution as the critical values of the nonlinear terms are exceeded.
The next section of the article is devoted to the mathematical analysis, which consist of the model formulation, dimensionless process and the numerical solution of the couple problem including thermal stability. Numerical results are presented in the third section of the paper while the article is concluded in section four.

Mathematical analysis
Consider the gravity-driven ow of a temperaturedependent viscous and incompressible uid down an inclined heated surface. The heated plate is assumed to be at the angle of inclination, ϑ, to the horizontal with uniform lm thickness δ as shown in Figure 1 below. The uid dynamic viscosity, thermal conductivity, and the internal heat generation depends on temperature. Using the aspect ratio condition, the balance equations for the wave-less ow can be written as [16]: Subject to free and adiabatic ow conditions Additional terms in (1)-(3) are temperature dependent internal heat generation and thermal conductivity. In equations (1)-(3), u, ρ, g represent the uid dimensional velocity, density, and gravitational acceleration respectively, the uid dynamic viscosity µ, thermal conductivity k depends on temperature T in an exponential manner while internal heat generation Q is a linear function as follows: where µ , k , Ta are the referenced dynamic viscosity, thermal conductivity, and temperature (associated to a) respectively. Also, α , β , Q are variation constants which could take positive or negative values depending on the uid under consideration. Using the following dimensionless variables and parameters Here the dimensionless parameters U, α, θ are dimensionless velocity, viscosity variation, and dimensionless temperature respectively while β, λ, η are thermal conductivity variation, viscous heating and internal heat generation parameter respectively. Moving forward, the boundary value problem (BVP) in (6)-(7) are coupled and nonlinear. The exact solution of the problem is practically not possible; therefore, it is more convenient to obtain the numerical solution. We implement the famous shooting method on the coupled system by converting the boundary value problem to its equivalent initial value problem as follows.
With (8), the coupled BVP becomes Where ( ) represents derivative with respect to x , equation (9) is subject to the initial conditions The system of equations (9) together with the initial conditions are elegantly obtained by using Runge-Kutta with the Fehlberg numerical integration approach. It is important to note that D , are the guess values that ensures that the boundary conditions at y = are met.
Next, the skin interaction and the wall heat transfer rate along the heated wall can be expressed as follows: With (5), (11) becomes Sf = τw δρg sin α = e −αθ dU dy y= (12)

Results and discussion
In this section, graphical representations of the numerical solutions are presented for various ow parameter values. In Figure 2, the temperature pro le is represented against the thermal conductivity varying parameter. From the plot, rise in the variable thermal conductivity parameter leads to increase in the rate of heat transfer at the free end due to increasing heat ux, however, the heat at inclined plate a ects the thermal conductivity of the uid layer closer to the wall and that explains the minimal convective heat transfer in that region. Results in Figure 3 shows that convective heat transfer rate elevates the uid velocity, and this follows the same trend with the temperature pro le. In Figure 4, the e ect of viscosity varying parameter on the temperature pro le is presented. Result shows that decreasing uid viscosity enhances the uid temperature. This is correct because of increased molecular interaction of the uid particles associated with decrease in the cohesive forces as the uid viscosity decreases. Similarly, in Figure 5, thermal e ect on the uid viscosity is presented. From the result, increased viscosity variation parameter weakens cohesive force binding the uid particles, as a result, the increased kinetic energy of the uid particles enhances the deformation rate as shown in the plot. Figures 6 and 7 represent the e ect of viscous heating on the uid temperature and velocity respectively. The  result shows heat generated from frictional interaction of uid particles elevates the uid temperature and the velocity respectively. Figures 8 and 9 depict the in uence of temperature dependent internal heat generation parameter on the temperature and velocity pro le respectively. From the plots, increasing the internal heat generation parameter helps elevate both uid temperature and the thin lm velocity along the heated plate. In Figure 10, the varia-  An increase in β values shows a decrease in the criticality values (αc), and this means that the rate of heat transfer at the wall is reduced as the viscosity of the uid is increased, the same scenario is observed in Figure 11.

Conclusion
A numerical approach to investigate the thin lm ow with temperature-dependent properties along heated inclined plates subjected to free and adiabatic surface has been presented. From the numerical computation, the temperature-dependent heat source, viscosity and thermal conductivity parameters are seen to enhance both