Film Flow Problem Research Articles

A numerical inquiry of the MHD radiative thin film nanofluid flow and heat transfer augmentation for Ni and Ta nanoparticles of different shapes dispersed in C 12 H 26–C 15 H 32 over an unsteady, curved stretching surface

Thin films are a versatile technology used in various industries, including light-emitting diodes, magnetic recording media, electronic device manufacturing, and optical coating. They are crucial for studying quantum phenomena, superlattices, and multiferroic materials. Their small size allows for better thermal transport studies due to the combination of thin film flow and nanoparticles. Therefore, this study aims to investigate thin film flow and heat transfer with the effects of thermal radiations, magnetic field, and velocity slip in association with tantalum ( Ta ) and nickel ( Ni ) nanoparticles that emerged in kerosene oil ( C 12 H 26 − C 15 H 32 ) toward a curved stretching surface. A review of the pertinent literature indicates that no research has previously been done on the present problem of an unsteady thin film flow across a curved stretching surface. The problem under consideration is modeled by choosing the curvilinear coordinate system. The controlling nonlinear system of partial differential equations is transformed into the system of ordinary differential equations by using an appropriate similarity transformation and then solved numerically using the BVP 4 C technique in MATLAB . The impact of significant physical parameters on velocity, temperature, skin friction, and Nusselt numbers are depicted through graphs and radar charts. The investigation highlights the significant influence of magnetohydrodynamics ( MHD ) , thermal radiation, volumetric fraction, and velocity slip on the enhancement of nanofluid temperature. It is also noted that cylindrical-shaped Ta nanoparticles exhibited the maximum velocity, while blade-shaped Ni nanoparticles showed the lowest. Moreover, the maximum and minimum temperatures were observed for blade-shaped Ni nanoparticles and cylindrical Ta nanoparticles, respectively. Additionally, the maximum and minimum values of skin friction are found for cylindrical Ni nanoparticles and blade-shaped Ta nanoparticles, respectively. Similarly, the maximum and minimum Nusselt numbers are recorded for cylindrical Ta nanoparticles and blade-shaped Ni nanoparticles, respectively.

An Algorithm for Approximating Third-Order Ordinary Differential Equations with Applications to Thin Film Flow and Nonlinear Genesio Problems

Third-order problems have been found to model real-life phenomena such as thick film fluid flow, boundary layer problems, and nonlinear Genesio problems, to name a few. This research focuses on the development of an algorithm using a basis function that combines an exponential function with a power series. This algorithm, called the Exponential Function Induced Algorithm (EFIA), was formulated using the collocation and interpolation techniques. The theoretical analysis of the algorithm was also carried out in this research. The outcome of the analysis showed that the algorithm is consistent, convergent, and zero-stable. The EFIA was applied to solving some real-life third-order problems like thin film flow and nonlinear Genesio problems. The numerical and graphical results obtained showed that the EFIA is accurate, has high precision, and is easy to implement. Algorithm, Exponentially-fitted function, Hybrid block method, Nonlinear Genesio equations, Thin film flow problem, Third-order

Modified Fourth Derivative Block Method and its direct applications to third-order initial value problems

A theoretical order eight Modified Fourth Derivative four-step block method (MFDFBM) has been derived, analysed and numerically applied to solve multiple problems originating from Fluid Dynamics, engineering and other sciences. The MFDFBM was derived by applying collocation and interpolation techniques to a power series approximation. Further introducing fourth derivative terms at each of the collocating points yields a block method with an improved order of accuracy. It was observed that the order of the block method increases with the number of fourth derivative terms introduced into the integration interval. Numerical experiments are presented to test MFDFBM on numerical examples, including non-linear homogeneous thin film flow (NHTFF) problems and two non-linear initial value problems(IVPs). The experiments confirm the good impact of adding the fourth derivative terms, which help improve the order of accuracy of the derived MFDFBM, thereby minimising error and agreeing with analytical solution up to at least seven decimal places.

A Numerical Framework for Entropy Generation Using Second-Order Nanofluid Thin Film Flow over an Expanding Sheet: Error Estimation and Stability Analysis

Thin film flow (TFF) problems received a lot of attention in recent times. Some frequent applications of TFF include polymer and metal extraction, elastic sheet drawing, food striating, heat exchanges, and device fluidization. Further improvement and enhancement of TFF need to be examined due to its practical applications. In the current analysis, viscoelastic nanofluid thin film flow through the vertical expanding sheet in the presence of a magnetic field with entropy function has been examined. The governing equations are transformed to first-order ODEs through similarity transformation and then solved numerically by using RK4 along with the shooting technique and ND Solve method. The impact of embedded parameters is discussed using graphs and tables. Physical quantities of interest are also discussed in detail. For the numerical solution, the error estimation and the residue error are calculated for the stability and confirmation of the mathematical model.

SELECTED SINGLE-STEP HYBRID BLOCK FORMULA FOR SOLVING THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH APPLICATION IN THIN FILM FLOW

This research paper examines the derivation of selected hybrid single-step block method for the numerical integration of third order ordinary differential equations. The method has the advantage of selecting only odd off-grid points within the interval of interest. The basis function for the formula is interpolated at three selected non-grid points within a single-step interval and collocated at all points. Further analysis of the basic numerical properties were established. The method was found to be A-stable, zero-stable and consistent. The small scale errors observed from numerical experiment indicate that the derived formula has better numerical approximations than some comparable methods in literature, while its application on practical thin film flow problem also showed improved solutions.

A faster optimal solver for thin film flows

A new, multigrid-based, algorithm is proposed for the solution of the discretized lubrication equations which are widely used to model a broad class of thin film flow phenomena. The approach is based upon the use of a Newton-Krylov solver for the nonlinear algebraic systems of equations that arise following mesh-based spatial discretizations and implicit time discretizations. The novel contribution is to propose a block-based preconditioner that includes two applications of algebraic multigrid (AMG) as a key component: thus allowing AMG techniques to be applied in the solution of thin film flow problems for the first time. An implementation of this preconditioned solver is demonstrated for a typical thin film scenario of free-surface flow down a non-smooth inclined plane, considering both steady-state and time-dependent configurations. Systematic computational testing is undertaken in comparison with two other state-of-the-art multigrid methods in order to demonstrate the substantial computational advantages of the proposed algorithm.

A particularly unstable film

Analysis of the classic problem of shallow film flow on an inclined plane is revisited for a Brownian suspension. The particle phase, considered in a two-fluid model, is predicted to cause pronounced changes to the instability characteristics of the flow. These are due to an indirect effect of the non-Newtonian rheology, with the normal stresses causing migration and a viscosity stratification which strongly alters the base state from its Newtonian counterpart. Both the short- and long-wavelength inertial stability boundaries are altered, and, more strikingly, an instability at zero Reynolds number is found.

Asymptotic Analysis of Thin Film Flows on Curved Surfaces

This paper presents an in-depth asymptotic analysis of thin film flows on curved surfaces, focusing on both spherical and cylindrical geometries. The study begins with a review of the mathematical preliminaries, including differential geometry, the governing Navier-Stokes equations for thin films, and asymptotic analysis techniques. The thin film flow problem is formulated on general curved surfaces, with assumptions and simplifications such as the lubrication approximation. The governing equations are then non-dimensionalized and analyzed using perturbation methods to obtain leading-order solutions. The paper includes case studies on thin film flows on spherical and cylindrical surfaces, using hypothetical data to demonstrate the application of the theoretical models. The leading-order solutions reveal significant insights into the effects of curvature on thin film dynamics, with numerical methods used to validate and compare the results. The findings highlight the critical role of curvature in determining film thickness and pressure distribution, with practical implications for applications in coating processes, biological systems, and environmental engineering. The study concludes with a discussion on the contributions to the field of thin film flows, including the development of analytical and numerical frameworks and the practical insights gained from the case studies. Future research directions are suggested, focusing on higher-order asymptotic analysis, complex geometries, non-Newtonian fluids, experimental validation, and application-specific studies.

MHD dissipative Casson nanofluid liquid film flow due to an unsteady stretching sheet with radiation influence and slip velocity phenomenon

Abstract The problem of non-Newtonian Casson thin film flow of an electrically conducting fluid on a horizontal elastic sheet was studied using suitable dimensionless transformations on equations representing the problem. The thin film flow and heat mechanism coupled with mass transfer characteristics are basically governed by the slip velocity, magnetic field, and the dissipation phenomenon. The present numerical analysis by the shooting method was carried out to study the detailed, fully developed heat and mass transfer techniques in the laminar thin film layer by solving the competent controlling equations with eight dominant parameters for the thin liquid film. Additionally, the predicted drag force via skin-friction coefficient and Nusselt and Sherwood numbers were correlated. In view of the present study, a smaller magnetic parameter or a smaller slip velocity parameter exerts very good influence on the development of the liquid film thickness for the non-Newtonian Casson model. Furthermore, a boost in the parameter of unsteadiness causes an increase in both velocity distribution and concentration distribution in thin film layer while an increase in the same parameter causes a reduction in the film thickness. Likewise, the present results are observed to be in an excellent agreement with those offered previously by other authors. Finally, some of the physical parameters in this study, which can serve as improvement factors for heat mass transfer and thermophysical characteristics, make nanofluids premium candidates for important future engineering applications.

Three Weighted Residuals Methods for Solving the Nonlinear Thin Film Flow Problem

In this paper, the methods of weighted residuals: Collocation Method (CM), Least Squares Method (LSM) and Galerkin Method (GM) are used to solve the thin film flow (TFF) equation. The weighted residual methods were implemented to get an approximate solution to the TFF equation. The accuracy of the obtained results is checked by calculating the maximum error remainder functions (MER). Moreover, the outcomes were examined in comparison with the 4th-order Runge-Kutta method (RK4) and good agreements have been achieved. All the evaluations have been successfully implemented by using the computer system Mathematica®10.

A hybrid analytical algorithm for thin film flow problem occurring in non-Newtonian fluid mechanics

In this work, we investigate thin film flow of a third grade fluid down a inclined plane. The solution of a nonlinear boundary value problem (BVP) is derived by using an effective well organized computational scheme namely homotopy perturbation Elzaki transform method. Furthermore, this model is also resolved by Elzaki decomposition technique. The outcomes achieved by these two approaches are consistent with each other and because of that this technique may be regarded as an optional and effective scheme for determining results of linear and nonlinear BVP. Moreover, the homotopy perturbation Elzaki transform method leads over the Elzaki decomposition method since the nonlinear problems are solved without utilization of Adomian polynomials.

Numerical Study of Axisymmetric Flow and Heat Transfer in a Liquid Film over an Unsteady Radially Stretching Surface

The main emphasis on this paper is to analyze the axisymmetric flow and heat transfer in a liquid film over an unsteady radially stretching surface in the presence of a transverse magnetic field. The similarity transformations are used to reduce the highly nonlinear governing partial differential equations for momentum and energy into a set of ordinary differential equations. A numerical scheme is developed for the reduced nonlinear differential equations for the velocity and temperature fields. The literature survey shows that the present problem of thin film flow over a radially stretching sheet has not been studied before. The features of the flow and heat transfer characteristic for different values of governing parameters such as unsteadiness parameter, Prandtl number, Eckert number, and magnetic parameter are thoroughly examined. This study noticed that, by increasing the magnetic parameter and unsteadiness parameter, film thickness decreases.

Non-uniqueness solutions for the thin Carreau film flow and heat transfer over an unsteady stretching sheet

The present paper probed the problem of thin liquid film flow and heat transfer in the Carreau fluid past a permeable stretching sheet. The similarity transformation reduced the partial differential equations into a system of ordinary differential equations, which was then solved numerically by a collocation method. The influence of the governing parameters towards the model was discussed and presented graphically. The non-uniqueness solutions are observable as the governing parameters vary.

The operational matrix of Legendre polynomials for solving nonlinear thin film flow problems

In this paper, the operational matrix with Legendre polynomials method will be implemented to get the approximate solution for the nonlinear thin film flow (NTFF) of third grade problem. This method provides an approximate solution by converting the nonlinear differential equation into nonlinear algebraic equations system which can be solved by using Mathematica® 10. To illustrate the accuracy and efficiency of the proposed method, the maximum error remainder was calculated. The results obtained from using the proposed method have proven to be accurate, reliable and effective. In addition, the approximate solution is compared with the fourth order Runge-Kutta method (RK4) and good agreement has been achieved.

Triple Solutions of Carreau Thin Film Flow with Thermocapillarity and Injection on an Unsteady Stretching Sheet

Thin films and coatings which have a high demand in a variety of industries—such as manufacturing, optics, and photonics—need regular improvement to sustain industrial productivity. Thus, the present work examined the problem of the Carreau thin film flow and heat transfer with the influence of thermocapillarity over an unsteady stretching sheet, numerically. The sheet is permeable, and there is an injection effect at the surface of the stretching sheet. The similarity transformation reduced the partial differential equations into a system of ordinary differential equations which is then solved numerically by the MATLAB boundary value problem solver bvp4c. The more substantial effect of injection was found to be the reduction of the film thickness at the free surface and development of a better rate of convective heat transfer. However, the increment in the thermocapillarity number thickens the film, reduces the drag force, and weakens the rate of heat transfer past the stretching sheet. The triple solutions are identified when the governing parameters vary, but two of the solutions gave negative film thickness. Detecting solutions with the most negative film thickness is essential because it implies the interruption in the laminar flow over the stretching sheet, which then affects the thin film growing process.

On Two-Derivative Runge–Kutta Type Methods for Solving u‴ = f(x,u(x)) with Application to Thin Film Flow Problem

A class of explicit Runge–Kutta type methods with the involvement of fourth derivative, denoted as two-derivative Runge–Kutta type (TDRKT) methods, are proposed and investigated for solving a special class of third-order ordinary differential equations in the form u ‴ ( x ) = f ( x , u ( x ) ) . In this paper, two stages with algebraic order four and three stages with algebraic order five are presented. The derivation of TDRKT methods involves single third derivative and multiple evaluations of fourth derivative for every step. Stability property of the methods are analysed. Accuracy and efficiency of the new methods are exhibited through numerical experiments.

Solution of Third Grade Thin Film Flow Using Algorithmic Differentiation

We solve the steady thin film flow problem for a third grade fluid by using Taylor series and shooting. Neither finite difference formulas nor lengthy analytical expressions are used for calculating the derivatives needed. Instead, exact derivatives are computed directly through algorithmic differentiation, which leads to recursive formulas for the derivatives. The method avoids round-off effects and the use of symbolic manipulation systems. Therefore, the method requires much less computational effort when compared to other existing methods for producing results of comparable accuracy. Our numerical results are in excellent agreement with the several approximate solutions obtained previously.

Numerical treatment of a physical problem in fluid film flow using the differential transformation method

The problem of thin liquid film flow over a heated horizontal stretching surface permitted by uniform transverse magnetic field in the presence of variable heat flux is studied numerically. The ordinary differential equations describing the physical problem are solved approximately by the differential transformation method (DTM). A study of the convergence analysis of the proposed method is given. The effects of the various parameters governing the thin film flow and heat transfer in this study are discussed and presented graphically. Our paper contributes to the existing literature by introducing the numerical treatment using the DTM for the nonlinear system of ordinary differential equations, which describe a thin film fluid flow affected by the variable heat flux.

A Potential Field Description for Gravity-Driven Film Flow over Piece-Wise Planar Topography

Models based on a potential field description and corresponding first integral formulation, embodying a reduction of the associated dynamic boundary condition at a free surface to one of a standard Dirichlet-Neumann type, are used to explore the problem of continuous gravity-driven film flow down an inclined piece-wise planar substrate in the absence of inertia. Numerical solutions of the first integral equations are compared with analytical ones from a linearised form of a reduced equation set resulting from application of the long-wave approximation. The results obtained are shown to: (i) be in very close agreement with existing, comparable experimental data and complementary numerical predictions for isolated step-like topography available in the open literature; (ii) exhibit the same qualitative behaviour for a range of Capillary numbers and step heights/depths, becoming quantitively similar when both are small. A novel outcome of the formulation adopted is identification of an analytic criteria enabling a simple classification procedure for specifying the characteristic nature of the free surface disturbance formed; leading subsequently to the generation of a related, practically relevant, characteristic parameter map in terms of the substrate inclination angle and the Capillary number of the associated flow.

An optimal homotopy asymptotic method applied to the nonlinear thin film flow problems

Purpose This paper aims to discuss the approximate solution of the nonlinear thin film flow problems. A new analytic approximate technique for addressing nonlinear problems, namely, the optimal homotopy asymptotic method (OHAM), is proposed and used in an application to the nonlinear thin film flow problems. Design/methodology/approach This approach does not depend upon any small/large parameters. This method provides a convenient way to control the convergence of approximation series and to adjust convergence regions when necessary. Findings The obtained solutions show that the OHAM is more effective, simpler and easier than other methods. The results reveal that the method is explicit. By applying the method to nonlinear thin film flow problems, it was found to be simpler in applicability, and more convenient to control convergence. Therefore, the method shows its validity and great potential for the solution of nonlinear problems in science and engineering. Originality/value The proposed method is tested upon nonlinear thin film flow equation from the literature and the results are compared with the available approximate solutions including Adomian decomposition method (ADM), homotopy perturbation method, modified homotopy perturbation method and HAM. Moreover, the exact solution is compared with the available numerical solutions. The graphical representation of the solution is given by Maple and is physically interpreted.