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The purpose of this paper is to first deduce a new form of the exact analytic solution of the well-known nonlinear second-order differential equation subject to a set of mixed nonlinear Robin and Neumann boundary conditions that model the thin film flows of fourth-grade fluids, and second to compare the approximate analytic solutions by the Adomian decomposition method (ADM) with the new exact analytic solution to validate its accuracy for parametric simulations of the thin film fluid flows, even for more complex models of non-Newtonian fluids in industrial applications.

The approach to calculating a new form of the exact analytic solution of thin film fluid flows rests upon a sequence of transformations including the modification of the classic technique due to Scipione del Ferro and Niccolò Fontana Tartaglia. Next the authors establish a lemma that justifies the new expression of the exact analytic solution for thin film fluid flows of fourth-grade fluids. Second, the authors apply a modification of the systematic ADM to quickly and easily calculate the sequence of analytic approximate solutions for this strongly nonlinear model of thin film flow of fourth-grade fluids. The ADM has been previously demonstrated to be eminently practical with widespread applicability to frontier problems arising in scientific and engineering applications. Herein, the authors seek to establish the relative merits of the ADM in the context of the thin film flows of fourth-grade fluids.

The ADM is shown to closely agree with the new expression of the exact analytic solution. The authors have calculated the error remainder functions and the maximal error remainder parameters in the error analysis to corroborate the solutions. The error analysis demonstrates the rapid rate of convergence and that we can approximate the exact solution as closely as we please; furthermore the rate of convergence is shown to be approximately exponential, and thus only a low-stage approximation will be adequate for engineering simulations as previously documented in the literature.

This paper presents an accurate work for solving thin film flows of fourth-grade fluids. The authors have compared the approximate analytic solutions by the ADM with the new expression of the exact analytic solution for this strongly nonlinear model. The authors commend this technique for more complex thin film fluid flow models.

The aim of this paper is to study the effect of deterministic roughness and small elastic deformation of surface on flow rates, load capacity and coefficient of friction in Rayleigh step bearing under thin film lubrication.

Reynolds equation, pressure-density relationship, pressure-viscosity relationship and film thickness equation are discretized using finite difference method. Progressive mesh densification (PMD) method is applied to solve the related equations iteratively.

The nature and shape of roughness play a significant role in pressure generation. It has been observed that square roughness dominates the pressure generation for all values of minimum film thickness. Deformation more than 100 nm in bounding surfaces influences the film formation and pressure distribution greatly. Divergent shapes of film thickness in step zone causes a delay of pressure growth and reduces the load capacity with decreasing film thickness. The optimum value of film thickness ratio and step ratios have been found out for the maximum load capacity and minimum coefficient of friction, which are notably influenced by elastic deformation of the surface.

It is expected that these findings will help in analysing the performance parameters of a Rayleigh step bearing under thin film lubrication more accurately. It will also help the designers, researchers and manufacturers of bearings.

Most of the previous studies have been limited to sinusoidal roughness and thick film lubrication in Rayleigh step bearing. Effect of small surface deformation due to generated pressure in thin film lubrication is significant, as it influences the performance parameters of the bearing. Different wave forms such as triangular, sawtooth, sinusoidal and square formed during finishing operations behaves differently in pressure generation. The analysis of combined effect of roughness and small surface deformation has been performed under thin film lubrication for Rayleigh step bearing using PMD as improved methods for direct iterative approach.

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.

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.

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.

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.

The purpose of this paper is to investigate numerically the effect of rheology on the leveling of thin fluid films on horizontal solid substrates.

A mathematical model based on the lubrication approximation which defines non-Newtonian rheology using a Power-law model is presented. The rheology is described by two parameters: the consistency factor and the flow behavior index. The resulting highly non-linear coupled set of equations is discretized using Finite-Difference and the resulting algebraic system is solved via an efficient Multigrid algorithm.

Importantly, the non-dimensionalization process leads to a pair of Partial Differential Equations which depends on one parameter only, the flow behavior index. The authors show that the consistency factor only affects the time scale of the leveling process, hence stretching or contracting the time line. Results for the leveling of sinusoidal perturbations of the fluid film highlights important differences between the leveling of shear-thinning and shear-thickening fluids. In a normalized time frame, the onset of leveling occurs earlier for the shear-thinning fluid than for the shear-thickening one. However, the dimensionless leveling rate is higher for the shear-thickening fluid than the shear-thinning one. This results in a “threshold thickness” which delimits two regimes: the shear-thinning fluid levels to a thickness above this threshold faster than the shear-thickening fluid but the opposite is true for a film thickness below this threshold. An important aspect of this study is the verification of the numerical implementation using the Method of Manufactured Solutions (MMS), a first in the context of thin film studies. The paper also highlights differences between the leveling of two-dimensional and three-dimensional thickness perturbations.

The study of the leveling of disturbances at the free surface of a liquid film using a Power-law rheological model does not appear to have been covered in the literature. Also, the paper uses the MMS to test the validity of the implementation. This appears to be the first time it has been used in the context of the lubrication approximation. Finally, unlike most prior studies, the work does away with the planar assumption.

This paper aims to numerically investigate the two-dimensional unsteady laminar magnetohydrodynamic bioconvection flow and heat transfer of an electrically conducting non-Newtonian Casson thin film with uniform thickness over a horizontal elastic sheet emerging from a slit in the presence of viscous dissipation. The composite effects of variable heat, mass, nanoparticle volume fraction and gyrotactic micro-organism flux are considered as is hydrodynamic (wall) slip. The Buongiorno nanoscale model is deployed which features Brownian motion and thermophoresis effects. The model studies the manufacturing fluid dynamics of smart magnetic bio-nano-polymer coatings.

The coupled non-linear partial differential boundary-layer equations governing the flow, heat and nano-particle and micro-organism mass transfer are reduced to a set of coupled non-dimensional equations using the appropriate transformations and then solved as an nonlinear boundary value problem with the semi-numerical Liao homotopy analysis method (HAM).Validation with a generalized differential quadrature (GDQ) numerical technique is included.

An increase in velocity slip results in a significant decrement in skin friction coefficient and Sherwood number, whereas it generates a substantial enhancement in Nusselt number and motile micro-organism number density. The computations reveal that the bioconvection Schmidt number decreases the micro-organism concentration and boundary-layer thickness which is attributable to a rise in viscous diffusion rate. Increasing bioconvection Péclet number substantially elevates the temperatures in the regime, thermal boundary layer thickness, nanoparticle concentration values and nano-particle species boundary layer thickness. The computations demonstrate the excellent versatility of HAM and GDQ in solving nonlinear multi-physical nano-bioconvection flows in thermal sciences and furthermore are relevant to application in the synthesis of smart biopolymers, microbial fuel cell coatings, etc.

The numerical study is valid for two-dimensional, unsteady, laminar Casson film flow with nanoparticles over an elastic sheet in presence of variable heat, mass and nanoparticle volume fraction flux. The film has uniform thickness and flow is transpiring from slit which is fixed at origin.

The study has significant applications in the manufacturing dynamics of nano-bio-polymers and the magnetic field control of materials processing systems. Furthermore, it is relevant to application in the synthesis of smart biopolymers, microbial fuel cell coatings, etc.

The originality of the study is to address the simultaneous effects of unsteady and variable surface fluxes on Casson nanofluid transport of gyrotactic bio-convection thin film over a stretching sheet in the presence of a transverse magnetic field. Validation of HAM with a GDQ numerical technique is included. The present numerical approaches (HAM and GDQ) offer excellent promise in simulating such multi-physical problems of interest in thermal thin film rheological fluid dynamics.

The purpose of this paper is to analyze heat and mass transport mechanism of unsteady MHD thin film flow of aluminium–copper/water hybrid nanofluid influenced by thermophoresis, Brownian motion and radiation.

The authors initially altered the time dependent set of mathematical equations into dimensionless form of equations by using apposite transmutations. These equations are further solved numerically by deploying Runge–Kutta method along with shooting technique.

Plots and tables for skin friction coefficient, Nusselt number, Sherwood number along with velocity, temperature and concentration profiles against pertinent non-dimensional parameters are revealed. The study imparts that aluminium–copper hybrid nanoparticles facilitate higher heat transfer rate compared to mono nanoparticles. It is noteworthy to disclose that an uplift in thermophoresis and Brownian parameter depreciates heat transfer rate, while concentration profiles boost with an increase in thermophoretic parameter.

The current study targets to investigate heat transfer characteristics of an unsteady thin film radiative flow of water-based aluminium and copper hybrid nanofluid. The high thermal and electrical conductivities, low density and corrosion resistant features of aluminium and copper with their wide range of industrial applications like power generation, telecommunication, automobile manufacturing, mordants in leather tanning, etc., have prompted us to instil these particles in the present study.

The present study has many practical implications in the industrial and manufacturing processes working on the phenomena like heat transfer, magnetohydrodynamics, thermal radiation, nanofluids, hybrid nanofluids with special reference to aluminium and copper particles.

To the best extent of the authors’ belief so far no attempt is made to inspect the flow, thermal and mass transfer of water-based hybridized aluminium and copper nanoparticles with Brownian motion and thermophoresis.

The purpose of this paper is to study the thin film flow of a fourth grade fluid subject to slip conditions in order to understand its velocity profile.

An exact expression for flow velocity is derived in terms of hyperbolic sine functions. The practical usage of the exact flow velocity is restrictive as it involves very complicated integrals. Therefore, an approximate solution is also derived using a Galerkin finite element method and numerical error analysis is performed.

The behavior of fluid velocity with respect to various flow parameters is discussed. The results are not restrictive to small values of flow parameters unlike those obtained earlier using homotopy analysis method and homotopy perturbation method.

An approximate solution based on finite element technique is derived.

A novel interconnect technology, introducing thin film on a laminate substrate base, is presented. A specially constructed laminate board is used as a substrate for the thin film build‐up process. The main characteristics of the laminate core substrate are the z‐axis electrical connections, the absence of holes in the substrate and the very flat nature of the top surface. As a result, the base substrate can be processed further in a thin film processing line. The manufacturing and properties of these substrates are discussed.

This paper aims to study thermocapillarity‐induced flow of thin liquid films covering heated horizontal walls with 2D topography.

A numerical model based on the 2D solution of heat and fluid flow within the liquid film, the gas above the film and the structured wall is developed. The full Navier‐Stokes equations are solved and coupled with the energy equation by a finite difference algorithm. The movable gas‐liquid interface is tracked by means of the volume‐of‐fluid method. The model is validated by comparison with theoretical and experimental data showing a good agreement.

It is demonstrated that convective motion within a film on a structured wall exists at any nonzero Marangoni number. The motion is caused by surface tension gradients induced by temperature differences at the gas‐liquid interface due to the spatial structure of the heated wall. These simulations predict that the maximal flow velocity is practically independent from the film thickness, and increases with increasing temperature difference between the wall and the surrounding gas. It is found that an abrupt change in wall temperature causes rupture of the liquid film. The thermocapillary convection notably enhances heat transfer in liquid films on heated structured walls.

Our solutions are restricted to the case of periodic wall structure, and the flow is enforced to be periodic with a period equal to that of the wall.

The reported results are useful for design of the heat transfer equipment.

New effects in thermocapillary convection are presented and studied using a developed numerical model.

This paper aims to explore the variable properties of a flow inside the thin film of a unsteady Maxwell fluid and to analyze the effects of shrinking and stretching sheet.

The governing mathematical model has been developed by considering the boundary layer limitations. As a result of boundary layer assumption, a nonlinear partial differential equation is obtained. Later on, similarity transformations have been adopted to convert partial differential equation into an ordinary differential equation. A well-known homotopy analysis method is implemented to solve the problem. MATHEMATICA software has been used to visualize the flow behavior.

It is observed that variable viscosity does not have a significant effect on velocity field and temperature distribution either in shrinking or stretching case. It is noticed that Maxwell parameter has no dramatic effect on the flow of thin liquid fluid. It has been seen that heat flow increases by increasing the conductivity with temperature in both cases (shrinking/stretching). As a result, fluid temperature goes down when than delta = 0.05 than delta = 0.2.

To the best of authors’ knowledge, nobody has conducted earlier thin film flow of unsteady Maxwell fluid with variable fluid properties and comparison of shrinking and stretching sheet.