Long-wave Perturbation Method Research Articles

Thin film development on a double layer of fluids over a stretching sheet

Abstract The thin film development on a double layer of fluids in which the first layer is Newtonian and the second layer is of second-grade fluid on a stretching sheet is analyzed in this paper. The governing equations describing the flow are solved using the Long-wave perturbation method. The coupled partial differential equations thus obtained are solved numerically by the finite volume approach where the spatial derivatives are approximated using the upwind difference scheme and the time derivatives are by forward difference. The study focuses on analyzing the impact of surface tension parameters, viscosity ratio parameters, and Reynolds numbers on the flow dynamics. From the analysis, it is observed that the film height decreases for higher surface tension parameters and Reynolds numbers of both fluid layers and the viscosity ratio parameter $m>1$.

Stability Analysis of Thin Power-Law Fluid Film Flowing down a Moving Plane in a Vertical Direction

This paper analyses the stability of thin power-law fluid flowing down a moving plane in a vertical direction by using the long-wave perturbation method. Linear and nonlinear stability are discussed. The linear stable region increases as the downward speed increases and the power-law index increases. More accurate results are obtained on the coefficients of the nonlinear generalized kinematic equation in the power-law part. The regions of sub-critical instability and absolute stability are expanded when the downward movement of plane increases, or the power-law index increases, and meanwhile the parts of supercritical stability and explosive supercritical instability are compressed. By substituting the power-law index and moving speed into the generalized nonlinear kinematic equation of the power-law film on the free surface, the results can be applied to estimate the stability of the thin film flow in the field of engineering.

Effect of odd-viscosity on the dynamics and stability of a thin liquid film flowing down on a vertical moving plate

In this study, we focus on the problem of hydrodynamic instability of a thin, viscous, Newtonian liquid film with broken time-reversal-symmetry flowing down along the surface of a vertical moving plate. The presence of odd viscosity gives rise to new terms in the pressure gradient of the flow. Utilizing the classical long-wave perturbation method, we obtain the analytical solutions as well as derive the nonlinear evolution equation of Benney-type in terms of film thickness h(x,t) which is significantly modified due to the presence of odd viscosity in the liquid. We solve the linear model by using the normal mode approach and for three different conditions, namely, the quiescent, up-moving and down-moving plate velocity. The linear study shows that the effect of the down-moving motion of the vertical plate is to enhance the stability of the film flow whereas the up-moving motion of the vertical plate tends to reduce it. Further, the study shows that odd viscosity always has a stabilizing effect on the flow field. In addition, the Orr–Sommerfeld equation is also derived and solved analytically to obtain the critical Reynolds number. Finally, we show the numerical solution of the evolution equation in a periodic domain which clearly demonstrates the role of odd-viscosity on the dynamics of the plate motions of thin film flows coating in isothermal environments. Our study clearly shows how odd viscosity influences the stability of the flow.

Modelling and nonlinear boundary stabilization of the modified generalized Korteweg\u2013de Vries\u2013Burgers equation

In this paper, we study the modelling and nonlinear boundary stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is [0,1]. First, the MGKdVB equation is derived using the long-wave approximation and perturbation method. Then, two nonlinear boundary controllers are proposed for this equation and the L^{2} -global exponential stability of the solution is shown. Numerical simulations are given to illustrate the efficiency of the developed control schemes.

Hydromagnetic Stability of a Thin Viscoelastic Magnetic Fluid on Coating Flow Using Landau Equation

AbstractThis paper investigates the weakly nonlinear stability of a thin axisymmetric viscoelastic fluid with hydromagnetic effects on coating flow. The governing equation is resolved using long-wave perturbation method as part of an initial value problem for spatial periodic surface waves with the Walter's liquid B type fluid. The most unstable linear mode of a film flow is determined by Ginzburg-Landau equation (GLE). The coefficients of the GLE are calculated numerically from the solution of the corresponding stability problem on coating flow. The effect of a viscoelastic fluid under an applied magnetic field on the nonlinear stability mechanism is studied in terms of the rotation number, Ro, viscoelastic parameter, k, and the Hartmann constant, m. Modeling results indicate that the Ro, k and m parameters strongly affect the film flow. Enhancing the magnetic effects is found to stabilize the film flow when the viscoelastic parameter destabilizes the one in a thin viscoelastic fluid.

Long interfacial waves on the upper convected Maxwell (UCM) fluid film

The problem of hydrodynamic instability of a very thin UCM liquid film flowing down on the outer surface of an inclined plane is investigated. In order to improve the accuracy of numerical results, the viscoelastic parameters and van der Waals forces have been included into the governing equations. Also, the analytical solutions are obtained by utilizing the long-wave perturbation method. The influence of some physical parameters is discussed in both linear and non-linear steps of the problem. It has been revealed that the instability of the film flow is weakened when Reynolds number, viscoelastic property and attractive van der Waals force are reduced. Further, the thin film flow can be stabilized by short-range Born repulsion. The results show that both supercritical stability and subcritical instability may take place within the film flow system. Moreover, the presence of non-linear terms leads to an obvious difference in the behavior of some physical parameters.

Nonlinear evolution of the travelling waves at the surface of a thin viscoelastic falling film

The problem of hydrodynamic instability of a thin condensate viscoelastic liquid film flowing down on the outer surface of an axially moving vertical cylinder is investigated. In order to improve the accuracy of numerical results, the viscoelastic and heat transfer parameters have been included into the governing equations. Also, the analytical solutions are obtained by utilizing the long-wave perturbation method. The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem. It has been revealed that the stability of the film flow is weakened when the radius of cylinder and the temperature difference are reduced. Moreover, it is found that the increment of down-moving motion of the cylinder can enhance the flow stability. Further, the thin film flow can be destabilized by the viscoelastic property. The results show that both supercritical stability and subcritical instability can take place within the film flow system given appropriate conditions. Moreover, the absence of Reynolds number leads to an obvious difference in the behavior of some physical parameters.

Viscous falling film instability around a vertical moving cylinder

The present work discusses both the linear and nonlinear stability conditions of a viscous falling film down the outer surface of a solid vertical cylinder which moves in the direction of its axis with a constant velocity. After studying the linear conditions, a generalized nonlinear kinematic model is then derived to present the physical system. Applying the boundary conditions, analytical solutions are obtained using the long-wave perturbation method. In the first step, the normal mode method is used to characterize the linear behaviors. In the second step, the nonlinear film flow model is solved by using the method of multiple scales, to obtain Ginzburg-Landau equation. The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem, and the results are displayed in many plots showing the stability criteria in various parameter planes.

Finite amplitude long-wave instability of a thin viscoelastic fluid during spin coating

This paper investigates the stability of a thin incompressible viscoelastic fluid designated as Walters’ liquid B″ during spin coating. The long-wave perturbation method is proposed to derive a generalized kinematic model of the film flow. The method of normal mode is applied to study the linear stability. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. Using the multiple scales method, the weakly nonlinear stability analysis is studied for the evolution equation of a film flow. The Ginzburg–Landau equation is determined to discuss the threshold conditions of the various critical flow states. The study reveals that the rotation number and the radius of the rotating circular disk generate the destabilizing effects. Moreover, the viscoelastic parameter k indeed plays a more significant role in destabilizing the film flow than a thin Newtonian fluid during spin coating [27].

Hydromagnetic Stability Analysis of a Film Coating Flow Down a Rotating Vertical Cylinder

ABSTRACTThe influence of both the Rossby number and the Hartmann number on the hydromagnetic stability of a thin liquid film flowing down along the surface of a vertical cylinder is investigated. The long-wave perturbation method is employed to solve for generalized nonlinear kinematic equations with a free film interface. The normal mode approach is used to compute the stability solution for the film flow. The modeling results indicate that the stability of the liquid film is enhanced by increasing the strength of the magnetic field or reducing the speed at which the cylinder rotates. By contrast, the flow becomes relatively more unstable as the cylinder radius is increased at larger values of the Rossby number. Notably, this finding is the opposite of that observed for film flows along a stationary vertical cylinder.

Weakly Nonlinear Stability Analysis of a Thin Magnetic Fluid during Spin Coating

This paper investigates the stability of a thin electrically conductive fluid under an applied uniform magnetic filed during spin coating. A generalized nonlinear kinematic model is derived by the long-wave perturbation method to represent the physical system. After linearizing the nonlinear evolution equation, the method of normal mode is applied to study the linear stability. Weakly nonlinear dynamics of film flow is studied by the multiple scales method. The Ginzburg-Landau equation is determined to discuss the necessary conditions of the various critical flow states, namely, subcritical stability, subcritical instability, supercritical stability, and supercritical explosion. The study reveals that the rotation number and the radius of the rotating circular disk generate similar destabilizing effects but the Hartmann number gives a stabilizing effect. Moreover, the optimum conditions can be found to alter stability of the film flow by controlling the applied magnetic field.

Effect of electric field on the stability of an oscillatory contaminated film flow

The stability of the viscous liquid film on an oscillating plane is investigated in the presence of both insoluble surfactant on the film surface and uniform electric field, acting normal to the plane. In this problem main motivation is to study the combine effect of surfactant and electric field on the stability of the liquid film. Here liquid is treated as a perfect conductor and the air above the liquid film is also treated as a perfect dielectric. The linear stability analysis is performed using the long-wave perturbation method based on Floquet theory. It is observed that two Floquet modes exist due to the presence of surfactant and both modes can be unstable. The growth rate corresponding to the Floquet modes increase with the presence of an electric field and decrease with the presence of surfactant.

Weakly Nonlinear Stability Analysis of a Thin Liquid Film With Condensation Effects During Spin Coating

This paper investigates the stability of a thin liquid film with condensation effects during spin coating. A generalized nonlinear kinematic model is derived by the long-wave perturbation method to represent the physical system. The weakly nonlinear dynamics of a film flow are studied by the multiple scales method. The Ginzburg–Landau equation is determined to discuss the necessary conditions of the various states of the critical flow states, namely, subcritical stability, subcritical instability, supercritical stability, and supercritical explosion. The study reveals that decreasing the rotation number and the radius of the rotating circular disk generally stabilizes the flow.

Nonlinear hydromagnetic stability analysis of a pseudoplastic film flow

The long-wave perturbation method is employed to investigate the nonlinear hydromagnetic stability of a thin electrically-conductive pseudoplastic liquid film flowing down the surface of a vertical wall in a magnetic field. The validity of the numerical results is improved through the introduction of the flow index and the magnetic force into the governing equation. In contrast to most previous studies presented in the literature, the solution scheme employed in this study is based on a numerical approximation approach rather than an analytical method. The solution procedure commences by employing the normal mode approach to analyze the linear stability of the film flow. The multiple-scales method is then applied to obtain the weak nonlinear dynamics of the thin-film system for stability analysis. The modeling results reveal that the pseudoplastic film flow system may exhibit both subcritical instability and supercritical stability states. The flow stability can be enhanced by increasing the intensity of the magnetic field and the flow index, respectively. In general, the optimum conditions can be found through the use of a system to alter stability of the film flow by controlling the applied magnetic field .

Hydromagnetic stability of a thin electrically conducting fluid film flowing down the outside surface of a long vertically aligned column

This article considers the stability of a thin electrically conducting fluid film flowing down the outer surface of a long vertical cylinder in the presence of an applied magnetic field. Using the long-wave perturbation method to solve the generalized non-linear kinematic equations with free film interface, the normal mode approach is first used to compute the linear stability solution. The method of multiple scales is then used to obtain the weak non-linear dynamics. The results indicate that both subcritical instability and supercritical stability conditions are possible. The degree of instability in the film flow is intensified by the lateral curvature of the cylinder. The results also show that increasing the strength of the magnetic field tends to enhance the stability.

Weakly Nonlinear Hydrodynamic Stability of the Thin Newtonian Fluid Flowing on a Rotating Circular Disk

The main object of this paper is to study the weakly nonlinear hydrodynamic stability of the thin Newtonian fluid flowing on a rotating circular disk. A long‐wave perturbation method is used to derive the nonlinear evolution equation for the film flow. The linear behaviors of the spreading wave are investigated by normal mode approach, and its weakly nonlinear behaviors are explored by the method of multiple scales. The Ginzburg‐Landau equation is determined to discuss the necessary condition for the existence of such flow pattern. The results indicate that the superctitical instability region increases, and the subcritical stability region decreases with the increase of the rotation number or the radius of circular disk. It is found that the rotation number and the radius of circular disk not only play the significant roles in destabilizing the flow in the linear stability analysis but also shrink the area of supercritical stability region at high Reynolds number in the weakly nonlinear stability analysis.

Long-Wave Perturbation Method to Investigate Nonlinear Stability of the Thin Power Law Liquid Film Flowing Down on a Vertical Cylinder

ABSTRACTThe influence of both the flow index and the cylinder size on the nonlinear hydrodynamic stability of a thin power law liquid film flowing down along the surface of a vertical cylinder is investigated. The long-wave perturbation method is employed to solve for generalized nonlinear kinematic equations with a free film interface. The normal mode approach is first used to compute the linear stability solution for the film flow. The method of multiple scales is then used to obtain the weakly nonlinear dynamics of the film flow for stability analysis. The stability criteria are discussed theoretically and numerically and stability diagrams are obtained. The modeling results indicate that by increasing the flow index and increasing the radius of the cylinder the film flow can become relatively more stable as traveling down along the vertical cylinder.

Finite-Amplitude Long-Wave Instability of a Thin Power-Law Liquid Film Flowing Down a Vertical Column in a Magnetic Field

The long-wave perturbation method is employed to investigate the nonlinear hydromagnetic stability of a thin electrically conductive power-law liquid film flowing down a vertical cylinder. In contrast to most previous studies presented in literature, the solution scheme employed in this study is based on a numerical approximation approach rather than an analytical method. The modeling results reveal that the stability of the film flow system is weakened as the radius of the cylinder is reduced. However, the flow stability can be enhanced by increasing the intensity of the magnetic field and the flow index.

Hydromagnetic instability of a power-law liquid film flowing down a vertical cylinder using numerical approximation approach techniques

The long-wave perturbation method is employed to investigate the hydromagnetic stability of a thin electrically-conductive power-law liquid film flowing down the external surface of a vertical cylinder in a magnetic field. The validity of the numerical results is improved through the introduction of the flow index and the magnetic force into the governing equation. In contrast to most previous studies presented in the literature, the solution scheme employed in this study is based on a numerical approximation approach rather than an analytical method. The normal mode approach is used to analyze the stability of the film flow. The modeling results reveal that the stability of the film flow system is weakened as the radius of the cylinder is reduced. However, the flow stability can be enhanced by increasing the intensity of the magnetic field and the flow index, respectively. In general, the optimum conditions can be found through the use of a system to alter stability of the film flow by controlling the applied magnetic field.

Finite-amplitude long-wave instability of Bingham liquid films

The long-wave perturbation method is employed to investigate the weakly nonlinear hydrodynamic stability of a thin Bingham liquid film flowing down a vertical wall. The normal mode approach is first used to compute the linear stability solution for the film flow. The method of multiple scales is then used to obtain the weak nonlinear dynamics of the film flow for stability analysis. It is shown that the necessary condition for the existence of such a solution is governed by the Ginzburg–Landau equation. The modeling results indicate that both the subcritical instability and supercritical stability conditions can possibly occur in a Bingham liquid film flow system. For the film flow in stable states, the larger the value of the yield stress, the higher the stability of the liquid film. However, the flow becomes somewhat unstable in unstable states as the value of the yield stress increases.