Stability analysis on the flow of thin second-grade fluid over a heated inclined plate with variable fluid properties

https://doi.org/10.1016/j.ijnonlinmec.2021.103711Get rights and content

Abstract

This work investigates the thin film instability of a non-Newtonian second-grade fluid flowing down a heated inclined plane subject to linear variation of physical properties such as density, viscosity, thermal diffusivity, and surface tension concerning temperature. A nonlinear evolution equation for the description of the free surface is derived using long-wave approximation. The critical conditions for the onset of instability, such as critical Reynolds number, critical wave number, and the linear phase speed of wave propagation, are obtained by linear stability analysis through the normal mode approach. Further, the investigation on the weakly nonlinear stability characteristic of the fluid using a multi-scale approach reveals that both supercritical stability and subcritical instability exist. The influence of the non-Newtonian parameter and the variable fluid properties on subcritical, supercritical, unconditional, and explosive zones are discussed. It also discusses the amplitude of disturbances in subcritical and supercritical regions. The results show the destabilizing nature of the second-grade parameter.

Introduction

Studies on the dynamics and stability characteristics of Newtonian/non-Newtonian thin-film flowing down a vertical/inclined plane have ample scope in chemical and nuclear engineering, in industries such as magnetic film coating, painting, etc. and in interface heat and mass transfer processes in chemical technology. Researches on the instabilities of thin-film flow, which manifests as waves propagating on the free surface, is of great importance for maintaining the uniform thickness of liquid layers in the coating industry. Kapitza and Kapitza [1], who first experimentally observed the occurrence of interfacial wave structures, which is the manifestation of instabilities of fluid flow, was the pioneering work in this research area. An extensive and in-depth study on the instability of thin films can be found in [2], [3]. Benjamin [4] performed a theoretical analysis on the onset of instability, in which the governing perturbation equations were linearized to investigate the uniform flow stability. Later, Yih [5] studied the instability on a vertically falling layer and found that the parallel flow is unstable to periodic perturbations of long wavelength. The work provides a cutoff Reynolds number Rec=56cot(α), where α is the angle of inclination with horizontal, through the linear stability analysis, which reveals the fact that if Reynolds number of the flow system exceeds the critical value, then the basic flat film solution becomes unstable. Benney [6] extended the linear stability theory to non-linear analysis, which accounts for the effects of non-linear evolution of waves using lubrication theory and small wave number approximation. The existence of subcritical and supercritical instability is reported in the solution of Benney type equations in [7], [8], [9].

In the stability analysis study, a few researchers have considered the variation of various physical properties as a function of temperature. Many research works in this area have assumed the fluid properties like density, surface tension, thermal diffusivity, etc. to be invariant. But experiments have shown that the assumed uniform fluid properties may change naturally with the variation of temperature. Dandapat et al. [10] analysed variable viscosity over an unsteady stretching sheet and emphasized that temperature-dependent thermal conductivity has a significant influence in the fluid flow analysis. Lai and Kulacki [11] and Chiam [12] examined the impact of viscosity as a function of temperature in fluid flows over the vertical surface and stretching sheet, respectively. The temperature-dependent physical fluid properties have practical applications in bio-sciences and medical fields. For example, Sinha and Misra [13] studied the effect of viscosity variation due to temperature change in the flow of magnetohydrodynamic fluid (MHD) through a dually stenosed artery. Yih and Seagrave [14] modified the governing Orr–Sommerfeld equations by successive perturbations to include the effects of temperature on viscosity. The hydrodynamic stability analysis of thin liquid film flowing down an inclined plane shows a cooling wall destabilizes while a heated wall stabilizes it.

Hwang and Weng [15] examined the nonlinear instability of fluid flow by considering the influence of viscosity variation. This study reveals the existence of both the supercritical stable and subcritical unstable states. It is further reported that the high Prandtl number flow enhances the stability of the flow. Mukhopadhyay et al. [16] have explored the long-wave instability of the Newtonian liquid film model flow over an inclined planar surface with a linear change in fluid properties for low Biot number. Reisfeld and Bankoff [17] examined the instability of a heated volatile fluid film concerning surface tension and Vander Waal forces presuming the change in viscosity with temperature linearly. Pascal et al. [18] investigated the flow instability of a viscous film flow along an inclined planar surface with a linear change in various physical properties. D’Alessio et al. [19] analysed the effect of variable fluid properties on the thin film flow by linear stability analysis using a long wave perturbation method. A linear stability study was conducted by Sadiq and Usha [20], which shows that when the permeability of the medium is increased, the thin fluid film flow over an inclined porous substrate has a destabilizing effect. In this study, the critical conditions for the outbreak of instability are further discussed.

The multiple-scale method is widely used in the weakly nonlinear stability analysis to evaluate the spectrum of wavenumbers for supercritical and subcritical instabilities. Thiele and Goyeau [21] have obtained the critical Marangoni and Reynolds number from the linear stability analysis on the thin film flowing along a heated porous substrate. A nonlinear analysis using continuation techniques yields static surface structures extending from surface waves to large amplitude structures in sliding drops or ridges. Usha et al. [22] have studied the effects of temperature-dependent surface tension and viscosity on the flow characteristics of a thin film of an incompressible non-volatile viscous fluid on a heated rotating disk. Authors have found that the growth of infinitesimal disturbances for small wavenumbers decreases and is temporarily stable for the range of Biot number values for large wavenumbers. Goussis and Kelly [23] analysed the impact of variable viscosity that depends exponentially upon the temperature and established a cutoff Prandtl number through a linear stability analysis. It is reported that flow is linearly stable to long waves above the cutoff Prandtl number.

Most of the available works in literature have presumed the fluid to be Newtonian. However, many practical importance fluids like fluids in the flow of liquefied metals/lava, coating industry, biological fluids, and fluids applicable in plastic manufacturing, polymeric fluids, and food products are non-Newtonian. These fluids have flow characteristics which do not follow a linear viscous flow model. Fluids of differential type is a significant part of viscoelastic fluids [24]. Markovitz and Coleman [25] have shown that a 5.4% preparation of polyisobutylene in cetane at 30°C has the characteristics of a second-order fluid. The peculiarity of this type of fluids is to lose memory sharply. Many works [26], [27] have been done on the significance of second-grade parameter on a thin film flow characteristics.

Linear stability analysis of second-order viscoelastic fluid flowing along an inclined plane was considered by Gupta [28]. A perturbation method is used, and it has been shown that viscoelasticity causes large wavelength disturbances to be unstable in the flow. The behaviour of the system to three-dimensional instabilities for second-order fluids have been investigated by Gupta and Rai [29]. Elgazery et al. [30] studied the impact of variable thermal conductivity and viscosity on the unsteady mass and heat transmission in a power-law fluid flow past a semi-infinite vertical sheet through a porous channel with the influence of a magnetic field. The thermal conductivity and fluid viscosity are varied linearly with temperature. Bin Hu [31] studies the effect of surface tension on the flow of a power-law fluid driven by gravity. Here, the authors discussed optimal flow conditions and fingering instability. Elgazery and Hassan [32] analysed the effects of variable viscosity, thermal diffusivity, and magnetic flux on the fluid flow and heat transmission process in a laminar non-Newtonian fluid film on a porous stretching sheet.

The study of linear and non-linear effects on the stability mechanism of temperature-dependent second-grade fluid flow in different geometries, such as shear flow in channels, tubes, flow past surfaces, etc., is a broad spectrum of significant and industrially important research area. Few researchers have considered the linear and non-linear effects on a thin second-grade fluid’s stability mechanism on different geometries. Rajagopal et al. [33] investigated the stability of a second-grade fluid between two infinite parallel plates rotating about non-coincidental axes and is normal to the plates for finite-amplitude disturbances and found that the region of stability decreases with an increase in the viscoelastic parameter. Dandapat and Gupta [34]’s work reveals the significance of viscoelastic and cross-viscosity parameters on a second-grade thin fluid film’s linear stability structure on a disk that is rotating uniformly about an axis and is normal to the plane of the disk. Further, the stability of a second-grade steady-state Rivlin–Ericksen magnetofluid flow between two parallel planes is studied by Carlsson and Lundgren [35]. The authors obtained neutral-stability curves that associate the effect of the second-grade viscoelastic parameter and the magnetic field. These works have not considered temperature-dependent fluid properties’ role on the second-grade fluid flow’s stability mechanism. We attempt to analyse the thermally dependent second-grade fluid’s stability on an inclined planer surface in the present work.

The present work is inspired by Patra et al. [36], who analysed the longwave approximation model of a thin second-grade fluid over an inclined plane. The authors have discussed the capillary ridge formation on the free surface of the second-grade fluid. Thus, a natural question arises on how the second-grade parameter and the temperature-dependent fluid properties influence the flow’s stability. This could have an interesting practical outcome to maintain the thin film flow’s uniformity relevant to coating industries. Therefore, the present work focuses on the analysis of the linear and weakly non-linear stability analysis of the thin second-grade fluid film that flows down a uniformly heated inclined plane subject to linear density variation, thermal diffusivity, dynamic viscosity, and surface tension.

Our novel contribution to this paper is finding stability and instability zones in the flow of a thin second-grade fluid film along a flat inclined planar surface, heated from below and driven by gravity. The fluid properties such as dynamical viscosity, density, thermal diffusivity, and surface tension vary linearly with temperature. The research on the importance of the second-grade viscoelastic parameter and the fluid variable properties on fluid flow instability mechanisms are discussed. The results show the presence of subcritical, supercritical, unconditional, and explosive zones within the framework of stability of the flow.

The paper is structured as follows. Section 2 starts with the formulation of the model and derive the free surface dynamic equation using longwave theory. In Section 3, we discuss the stability analysis of the thin film flow and report the critical Reynold’s number, critical wave number, and mode of perturbation of maximum growth by using the perturbation method. In addition, we extend the theory of stability to include the weakly non-linear analysis using multiple-scale techniques to demonstrate the existence of supercritical and subcritical stability zones. The influence of second-grade parameter and variable fluid properties on different stability and instability zones, amplitude of disturbances, and the non-linear velocity of wave propagation is discussed in the penultimate section. The paper closes with concluding remarks and future scope.

Section snippets

Mathematical formulation

We consider the flow of a two-dimensional thin non-Newtonian second-grade liquid film along a flat inclined planar surface of inclination angle α (0<απ2) with the horizontal axis. In Fig. 1, a sketch of the flow is given, where the x-axis is considered along the downhill inclination plane, and the z-axis is chosen along with the normal (upward) orientation. We assume the liquid to be non-volatile so that the effect of latent heat due to evaporation is neglected. The flow surface on which the

Stability analysis

We assume that the dimensionless film thickness (local thickness) is equal to one to discuss the stability analysis, since the minimal variation of the film thickness, is contained in the basic fluid flow. Let the film thickness h(t,x) (dimensionless) be written as: h(t,x)=1+η(t,x),where η1 is the dimensionless function representing the perturbation on the basic flow. We apply the transformation t=ϵt̃ and x=ϵx̃,in order to remove the ϵ dependency of A(h),B(h), and C(h). Using Eqs. (54), (55)

Conclusions

We performed the linear and weakly nonlinear stability analysis of a thin second-grade fluid flowing along a flat inclined planar surface, heated from below and driven by gravity. These include the fluid properties such as dynamical viscosity, density, thermal diffusivity, and surface tension that vary linearly with temperature. Using the longwave approximation method, we obtained the thin film equation for the description of thin-film height. We expressed the linear time growth rate of

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors gratefully acknowledge the anonymous reviewer for his/her valuable comments, and suggestions. The authors also acknowledge with gratitude towards the financial help received from the Council of Scientific and Industrial Research (CSIR), India , India for the Ph.D. work vide letter No. 09/874(0034)/2019-EMR-I, dated 04/07/2019.

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