Slippage effect on the flow stability induced by an inclined temperature gradient

Abstract

The effect of slippage on the linear hydrodynamic flow stability of a shallow fluid layer induced by an inclined temperature gradient was studied for fluids with Prandtl number of 0.1, 0.7, 10 and 500, related with applications. The basic flow is induced by the horizontal component of the inclined temperature gradient. The slippage in the horizontal walls provide changes in the instability modes giving a completely different map of the stability diagrams. The results suggest that slippage has an destabilizer effect in comparison with the case of no slip boundary condition, which can be seen in the reduction of stable areas for most of the assessed cases. However, for moderate and high Prandtl numbers, the slippage promotes the appearance of the oscillatory transverse mode To, giving as a result, additional codimension-two points.

Introduction

The slippage effect is a destabilizing agent of interest to a number of applications where heat transfer and fluid motion occurs, as in microfluidic technology in which the idealized no-slip boundary condition may not be held [1,2]. For example, in microfluidic systems involving biological applications, the thermal convection is of particular importance, especially those concerned with the Polymerase Change Reaction (PCR) for DNA amplification [[3], [4], [5]].

In practice, to guarantee a temperature gradient directed in only one direction is difficult. However, if an inclined temperature gradient is considered (vertical and horizontal applied simultaneously) the assessment is more realistic. Different phenomena have presented this condition, such as thermocapillary [6,7], porous media [8] and thermogravitational flow [[9], [10], [11], [12]] where the analysis has been made for shallow layers, their findings suggest that several modes of instability compete to be the first unstable. For fluids under the inclined temperature gradient, the Prandtl number plays a crucial role, since the most critical instability mode depends on it.

For Newtonian fluids, it has been demonstrated the compliance of the slip boundary condition through experimentation on a microscopic scale. The simulation and experimental flow of stearic acid diluted into hexadecane [13] through capillaries have resulted in the occurrence of the slippage. Among simulations, the research of Barrat et al. [14] comes into conclusion by using molecular dynamics simulations that when the contact angle is large enough, the slip boundary condition is suitable in the case of a liquid that partially wets the solid. Nanotechnology is not exempt from this phenomenon; the use of nanofibers provides slip on surrounding air used in air filtration materials (AFM) to combat air pollution [15]. However, the slippage brings consequences in the performance in MHD micropumps [16,17], which are of considerable impact in micro-total analysis systems (μTAS). Following the slippage on gas flow, Arkilic et al. [18] demonstrated that the mass flow rate through the microchannel can be accurately predicted by using a slip flow boundary condition at the walls of long microchannels when helium is flowing through them. Those results were also confirmed by Liu et al. [19]. In addition, the experimental study of gas flow (nitrogen, helium and argon) in silicon microchannels conducted by Harley et al. [20] demonstrated the agreement of experimental data with respect to the predictions of the theory by using an isothermal slip-flow model.

The complexity of the slip flow is wide, and cover different areas, for instance: in aerospace appplications, the slip flows are present in small air vehicles and subsonic rarefied flows in extraterrestrial atmospheres [21]. Numerical studies have been carried out regarding this topic, among them, Rahman et al. [21] carried out an assessment on the convective flow with slip condition for slightly rarefied fluids. For marine vehicles, the slippage caused by entrapped gas between the liquid-solid interface [22] and the riblets surfaces [23] can provide a drag reduction. Chemical engineering has not been the exception, Hamza [24] found that the slippage effect influences strongly the convection flow of an exothermic fluid. For electrochemical applications, it is shown that the electroconvective slippage influences the transition condition from electroconvective instability to Rayleigh-Bénard instability [25].

Moreover, slip on the wall makes thinner the boundary layers, delaying the transition to turbulence and modifying the heat transfer as settled by Martin and Boyd [26]. The combined effect of slippage and heat transfer contributes to modify the flow regime in microchannels [27,28] and plates [29]. The slippage effect on some surfaces can be controllable through several techniques. For hydrophobic surfaces at the micro-scale Karatay et al. [30] have controlled the slippage on the surface by modifying the geometry of the bubbles attached to the boundaries.

In terms of stability, Ghosh et al. [31] shown the stabilizing effect of slippage on his research related to the linear stability characteristics of pressure-driven in a channel with velocity slip at the wall. Xu et al. [32], on his research addressed the slip phenomena by the boundary modelling, by performing a numerically-experimental study to evaluate the extent of the slippage and transition of the length scales from slip-dominated to no-slip regime, using a general linear stability analysis and experiment of a layer of Polymethylmethacrylate (PMMA) over a sublayer of Polystyrene (PS). Besides these works, a very important application of the influence of an inclined temperature gradient over the material properties is in the float glass process, which has been proposed by Pilkington [33]. It consists of a plate of molten glass that is floating on a layer of molten tin. The convection flows in the tin bath can produce inappropriate temperatures in the formation and cooling of the glass sheet, being also important to control the convection currents that appear on the forming glass layer to prevent the development of surface distortion patterns of the glass sheet [34,35]. It is highly relevant for the industries related to glass-forming liquids and for the related research, that deals with Newtonian liquids [36,37] the understanding and control of these phenomena; as a result, better quality controls and safety process can be obtained.

On this paper, the results of the linear problem of natural convection under an inclined temperature gradient for the slip flow condition against those determined with the no-slip boundary condition for four different Prandtl numbers are presented. The calculated curves of criticality modes cover both areas of the vertical Rayleigh number $R{a}_{VC}$ (positive for cavities heated from below, and negative for cavities heated by the upper wall). This research is organized as follows: in Section 2, a description of the physical system is given along with the governing equations of mass, motion, and energy with their linear version to be used in the numerical analysis. Section 3 describes the linear stability analysis performed. In 4 is presented a description of the numerical method. The results of the critical vertical Rayleigh number ($R{a}_{VC}$), wavenumber (${\alpha }_C$), and frequency of oscillation (${\sigma }_C$) against the horizontal Rayleigh number ($R{a}_H$) are shown in Section 5. Section 6 presents the discussion of the results by means of the velocity and temperature profiles. Finally, in Section 7 the conclusions of the research are given.