PHYSICAL REVIEW E 97, 043105 (2018) Marangoni instability in a thin film heated from below: Effect of nonmonotonic dependence of surface tension on temperature Rajkumar Sarma and Pranab Kumar Mondal * Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Assam, 781039, India (Received 9 January 2018; published 9 April 2018) We investigate Marangoni instability in a thin liquid film resting on a substrate of low thermal conductivity and separated from the surrounding gas phase by a deformable free surface. Considering a nonmonotonic variation of surface tension with temperature, here we analytically derive the neutral stability curve for the monotonic and oscillatory modes of instability (for both the long-wave and short-wave perturbations) under the framework of linear stability analysis. For the long-wave instability, we derive a set of amplitude equations using the scaling k ∼ (Bi) 1/2 , where k is the wave number and Bi is the Biot number. Through this investigation, we demonstrate that for such a fluid layer upon heating from below, both monotonic and oscillatory instability can appear for a certain range of the dimensionless parameters, viz., Biot number (Bi), Galileo number (Ga), and inverse capillary number (). Moreover, we unveil, through this study, the influential role of the above-mentioned parameters on the stability of the system and identify the critical values of these parameters above which instability initiates in the liquid layer. DOI: 10.1103/PhysRevE.97.043105 I. INTRODUCTION Marangoni convection in a fluid layer, which finds appli- cations in areas like thin film evaporators, crystal growth, phase separation process, etc., has been a focus of interest for researchers for the past few decades [1–3]. Originating from the variation of surface tension of fluids, this convec- tion phenomenon is typically encountered in microfluidic systems as well [4–6]. Albeit both the Marangoni and the Bénard (induced by buoyancy) convection can occur in a heated liquid layer, several investigations reveal that for a sufficiently thin layer, Marangoni convection predominates over the Bénard convection [7–10] attributed primarily to the dominance of thermocapillarity over the buoyancy effect. Due to the involvement of the surface effects (surface tension) rather than the volumetric ones (buoyancy), such convection phenomenon can even occur in the microgravity environment as demonstrated by previous researchers in this field [11–14]. However, when both the thermocapillary and buoyancy effects are present together, the phenomenon is collectively called the “Marangoni-Bénard” convection as reported in a number of studies [15–18]. Surface tension is a property of liquids that can vary with temperature, concentration, electrochemical potential, etc. For a pure liquid, the surface tension is a sole function of temperature, whereas for binary fluids, surface tension can vary with both the temperature and concentration of the fluid [19–21]. Therefore, for pure liquids, the Marangoni convection is induced under the sole influence of thermocapillary effect, while, for binary fluids, such convection process occurs by the combined action of thermocapillary and solutocapillary effects * mail2pranab@gmail.com [22–24]. Several experimental investigations reported in the literature suggest that, for most of the fluids, surface tension has a linear relationship with temperature. With an increase in temperature, the surface tension of such fluids decreases monotonically [25]. As such, the literature is rich with the analysis of Marangoni convection for such kind of fluids [9,26,27]. Another important aspect is the nonlinear variation of surface tension with temperature. In this context, we would like to mention here that experiments carried out following well-developed methods reveal that for a certain class of fluids, surface tension varies nonmonotonically with temperature. This particular behavior of surface tension is demonstrated by fluids like long chain aqueous solutions of alcohol, water-oil- surfactant systems, nematic liquid crystals, ionic liquids, etc. [28–33]. These fluids are frequently called “self-rewetting” fluids due to their unique behavior in the boiling process [34]. Several investigations have revealed that such fluids enhance the heat transfer rate in thermal management systems, especially in heat pipes [35–37]. While a series of papers was devoted to the Marangoni convection in a liquid layer with linear dependency of surface tension with temperature [9,27,38,39], considering the nonmonotonic behavior of sur- face tension with temperature, a few studies on Marangoni convection are available in the literature as well [40–42]. In the study of Marangoni convection in a heated liquid layer resting atop a substrate, generally two cases are con- sidered: conducting and insulating substrates for temperature perturbations [9,43]. For the conducting case, a fixed liquid temperature prevails at the substrate, whereas for the insulating case, the normal component of heat flux remains fixed at the substrate. For both the conducting and insulating substrates (for temperature perturbations), two modes of instability can occur in a liquid layer, viz., the monotonic mode (stationary convection) and the oscillatory mode (overstability) [44–46]. 2470-0045/2018/97(4)/043105(13) 043105-1 ©2018 American Physical Society