Thermocapillary Stresses Research Articles

Thin film flow inside a uniformly heated/cooled inclined rotating cylinder

The present study investigates the flow of a thin film inside a uniformly heated/cooled cylinder rotating at various inclinations. The governing equations for mass, momentum, and energy are simplified using lubrication approximation and asymptotic analysis. The resulting spatiotemporal equation for film height evolution incorporates dimensionless parameters representing gravity, viscous drag, surface tension, and thermocapillary stress. Two-dimensional, steady-state solutions are derived, revealing that under the dominance of gravity over viscous drag, a liquid pool forms at the bottom of the horizontally rotating cylinder. This pool transforms into a more uniformly distributed thin film as the cylinder is rotated at slopes. Three-dimensional solutions show a ring structure for horizontally rotating cylinder, whereas it disappears for a cylinder rotating at slopes. Introducing infinitesimal axial perturbations to the steady solutions allows for the examination of their stability. The study finds that the thin film flow becomes unstable for a sloped cylinder. The gravitational force stabilizes (destabilizes) the flow for horizontal (vertical) cylinders. It is reported that thermocapillary stress has a stabilizing effect for a uniformly cooled cylinder. Additionally, the outcomes obtained through linear stability analysis have been corroborated through nonlinear computations.

Modeling the stability of thin liquid film flows on a uniformly heated slippery inclined substrate: A realistic approach

We investigate the stability of gravity-driven, Newtonian, thin liquid film falling down a uniformly heated slippery rigid inclined wall. The authors of previous research works considered specified temperature (ST) boundary condition to study the effects of slip length. However, the ST boundary condition does not include the effects of heat fluxes at wall–air and wall–liquid interfaces and so fails to incorporate the real situation. Consequently, we consider heat flux/mixed-type boundary condition as the thermal boundary condition on the rigid plate. This boundary condition involves the heat flux from the rigid plate to the surrounding liquid and the heat losses from the wall to the ambient air. Using long-wave expansion method, we construct a highly nonlinear evolution equation in terms of the film thickness at any instant. Using normal mode approach, the linear study reveals the stabilizing (destabilizing) behavior of the wall film Biot number (dimensionless slip length). It is found that the destabilizing tendency of the slip length is more in the absence of thermocapillary stress. The linear study reveals that the destabilizing role of MB may be controlled to some extent by increasing the wall film Biot number Bw. Using asymptotic expansions of the flow variables in terms of the small wave number k, the Orr–Sommerfeld boundary value problem gives an onset of instability in terms of critical Reynolds number. It slightly differs from that of the same as obtained by Benney's long-wave expansion method, due to the consideration of small free surface Biot number [B=O(ϵ)]. For arbitrary wave numbers, using Chebyshev spectral collocation method, the effect of Marangoni number (Ma), slip length (δ), and wall film Biot number (Bw) on the H, S, P, and shear modes of instability are discussed in detail. Near the threshold, both Ma and δ show the destabilizing effect on H mode of instability, whereas Bw gives the stabilizing effect. Interestingly, their roles on H mode of instability becomes diametrically opposite far from the onset of instability. For S mode, both Ma and Bw display the destabilizing effect, whereas δ plays the dual role. For P mode, both Ma and δ show the destabilizing effect, whereas Bw plays the stabilizing role. The slip length (δ) plays the stabilizing role, in the case of shear mode. In the absence of thermocapillary effect, the vorticity balance at the liquid–air interface explains that the amplitude of the vorticity perturbation amplifies the surface deformation due to the presence of inertia and the slip length. In the absence of the slip length, a weakly nonlinear study transforms the evolution equation to the famous Kuramoto–Sivashinsky (KS) equation.

Thermal Resistance of Heated Superhydrophobic Channels With Thermocapillary Stress

Abstract A pressure-driven channel flow between a longitudinally ridged superhydrophobic surface (SHS) and solid wall is studied, where a constant heat flux enters the channel from either the SHS or solid wall. First, a model is developed which neglects thermocapillary stresses (TCS) in the transverse direction. The caloric, convective, and total thermal resistance are evaluated, and their dependence on the shape of the liquid–gas interface (meniscus), gas ridge width, texture period, channel height, streamwise TCS, Péclet number, and channel length is established. The caloric resistance is minimized with menisci that protrude into the gas cavity, large slip fractions, small channel heights, and small streamwise TCSs. When heating from the SHS, the convective resistance increases, and therefore, a design compromise exists between caloric and convective resistances. However, when heating from the solid wall, the convective resistance remains the same and SHSs that minimize caloric resistance are optimal. We investigate both water and Galinstan for microchannel applications and find that both configurations can have a lower total thermal resistance than a smooth-walled channel. Heating from the solid wall is shown to always have the lowest total thermal resistance. Numerical simulations are used to analyze the effect of transverse TCSs. Our model captures much of the physics in heated superhydrophobic channels but is computationally inexpensive when compared to the numerical simulations.

Thermocapillary dynamics of a surfactant-laden droplet with internal thermal singularity

Thermocapillary droplets with internal thermal singularities have potential applications in drug delivery and cell analysis. Inspired by the work of Pak et al. (J. Fluid Mech., vol. 753, 2014, pp. 535–552), which was investigated for a surfactant-laden non-deformable droplet in an isothermal Poiseuille flow, we have explored the droplet dynamics by taking account of additional internal thermal singularities, namely monopole and dipole. A generalized mathematical model is developed, which is solved by using the solenoidal decomposition to describe the flow field in any arbitrary Stokes flow, and results are shown extensively for the case of a non-isothermal Poiseuille flow. Under small Péclet number ( $Pe_s$ ) limit, the droplet with an off-centred monopole or a dipole oriented along the flow direction shows cross-stream migration at $O(Pe_s^2)$ . However, a dipole oriented perpendicular to the flow direction results in an $O(1)$ effect due to thermocapillarity, and from $O(Pe_s)$ onwards, we observe the combined impact of thermocapillary and surfactant-induced Marangoni stresses. As a surprise, we see cross-stream migration of the droplet from the Poiseuille flow centreline in a non-isothermal field, in contrast to existing findings which rule out any cross-stream migration. We show the trade-off between thermal Marangoni number ( $Ma_T$ ) and surfactant Marangoni number ( $Ma_\varGamma$ ). Our findings on droplet dynamics inspire new possibilities for microfluidics-based design.

Rimming flow in a rotating horizontal cylinder with phase change at the interface

The two-dimensional solutions and stability analysis are presented for an evaporating thin viscous liquid film flowing inside a uniformly heated rotating horizontal cylinder. A non-linear, fourth-order, partial differential evolution equation is obtained by simplifying mass, momentum, and energy conservation equations within the lubrication approximation. The effect of evaporation, gravity, viscous drag, surface tension, thermocapillary stress, and intermolecular forces has been taken into account. The numerical solutions of the model are validated against the existing experimental as well as the numerical results, along with the analytical result for the limiting cases of the present model. The film thickness model is solved to elucidate two-dimensional spatiotemporal solutions and their stability for a wide range of thermal and other parameters. The evaporative mass flux at the liquid–air interface results in unsteady solutions which are oscillatory in nature, and the amplitude of the oscillations increases with an increase in the evaporative flux. The film ruptures after some time and the rupture time is found to be inversely proportional to the evaporation number, a non-dimensional number quantifying the rate of evaporation. The linear stability analysis shows that the thermocapillary stresses as well as the long-range molecular forces destabilize the film. A negative disjoining pressure is shown to reduce the rupture time and vice versa. Evaporation (condensation) plays a destabilizing (stabilizing) role in the thin film flow. Non-linear computations are carried out for the steady profiles, validating the growth rates obtained from the linear stability analysis.

Investigation of sessile droplet evaporation using a transient two-step moving mesh model

The evaporation of droplets on surfaces is a ubiquitous phenomenon essential in nature and industrial applications ranging from thermal management of electronics to self-assembly-based fabrication. In this study, water droplet evaporation on a thin quartz substrate is analyzed using an unsteady two-step arbitrary Lagrangian-Eulerian (ALE) moving mesh model, wherein the evaporation process is simulated during the constant contact radius (CCR) and contact angle (CCA) modes. The numerical model considers mass transfer in the gas domain, flow in the liquid and gas domains, and heat transfer in the solid, liquid, and gas domains. Besides, the model also accounts for interfacial force balance, including thermocapillary stresses, to obtain the instantaneous droplet shape. Experiments involving droplet evaporation on unheated quartz substrates agree with model predictions of contact radius, contact angle, and droplet volume. Model results indicating temperature and velocity distribution across an evaporating water droplet show that the lowest temperatures are at the liquid-gas interface, and a single vortex exists for the predominant duration of the droplet's lifetime. The temperature of the unheated substrate is also significantly reduced due to evaporative cooling. The interfacial evaporation flux distribution, which depends on heat transfer across the droplet and advection in the surrounding medium, shows the highest values near the three-phase contact line. In addition, the model also predicts evaporation dynamics when the substrate is heated and exposed to different advection conditions. Generally, higher evaporation rates result from higher substrate heating and advection rates. However, substrate heating and advection in the surrounding gas have minimal effects on the relative durations of CCR and CCA modes for a given receding contact angle. Specifically, in this case, a 40× increase in substrate heating rate or 7.5× increase in gas velocity can only change these relative durations by 3%. This study also highlights the importance of surface wettability, which affects evaporation dynamics for all the conditions explored by the numerical model.

Numerical study of melt hydrodynamics in pulsed laser processing of nickel-based superalloy

Pulsed laser processing (PLP) of materials has wide-spread applications in the industry, yet the underlying complex material response to the incident laser pulse especially the melt flow mechanism is not fully understood. Knowledge of melt hydrodynamics can provide systematic theoretical guidance for the improvement of PLP quality. In this paper, a two-dimensional numerical model was built to simulate melt flow during PLP of nickel-based superalloy. The driving forces for melt hydrodynamics at different stages of laser pulse irradiation were studied comprehensively. Numerical results reveal that thermo-capillary stress drives radially-outward melt flow at the initial stage of laser pulse irradiation. At the middle stage, evaporation-induced recoil pressure is the dominant driving force for melt pool deformation, creating a depression at the center and a hump at the periphery. Thermo-capillary force has a minor effect on melt pool deformation but a large contribution to the flow velocity field. At the post-irradiation stage, the peripheral molten hump flows back rapidly and the melt pool experiences some gradually damping oscillations under the action of Young-Laplace stress. Moreover, it was demonstrated that laser power density plays an important role in deciding the melt flow characteristics and the relative contributions of different driving forces. The results in this study provide an unambiguous physical picture of the melt hydrodynamics in pulsed laser processing.

Spatiotemporal stability of a thin film in the presence of thermal and solutal Marangoni stresses

The stability analysis of a gravity-driven thin liquid film over a uniformly heated substrate is carried out, and the effect of an insoluble surfactant is studied. The heating induces a temperature variation at the liquid–gas interface that generates a gradient in surface tension and creates a thermocapillary flow at the liquid–gas interface. When the film is perturbed the thermocapillary stress leads to the growth of the perturbation. The governing equations for the evolution of the film thickness and surfactant concentration are simplified within the lubrication approximation. Four non-dimensional groups appear in the model that affect the film dynamics and stability, namely, Marangoni numbers, M and Σ, the surface Peclet number, Pes, and Biot number, Bi. The critical conditions are explored in terms of these non-dimensional parameters for the film to become temporally unstable. Further, spatiotemporal stability analysis is performed to characterise the instability as absolute and convective instability. A critical Marangoni number is found beyond which the system is absolutely unstable independent of the inclination angle. This critical Marangoni number seems to increase with the increase in solutocapillary effect.

Analysis of Marangoni Effects on the Non-isothermal Immiscible Rayleigh-Taylor Instability

• Hydrodynamics and heat transfer of immiscible Rayleigh-Taylor Instability (RTI) with thermocapillary effect are studied. • A critical range of Marangoni numbers that result in the suppression of the RTI is found. • The dependence of the critical range of Marangoni numbers on the density of the fluids has been explained. • The contribution of the forces was found to vary non-monotonically with the Marangoni number. • All the above-mentioned nonmonotonicities are explained in terms of the vorticity and enstrophy of the system. Non-isothermal Rayleigh-Taylor Instability (RTI) has been investigated for incompressible, Newtonian, immiscible flows. Thermal gradients were triggered by imposing a cold and a hot temperature at the top and the bottom side of the domain, respectively. The effects of thermocapillary stresses resulting from temperature-induced interfacial tension gradients were investigated. The governing equations together with a conservative phase-field equation to track the interface, were solved numerically using the Lattice Boltzmann Method. The investigation focused on analyzing the effects of the Marangoni number and fluids density ratio and comparison of the flows of the isothermal and the non-isothermal cases when Marangoni effects are neglected. The results revealed that thermocapillary effects can change the instability characteristics by slowing down the usual downward flow or even inducing and sustaining an upward flow. This reversed flow phenomenon was found to occur only within an interval of the Marangoni numbers, outside of which the typical downward flow is observed again. The effects of the density ratio on this reversed flow phenomenon were analyzed. The phenomenon was characterized, and the underlying physics were explained by analyzing the flow dynamics and examining the relative contribution of the forces acting on the interface. In particular, the competing effects of the temperature gradients on the components of the interfacial tension along and normal to the interface, were found to play a major role. This study offers a new perspective for the control and orientation of the interface deformation and motion via thermocapillary stresses.

Marangoni Patterns in a Non-Isothermal Liquid with Deformable Interface Covered by Insoluble Surfactant

Marangoni patterns are created by instabilities caused by thermocapillary and solutocapillary stresses on the deformable free surface of a thin liquid layer. In the present paper, we consider the influence of the insoluble surfactant on the selection and modulational instability of stationary Marangoni patterns near their onset threshold. The basic governing parameters of the problem are the Biot number characterizing the heat-transfer resistances of and at the surface, the Galileo number indicating the role of gravity via viscous forces, and the elasticity number specifying the influence of insoluble surfactant on the interfacial dynamics of the liquid. The paper includes a review of the previous results obtained in that problem as well as new ones.

Influence of the gas-phase Lewis number and thermocapillary stress on motion of a slowly evaporating droplet in Stokes flow

The influence of thermocapillary stresses on motion of a slowly evaporating single-component droplet in Stokes flow is investigated analytically for the situation where the environment does not have a temperature gradient in the far field. The conservation equations are solved in the liquid and gas phases and coupled at the gas–liquid interface by applying conditions for conservation of mass, species, momentum and energy. It is found that thermocapillary stresses may influence droplet motion by changing the the interface velocity, and that the gas-phase Lewis number of the evaporating component determines whether Marangoni effects increase or decrease droplet drag. If the Lewis number is less than unity, then thermal Marangoni effects increase droplet drag, while if the Lewis number is greater than unity, then thermal Marangoni effects decrease droplet drag. This is related to the sign of the temperature gradient along the droplet surface that is induced by convection. It is found that conditions may exist where a vaporizing droplet in a microgravity environment will exhibit continuous translational motion driven by thermocapillary effects.

Interfacial Flows and Interfacial Shape Modulation Controlled by the Thermal Action of Light Energy

The review covers the research on thermocapillary convection caused by the thermal action of laser radiation in single-layer and bilayer liquid systems of capillary thickness. The advantages of using optical radiation are the instantaneous delivery of thermal energy to a place on demand (a bulk phase, interfaces); low radiation power required; concentrating heat flux on a spot of a few micrometers; the production of arbitrary spatial distributions of radiation intensity; and, as a result, corresponding thermal fields at a liquid interface and their fast reconfiguration. Thermocapillary stresses at the liquid interfaces lead to the transfer of the liquid and a change in the shape of the interface, in accordance with the distribution of the light-induced thermal field. Studies concerned with the methods of non-destructive testing of liquid media and solids, which are based on a photothermocapillary signal emitted by a laser-induced concave deformation of a thin layer, are considered. Features of thermocapillary deformation of a liquid–air interface caused by local heating of thin and thick (exceeding the capillary length) layers are demonstrated. A part of the review addresses the results of the study of thermocapillary rupture of films in the heating zone and the application of this effect in semiconductor electronics and high-resolution lithography. The works on the light-induced thermocapillary effect in bilayer (multilayer) liquid systems are analyzed, including early works on image recording liquid layer systems, liquid IR transducers, and nonlinear optical media.

Leidenfrost flows: instabilities and symmetry breakings

Leidenfrost drops were recently found to host strong dynamics. In the present study, we investigate both experimentally and theoretically the flow structures and stability inside a Leidenfrost water drop as it evaporates, starting with a large puddle. As revealed by infrared mapping, the drop base is warmer than its apex by typically 10 $^{\circ }$ C, which is likely to trigger bulk thermobuoyant flows and Marangoni surface flows. Tracer particles unveil complex and strong flows that undergo successive symmetry breakings as the drop evaporates. We investigate the linear stability of the base flows in a non-deformable, quasi-static, levitating drop induced by thermobuoyancy and the effective thermocapillary surface stress, using only one adjustable parameter. The stability analysis of nominally axisymmetric thermoconvective flows, parametrized by the drop radius $R$ , yields the most unstable, thus, dominant, azimuthal modes (of wavenumber $m$ ). Our theory predicts well the radii $R$ for the mode transitions and cascade with decreasing wavenumber from $m=3,\, m=2$ , down to $m=1$ (the eventual rolling mode that entails propulsion) as the drop shrinks in size. The effect of the escaping vapour is not taken into account here, which may further destabilize the inner flow and couple to the liquid/vapour interface to give rise to motion (Bouillant et al. Nat. Phys., vol. 14 (12), 2018, pp. 1188–1192; Brandão & Schnitzer Physical Review Fluids, vol. 5 (9), 2020, 091601).

Use of the reflective background oriented schlieren technique to measure free surface deformations in a thin liquid layer non-uniformly heated from below

To measure liquid film surface deformations, we use the Background Oriented Schlieren (BOS) technique with a reflective optical scheme, which includes a random dot pattern located above the liquid surface, and a camera that captures its reflection from the liquid interface. The substrate was covered with a special black coating to minimize stray light reflection, which allowed to provide clear visualization of the dot pattern reflected from the free surface even for very thin liquid films. Local slopes of the film surface cause the dot displacement on the pattern image relative to the pattern image captured in the absence of any free-surface deformations. Hence, the topography of the liquid film surface is encoded by the apparent displacement of the dot pattern. Using the cross correlation image processing, the displacement vector field was determined, which was then substituted in Poisson equation to reconstruct the free-surface profile of the liquid film. The main advantages of this technique are: applicability for liquids with surface temperature inhomogeneities, experimental simplicity, sensitivity and applicability for both transparent and opaque liquids. Using the reflective BOS technique and using substrate with black coating we measure the free-surface deformation field of a thin layer of silicone oil non-uniformly heated from below. Despite the fact that thermocapillary shear stresses tend to decrease the thickness of the liquid layer in hotter regions, in the experiment, a convex deformation of the liquid film surface was observed above the heater for a few seconds immediately after the start of heating. This phenomenon is explained by the expansion of the liquid induced by the heating, due to the dependence of the liquid density on temperature. The measurement results are compared with those obtained in a single point using the confocal sensor. A good agreement is achieved.

Self-Propulsion of a Volatile Drop on the Surface of an Immiscible Liquid Bath.

We demonstrate the self-propulsion of a volatile drop on the surface of a bath of an immiscible liquid. Evaporative heat pumping is converted into directed motion through thermocapillary stresses, which arise from the coupling between surface-tension-driven flows and temperature advection. A propulsive force arises from convection-sustained temperature gradients along the drop interface, resulting in a warmer pool of liquid being advected by the hydrodynamic flow in the underlying bath toward the back of the drop. The dependence of the drop speed on the activity source, i.e., the evaporation flux, is derived with scaling arguments and captures the experimental data.

Photothermally Reconfigurable Colloidal Crystals at a Fluid Interface, a Generic Approach for Optically Tunable Lattice Properties

Optically addressable colloidal assembly at fluid interfaces is a highly desired component to generate reconfigurable 2D materials but has rarely been achieved and only with specific interface engineering. Here we describe a generic method to get optically reconfigurable colloidal crystals at the air/water interface and emphasize a new mechanism to convert light into tunable lattice properties. We use light-absorbing anionic particles adsorbed at the air/water interface in the presence of minute amounts of cationic surfactant, which self-assembled into closely packed polycrystalline structures by collectively deforming the surrounding interface. Low-intensity irradiation of these colloidal crystals results in unprecedented control of the interparticle spacing in a preserved crystalline state while, at a higher intensity, cycles of melting/recrystallization with a controllable transition kinetics can be achieved upon successive on/off stimulations. We show that this photoreversible melting originates from an initial thermocapillary stress, expanding the colloidal assembly against the local confinement, and an increase in particles diffusivity imposing the transition kinetics. With this mechanism, local irradiation leads to highly dynamic patterns, including self-healing or self-fed "living" crystals, while multiresponsive assembly is also achieved by controlling particle organization with both light and magnetic stimuli.

Role of suction pressure in the stability of a gravity-driven thermoviscous liquid film flow down the interior surface of a cylinder.

This study aims to analyze the stability of a gravity-driven thin film flow in the heated/cooled interior surface of a vertical hollow cylinder. The model development involves simplifying the flow and energy equations using the usual thin-film approximation, where the average film thickness is considered to be much smaller than the radius of cylinder. A dispersion relation is then derived to study the temporal stability of the system in order to quantify the effect of various non-dimensional parameters present in the model, such as the thermoviscous number, Marangoni number, Biot number, and Bond number. Another non-dimensional parameter is introduced by considering an opposing suction pressure in the annulus region. The thermocapillary stress and the thermoviscous effect are shown to strongly affect the temporal stability of the flow. It is shown that although the suction pressure affects the velocity profile of the flow, it does not affect the temporal stability results. The suction pressure is then shown to have some effect on the spatiotemporal stability. Critical condition is presented for the transition between absolutely and convectively unstable systems, and parameter regimes are presented to quantify the effect of the above-mentioned parameters.

Competing Marangoni effects form a stagnant cap on the interface of a hydrogen bubble attached to a microelectrode

The formation of a stagnant cap on a bubble attached to a microelectrode is studied by means of numerical simulation including both thermo- and solutocapillary effects. Recently, strong Marangoni flow in an electrolyte around electrogenerated bubbles was observed by Yang et al. [1] and Massing et al. [2]. High local current density above the electrode led to Ohmic heating of the electrolyte near the bubble foot and resulted in thermocapillary convection. However, the experimentally observed Marangoni convection can be predicted better if a stagnant cap of surfactants on the top of the bubble is assumed. The present work provides evidence that supports this hypothesis and simulates the stagnant cap utilizing two methods. In the first method, a critical stagnation angle θs, which marks the border of the stagnant cap, is specified. At the top of the bubble above the stagnation θs the interface motion is suppressed whereas at the bottom of the bubble below the stagnation θs the thermocapillary effect dominates. In the more extensive second method, a transport equation for the surfactant concentration on the bubble interface is included. In this method, the thermo- and solutocapillary effects compete along the entire interface of the bubble. As a result, the top part of the bubble interface will stagnate. We quantify the rigidity of the bubble interface by a dimensionless number, the elasticity number. This elasticity number is the ratio of the solutocapillary stress due to surfactant variation to the thermocapillary stress due to temperature variation. The relevant temperature variation is due to Ohmic heating and is observed to scale with the square of the potential difference. As a consequence, the Marangoni velocity is also found to scale with the square of the potential difference.Additionally, the forces acting on the bubble before detachment are analysed. Special attention is given to the Marangoni force that is more dominant than reported previously because we included the force caused by the uneven pressure distribution along the bubble interface. The pressure distribution is uneven due to a secondary Marangoni vortex in the wedge between the bubble and the electrode. Furthermore, a general framework for Marangoni numbers is introduced to quantify the effect of each specific source of surface stress variation upon the spatial distribution of each variable governed by a convection-diffusion equation.

An integrated Eulerian-Lagrangian-Eulerian investigation of coaxial gas-powder flow and intensified particle-melt interaction in directed energy deposition process

In this work, a comprehensive framework consisting of coaxial gas-powder flow and melt-flow dynamics models is developed for the laser-assisted directed energy deposition process. Firstly, by using the coupled Eulerian-Lagrangian models, the dynamics of the carrier-shielding gases, the particle-stream trajectory, and the in-flight temperature rise of the powder-particles are predicted. This is done by solving the momentum transfer equations, the k-ε turbulence equations, the discrete phase equation and the energy equation. Thereafter, using the particle-stream characteristics above the melt-pool free-surface, an interface-tracking melt thermo-hydrodynamics model is developed in an open-source environment to track the incident powder-particles in the melt-pool. Using the developed framework, the spatial and temporal variation of the melt-pool morphology due to the powder-particle impingement is analyzed. The computational results of particles-melt interaction show an erratic flow pattern with an oscillatory behavior of the melt-pool free-surface. The incident powder-particle creates a crater on the free-surface of the melt-pool and generates a radially outward strong ripple-wave. The velocity field of the melt-pool shows a highly non-linear behavior which is extremely sensitive to the particle impact. It was found that the thermo-capillary stresses dominate only at the instances when there is no particle insertion and the flow is stabilized. Using the solidification parameters obtained from the melt-pool dynamics model, the influence of the fluctuating thermo-fluidic field on the resultant columnar dendritic microstructure is studied. The comparison of the numerically predicted deposition dimensions and the dendritic arm spacings shows a good correlation with the corresponding experimentally measured values from the in-house controlled experiments.

Thermocapillary instability on a film falling down a non-uniformly heated slippery incline

A gravity-driven, thin, incompressible liquid film flow on a non-uniformly heated, slippery inclined plane is considered within the framework of the long-wave approximation method. A mathematical model incorporating variation in surface tension with temperature has been formulated by coupling the Navier–Stokes equation, governing the flow, with the equation of energy. For the slippery substrate, the Navier slip boundary condition is used at the solid–liquid interface. An evolution equation is formed in terms of the free surface, which includes the effects of inertia, thermocapillary as well as slip length. Using the normal mode approach, linear stability analysis is carried out and a critical Reynolds number is obtained, which reflects its dependence on the Marangoni number Mn as well as slip length δ. This depicts that δ and Mn both have the destabilization effect on the flow field. The linear study also reveals that the inertia force has a negligible effect compare to the thermocapillary or slip. In addition, the study highlights a critical Marangoni number at which the instability sets in when the thermocapillary stress attains a critical value. The method of multiple scales is used to investigate the weakly nonlinear stability analysis of the flow. The study interprets that the variation of Mn and δ have substantial effects on different stable/unstable zones. It also shows that within a considered parametric domain, the unconditional stable zone completely vanishes for any value of Mn, when the slip length δ attains a critical value. The study also divulges that in the subcritical unstable (supercritical stable) zone the threshold amplitude (ζa) decreases (increases) with the increment of Mn and δ. Further, we discussed the spatial uniform solution of the complex Ginzburg–Landau equation for sideband disturbances. Employing the Crank–Nicolson method, the nonlinear evolution equation of the free surface is solved numerically in a periodic domain, considering the sinusoidal initial perturbation of small amplitude. The nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis. The evolution of the maximum hmax and minimum hmin thickness amplifies, for small change of Mn and δ. Further, it shows that the influence of the thermocapillary force amplifies the destabilizing nature of δ. The traveling wave solution confirms the existence of a fixed point for the considered parametric domain, chosen from the experimental data. Finally, the Hopf bifurcation of the fixed point exhibits that the nonlinear wave speed at the transcritical point increases as δ increases but decreases as Mn increases. The noteworthy result which arises from the study is that a transcritical Hopf bifurcation exists if the slip length δ>max16Mn−13,12Mn−23−Mn.