Critical Marangoni Number Research Articles

Thermocapillary effects during the melting of phase change materials in microgravity: Heat transport enhancement

An extensive numerical investigation of the heat transport enhancement achieved by the thermocapillary effect during the melting of Phase Change Materials in microgravity is presented. The phase change transition is analysed for the high Prandtl number paraffin n-octadecane in a two-dimensional rectangular container subjected to isothermal conditions along the lateral boundaries. The thermocapillary effect, supported by a free surface created as the solid/liquid front progresses, acts driving convection and as an effective mechanism for heat transport. For reference, the melting is first analysed under purely diffusive transport (conduction) and the heat transfer rate is shown to depend only on the Stefan number. Thermocapillary effects are then examined in the limiting cases of large and short containers. Results are presented for the representative aspect ratios Γ=12,2.25, where the thermocapillary flow is characterised by the appearance of oscillatory convection through the hydrothermal travelling wave and oscillatory standing wave modes, respectively, at a critical Marangoni number (Ma). The contribution of the thermocapillary effect to the heat transfer rate is quantified by the ratio of melting times between reference and thermocapillary simulations. For large Γ, heat transport can be greatly enhanced by a factor of up to 20, depending on Ma. For small Γ, this factor can be on the order of 5 when oscillatory flow is present. The analysis of the melting over a wide range of the governing parameters discovers the existence of an optimal curve Γ(Ma) that maximises the thermocapillary contribution.

Stability of surfactant-laden liquid film flow over a cylindrical rod

The stability of surfactant-laden liquid film flow over a cylindrical rod is examined in creeping flow limit using standard temporal linear stability analysis. The clean film flow configuration (i.e., in absence of surfactant) is well-known to become unstable owing to Rayleigh-Plateau instability of cylindrical liquid interfaces. Previous studies demonstrated that for a static liquid film (i.e., zero basic flow) coating a rod, the presence of interfacial surfactant decrease the growth of Rayleigh-Plateau instability, but is unable to suppress it completely. Further, the presence of interfacial surfactant is known to introduce an additional mode, referred to as surfactant mode in the present work. To the best of our knowledge, the stability of surfactant mode has not been analyzed in the context of cylindrical film flows. Thus, we reexamined the stability of surfactant-laden cylindrical liquid film flow to analyze the stability behavior of the above said two modes when the basic flow is turned on. The present study reveals that the incorporation of basic flow in stability analysis leads to the complete suppression of Rayleigh-Plateau instability due to the presence of interfacial surfactants as compared to the partial suppression obtained for a stationary liquid film. Three nondimensional parameters appear for this problem: Bond number (denoted as Bo) which characterizes the strength of basic flow, Marangoni number (denoted as Ma) which signifies the presence of surfactant, and ratio of rod radius to film thickness denoted as S. In creeping flow limit, the characteristic equation is quadratic with one root belonging to Rayleigh-Plateau mode and the other to surfactant mode. We first carried out an asymptotic analysis to independently capture the eigenvalues corresponding to both the modes in limit of long-wave disturbances. The long-wave results show that the Rayleigh-Plateau instability is completely suppressed on increasing the Marangoni number above a critical value while the surfactant mode always remains stable in low wave-number limit. The continuation of long-wave results to arbitrary wavelength disturbances show that the suppression of Rayleigh-Plateau instability mode still holds, however, the surfactant mode becomes unstable at sufficiently high values of Marangoni number. Further, this surfactant mode instability shifts toward low wave numbers with critical Marangoni number for instability scaling with wave number in a particular fashion. We used this scaling and carried out an asymptotic analysis to capture this instability in low wave-number limit. Depending on S and Bo, we observed the existence of a stable gap in terms of Ma where both the eigen-modes remain stable. Our results indicate that for a given Bond number, the width of stable gap in terms of Ma decreases with decrease in S and the stable gap vanishes when S is sufficiently small. The effect of increasing Bond number (or equivalently, the strength of basic flow) is found to be stabilizing for the film flow configuration.

Ferroconvection in an Anisotropic Porous Medium with Variable Gravity

A linear stability assessment was performed to study the impact of internal heating and variable gravity in an anisotropic porous medium of a ferrofluid layer system on the onset of Benard-Marangoni convection. The system is heated from below with both the lower and upper limits are considered as completely insulated to the disturbance of the temperature. The eigenvalue problem is solved by using regular perturbation technique to obtain the critical Marangoni number and also the critical thermal Rayleigh number. It is noted that the increase of value anisotropic permeability, Darcy number and also magnetic number will enhance the convection of the system while the increasing values of anisotropic thermal diffusivity will help to stabilize the system.

Stability of thin film flowing down the outer surface of a rotating non-uniformly heated vertical cylinder

We investigated an incompressible viscous liquid film flow over a rotating vertical cylinder of radius R and of infinite length rotating with a uniform angular velocity $$\varvec{\Omega }$$ about its axis. The surface of the vertical cylinder is non-uniformly heated where the temperature varies linearly in the downstream direction. The flow is assumed to be axisymmetric, and the component of the velocity along the azimuthal direction is assumed to be constant. The surface tension of the liquid is assumed to vary linearly with temperature such that as the temperature increases, the surface tension decreases. This gives rise to Marangoni stress over the free surface of the thin film. Using the long-wave approximation method, we derived a free surface evolution equation. For linear stability analysis, we used a normal mode approach and found that the Marangoni number plays a double role. There exists a critical Marangoni number $$\left( \mathrm{Mn}^{*}\right) $$ such that for $$\mathrm{Mn} \mathrm{Mn}^{*}$$ it plays a destabilizing role. We also found that as the rotation number $$\mathrm{Ro}$$ increases, the destabilizing zone increases but it decreases with the increment of the radius R of the cylinder. We further performed a weakly nonlinear analysis of the flow using the method of multiple scales. The study reveals that the Marangoni number Mn, the radius R and the rotation number $$\mathrm{Ro}$$ have substantial effects on different stability zones. The study also reveals that in the supercritical stable (subcritical unstable) zone, the threshold amplitude of the nonlinear disturbance increases (decreases) with the increment of Mn and $$\mathrm{Ro}$$ but decreases (increases) with the increment of R. The nonlinear wave speed in the supercritical stable zone decreases with the increment of Mn and $$\mathrm{Ro}$$, whereas it increases with the increment of R. We also examined the effect of thermocapillarity and rotation on the profile of the steady travelling wave solutions of the leading order part of the evolution equation.

Instability of thermocapillary–buoyancy convection in droplet migration

The instabilities of thermocapillary–buoyancy convection in droplet migration are examined by linear stability analysis. The droplet is flattened by gravity and placed on a unidirectional heated solid surface. The velocity and temperature distributions of basic flow are derived as a function of the migration velocity and the Bond number. The critical Marangoni number is obtained, which depends on the Prandtl number (Pr), the Bond number, and the migration velocity. The preferred modes at small and moderate Pr are oblique waves, which travel either upstream or downstream. For high Pr, the preferred modes include oblique and streamwise waves, while the amplitude of temperature on the surface is much smaller than that of the hot spot in the flow region. The instability mechanism is discussed and comparisons are made with liquid layers.

Convection in a Two-Layer Liquid System in a Finite Cylinder

This paper describes a cylindrical container of finite dimensions, filled with two quiescent immiscible heat-conducting liquids with a common flat interface. The side walls and bases of the vessel are solid, there are no external forces, and the contact angle of the interface with the side wall of the container is π/2. The interface has a surface tension whose strength linearly depends on temperature. When one of the container bases is heated to a critical temperature, there is movement inside the vessel. When modeling takes into account the energy spent on the interface deformation. The emerging spectral problem is solved by the modified Galerkin method. For various liquids, in the case of monotonous vibrations, the dependence of a critical Marangoni number on a container size and a temperature ratio, specified on the cylinder bases, is obtained, and a perturbed motion velocity field is constructed.

Surface deformations in dynamic thermocapillary convection under partial slip.

A prescribed, horizontal temperature gradient is imposed upon a horizontal liquid layer bounded from above by a deformable, liquid-gas interface and bounded from below by a partial-slip, rigid surface. A steady shear flow driven by thermocapillary motion emerges. This dynamic liquid layer is susceptible to the onset of oblique three-dimensional hydrothermal waves, purely two-dimensional hydrothermal waves, longitudinal traveling waves, and longitudinal rolls depending on the capillary number. A low capillary number analysis finds that surface deformations are destabilizing for all modes of instability. There is a preference for two-dimensional hydrothermal waves when there are surface deformations. Though longitudinal traveling waves are never selected as the preferred mode of instability, these waves offer a convenient way to understand the behavior of oblique hydrothermal waves, which are near-longitudinal. This is especially the case for low capillary numbers, but oblique hydrothermal waves instead tend to align themselves with the direction of flow as the capillary number increases. Surface deformations affect longitudinal waves most significantly out of all the modes of instability, especially for low Prandtl numbers. The typical length scales shorten and the critical Marangoni numbers increase with the capillary number for all types of modes. Notably, the system selects long waves near a critical Prandtl number when the interface is nondeformable and when the layer is subject to partial slip, but this is no longer the case when the upper surface is deformable.

Linear Stability Analysis of Thermocapillary Flow in Rotating Shallow Pools Heated from Inner Wall

Thermocapillary flow of silicon melt (Pr=0.011) in shallow annular pool heated from inner wall was simulated at the dimensionless rotation rate ω ranging from 0~7000. The effect of pool rotation on the stability of the thermocapillary flow was investigated. The steady axisymmetric basic state was solved by using the spectral element method; the critical stability parameters were determined by linear stability analysis; the mechanism of the flow instability was explored by the analysis of energy balance. A stability diagram, exhibiting the variation of the critical Marangoni number versus the dimensionless rotation rate ω was presented. The results reveal that only one Hopf bifurcation point appeared in the intervals of ω 3965, and the corresponding instability was caused by the shear energy, which was provided by the thermocapillary force and pool rotation, respectively. In addition, the competition between thermocapillary force and pool rotation leads to three Hopf bifurcation points in the range of 3020<ω<3965 with the increase of Marangoni number.

Experimental study on the effect of surface evaporation on the thermocapillary-buoyancy convection in a shallow annular pool

In order to understand effect of surface evaporation on thermocapillary-buoyancy convection, a series of experiments on thermocapillary-buoyancy convection in a shallow annular pool with different surface evaporation rates have been conducted. The annular pool is subjected to a radial temperature gradient, and the inner and outer radii are respectively 20 mm and 40 mm. The 0.65 cSt silicone oil has been selected as the working fluid for its relatively strong volatility. Results show that the surface evaporation rate depends mainly on Marangoni number and the average temperature of the working fluid. At a weak evaporation, the critical Marangoni number increases with the increase of the surface evaporation rate and the liquid pool depth, i.e. the surface evaporation and buoyancy effects can suppress the flow destabilization. When the three-dimensional flow appears, the temperature fluctuation from the hydrothermal waves and the free surface fluctuation of the liquid layer are regular and coincident. However, in the case of the rapid evaporation, the vapor recoil can directly trigger the flow instability. Furthermore, the temperature fluctuation is weakened significantly, and the surface fluctuation of the liquid layer become irregular due to the vapor recoil effect.

Thermocapillary convection of moderate Prandtl number fluid in a shallow annular pool heated from inner cylinder with surface heat dissipation

This paper presents a series of three-dimensional numerical simulations on thermocapillary convection of moderate Prandtl number (Pr = 6.7) fluid in a shallow annular pool heated from the inner cylinder with surface heat dissipation. The aspect ratio and radius ratio of the annular pool are respectively fixed at ε = 0.05 and η = 0.5. Biot number varies from 0 to 40 and Marangoni number varies from 0.1 × 105 to 6.0 × 105. Results show that there is a large temperature drop near the heated cylinder and the flow becomes more intense than those when the outer cylinder is heated at adiabatic free surface. With the increase of Marangoni number, the flow will transit from the axisymmetric steady flow to the three-dimensional oscillatory flow. The critical Marangoni number decreases first, and then increases with the increase of Biot number. After the flow destabilization, thermocapillary convection with adiabatic free surface exhibits the multicellular wave pattern when the inner cylinder is heated, which is different from that when the outer cylinder is heated. The temperature fluctuation pattern depends on the coupling effect of thermocapillary convection and heat exchange on the free surface. The wave number increases first, and then decreases with the increase of Marangoni number at a fixed Biot number. Furthermore, as Marangoni number increases, the temperature fluctuation on the free surface will radially extend from the inner cylinder to the outer cylinder gradually, and the amplitude of temperature fluctuation increases obviously.

Three-Dimensional Numerical Simulation on Thermocapillary Convection of Moderate Prandtl Number Fluid in an Annular Shallow Pool with Surface Heat Dissipation

In order to understand the effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in an annular shallow pool, we performed a series of three-dimensional numerical simulations by using the finite volume method. An annular shallow pool is full of 0.65cSt silicone oil with Prandtl number of 6.7. The radius ratio and the aspect ratio of the pool are respectively fixed at 0.5 and 0.1. Surface heat dissipation Biot number is varied from 0 to 50. Results indicate that the critical Marangoni number of flow destabilization mainly depends on the coupling effect of the thermocapillary force and surface heat dissipation on the free surface, which decreases first, and then increases gradually. When Biot number is small, the steady axisymmetric flow after the flow destabilization will bifurcate to the hydrothermal wave. With the increase of Marangoni number, the flow pattern evolution at a small Biot number is similar to that of an adiabatic free surface, and the fundamental oscillatory frequency increases gradually. With the increase of Biot number, the radial roll cells near the outer cylindrical wall gradually extend to the inner wall, and eventually occupy the whole liquid pool. When Biot number is large, the flow pattern after the flow destabilization is a radial rolling cell pattern with alternating azimuthal direction. Then it gradually evolves into azimuthal waves, and the fundamental oscillatory frequency has a slight decrease.

Effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in a shallow annular pool

This paper presents a series of three-dimensional numerical simulations on the effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in a shallow annular pool. The annular pool with a fixed aspect ratio of 0.05 and radius ratio of 0.5 is filled with 0.65 cSt silicone oil. Its Prandtl number is 6.7. Biot number of surface heat dissipation ranges from 0 to 50. The results show that when Marangoni number is small, thermocapillary convection is the axisymmetric steady flow. When Marangoni number exceeds the critical value, the basic flow will destabilize and transit to the hydrothermal waves that propagate along the clockwise or anticlockwise direction. The critical Marangoni number of the flow destabilization increases with the increase of Biot number. Furthermore, when Biot number increases, the radial temperature gradient on the free surface decreases near the inner cylindrical wall, but increases near the outer cylindrical wall, which results in the thermocapillary convective cell moving gradually from the inner to the outer cylindrical walls. After the flow destabilizes, the temperature fluctuation on the free surface mainly appears near the outer cylinder at a large Biot number.

Surface Heat Dissipation Dependence of Thermocapillary Convection of Moderate Prandtl Number Fluid in an Annular Pool

In order to understand surface heat dissipation dependence of thermocapillary convection for moderate Prandtl number fluid in a deep annular pool, a series of three-dimensional numerical simulations have been carried out by using the finite volume method. The radius ratio and the aspect ratio of an annular pool are fixed at 0.5 and 1.0, respectively. The working fluid is 0.65cSt silicone oil with Prandtl number of 6.7. Surface heat dissipation Biot (Bi) number is varied from 0 to 50. Results indicate that with the increase of Biot number, the radial temperature gradient near the inner cylindrical wall decreases, and near the outer cylindrical wall it increases, so the flow is enhanced. When 0 < Bi < 10, with the increase of Marangoni number, the axisymmetric steady flow first transits to the standing wave, and then to the azimuthal waves. The standing wave should be attributed to Marangoni-Benard instability. However, the azimuthal waves should be corresponded to hydraulic instability, which is mainly driven by the azimuthal motion of temperature fluctuation from the sudden change of flow direction near the bottom and the inner cylindrical wall. When Bi ≥ 10, when the flow destabilizes, the axisymmetric steady flow transits directly to the azimuthal waves. With the increase of Biot number, the critical Marangoni number of the flow destabilization increases. Furthermore, the fundamental frequency and the wave number of three-dimensional oscillatory flow increase gradually with the increase of Biot number.

Bénard-Marangoni instability in sessile droplet evaporating at constant contact angle mode on heated substrate

Marangoni instabilities in a sessile droplet evaporating at constant contact angle mode on a heated substrate were investigated through a series of experiments. The quasi-steady state flower-like patterns of Bénard-Marangoni instabilities were observed, which uniformly distributed along the azimuthal direction of droplet. With the receding of triple line, the cell number of the Bénard-Marangoni convection decreases one by one following a square function of the evaporating time while the flower-like pattern maintains the quasi-steady state continuously. Different from the classical polygonal cells in flat liquid layer, the shape of cells in droplet is circular-arc. The critical Marangoni numbers for the incipience as well as those for the disappearance of the Bénard-Marangoni patterns were determined experimentally. Besides numerical simulations were conducted and the detailed characteristics of the Bénard-Marangoni instability patterns inside the droplet are explained thoroughly.

Surfactant-induced migration of a spherical droplet in non-isothermal Stokes flow

The motion of viscous droplet in an unbounded arbitrary (non-axisymmetric) Stokes flow under the combined influence of bulk-insoluble surfactant Marangoni stresses and thermal Marangoni stresses is studied analytically in two limiting cases, namely, low and high surface Péclet numbers. This work considers a more general model where the nonlinear variation of the interfacial tension is due to both thermal and surfactant gradients. It is well known that linear thermocapillary stresses assist migration of the droplet, while surfactants resist when the ambient thermal field is along the direction of the ambient hydrodynamic field. We have observed that this behavior prevails even when a nonlinear combination of thermal and surfactant stresses is considered. However, since the retardation due to surfactants is marginal, when combined linear thermal and linear surfactant stresses are superimposed, the thermal forces dominate the overall migration. The present work derives closed form expressions for the drift and the migration velocity where the capillary stresses can be non-axisymmetric and along the axial or transverse direction. This would enable one to design parameter combinations to control the droplet migration for possible use in various applications. Since the results are for any arbitrary ambient flow, we have provided the corresponding analysis when the ambient hydrodynamic flow is due to Poiseuille flow. The corresponding results when the thermal gradients are axial or transverse to the flow direction are discussed. In this regard, we contribute some important findings on the cross migration of the droplet. We observe that the droplet can move towards or away from the centerline depending on the critical thermal Marangoni number. We have seen the variation of velocity fields in different planes when the ambient flow field is a Poiseuille flow. Furthermore, we have computed the power (rate of work) on the surface of the droplet. The obtained analytical results are compared with the existing literature in all possible limiting cases. Finally, we draw a striking analogy with flow through porous media that the centroid of the droplet migrates with a velocity that is thought of as the volume averaged velocity inside a resistive medium.

Thermorheological and magnetorheological effects on Marangoni-Ferroconvection with internal heat generation

Marangoni convectiveinstability in a ferromagnetic fluid layer in the presence of a spatial heat sourceand viscosity variation is examined by means of the classical linear stability analysis. The higher order Rayleigh-Ritz technique is used to compute the critical Marangoni number. The effective viscosity of the ferromagnetic liquid is taken to be a quadratic function of both the temperature and magnetic field strength. It is shown that the ferromagnetic fluid is significantly influenced by the effect of viscosity variation and is more prone to instability in the presence of heat source compared to that when viscosity is constant. On comparing the corresponding results of heat source and heat sink it is found that heat sink works in tandem with the effect of viscosity variation if magnetic field dependence of viscosity dominates over temperature dependence. If the temperature dependence of viscosity dominates, the effects of viscosity variation and heat sink are mutually antagonistic.

Instability of the Benard–Marangoni Convection in a Porous Layer Affected by a Non-Vertical Magnetic Field

The onset of the Benard–Marangoni convection in a horizontal porous layer permeated by a magnetohydrodynamic fluid with a nonlinear magnetic permeability is examined. The porous layer is assumed to be governed by the Brinkman model; it is bounded by a rigid surface from below and by a non-deformable free surface from above and subjected to a non-vertical magnetic field. The critical effective Marangoni number and the critical Rayleigh number are obtained for different values of the effective Darcy number, Biot number, Chandrasekhar number, nonlinear magnetic parameter, and angle from the vertical axis for the cases of stationary convection and overstability. The related eigenvalue problem is solved by using the first-order Chebyshev polynomial method.

Thermocapillary instabilities of liquid layers on an inclined plane

The thermocapillary instabilities of the liquid layers on an inclined plane with zero heat flux are examined by linear stability analysis. There is an angle between the gravity and the thermocapillary force. The basic flow depends on the ratio of the gravity effect to thermocapillary effect. When the ratio is positive, the gravity always increases the velocity and destabilizes the flow. When the ratio is negative, the influence of gravity on the flow stability depends on the Prandtl (Pr) number. At high and moderate Pr, the critical Marangoni number always decreases with the ratio. The vertical convection is the key to the instability mechanism at high Pr, while the horizontal convection is dominant at moderate Pr. By contrast, at small Pr, the critical Marangoni number has a maximum and the energy from the basic flow is important for the perturbation energy.

Linear-stability analysis of thermocapillary convection in an annular two-layer system with free surface subjected to a radial temperature gradient

The linear-stability analyses of thermocapillary flow in an annular two-layer system subjected to a radial temperature gradient were conducted. The annular pool with upper free surface and adiabatic bottom is filled with immiscible double layer liquid of 5cSt silicone oil / HT-70. The outer wall of the annular pool is heated and inner wall is cooled. The results indicate that the critical Marangoni number decreases with the increase of aspect ratio and radius ratio. The effect of the ratio of the lower liquid thickness to the total double layer thickness on the convection onset is complicated. The minimum critical Marangoni number is located near the thickness ratio of 0.125. Moreover, there is one Hopf bifurcation from the steady to oscillatory flow when the thickness ratio varies from 0.625 to 0.75. At the same time, two types of flow destabilizing mechanism are identified. Comparing with the results with upper rigid wall, it can be found that the stability of thermocapillary convection in annular two-layer system with the free surface reduces largely.

Flow pattern transition and destabilization mechanism of thermocapillary convection for low Prandtl number fluid in a deep annular pool with surface heat dissipation

In order to understand clearly the flow pattern transition and the destabilization mechanism of thermocapillary convection for low Prandtl number fluids in a deep annular pool with surface heat dissipation, we carried out a series of three-dimensional numerical simulations by using the finite volume method. The radius ratio and the aspect ratio of an annular pool are respectively fixed at 0.5 and 1.0. Prandtl number of the working fluid is 0.011. Because the total heat dissipation coefficient on the free surface for low Prandtl fluids is small, Biot number is varied from 0 to 1.0. Results indicate that thermocapillary convection experiences the transitions from axisymmetric steady state flow into three-dimensional steady flow, and then into three-dimensional oscillation flow with the increase of Marangoni number. The critical Marangoni number of flow pattern transition decreases slightly with the increase of Biot number, and the maximum temperature and velocity fluctuations appear near the lower part of outer wall. The azimuthal temperature fluctuation on the free surface gradually shrinks to the inner wall, and the temperature fluctuation region decreases. However, it remains almost unchanged near the bottom of the annular pool, but the fluctuation amplitude increases.