CIS-01517; No of Pages 13 Advances in Colloid and Interface Science xxx (2015) xxx–xxx

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Modulation of Marangoni convection in liquid films Tatiana Gambaryan-Roisman ⁎ Institute for Technical Thermodynamics and Center of Smart Interfaces, Technische Universität Darmstadt, Alarich-Weiss-Str. 10, 64287 Darmstadt, Germany

a r t i c l e

i n f o

Available online xxxx Keywords: Marangoni convection Liquid film Marangoni instability Wall topography Non-uniform heating Non-uniform thermal properties

a b s t r a c t Non-isothermal liquid films are subject to short- and long-wave modes of Marangoni instability. The short-wave instability leads to the development of convection cells, whereas long-wave instability is one of the primary causes of the film rupture. In this paper different methods for modulation of Marangoni convection and Marangoni-induced interface deformation in non-isotherm liquid films are reviewed. These methods include modification of substrates through topographical features, using substrates with non-uniform thermal properties, non-uniform radiative heating of the liquid–gas interface and non-uniform heating of substrates. All these approaches aim at promotion of temperature gradients along the liquid–gas interface, which leads to emergence of thermocapillary stresses, to the development of vortices and to the interface deformation. Finally, Marangoni convection in a liquid film supported by a substrate with periodic temperature distribution is modeled by solution of steady state creeping flow equations. This approach is justified for low Reynolds numbers and for Marangoni convection in liquids with high Prandtl numbers. The model predicts interaction between Marangoni convection induced by non-uniform wall heating and the Marangoni short-wave instability. © 2015 Elsevier B.V. All rights reserved.

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modulation of Marangoni convection: approaches . . . . . . . . . . . . . . . . . . . . 2.1. Non-uniform wall heating . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Wall topography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Non-uniform thermal properties of the substrate . . . . . . . . . . . . . . . . . 2.4. Non-uniform heating or cooling of the liquid–gas interface . . . . . . . . . . . . 3. Short-scale Marangoni convection in liquid films with periodic wall temperature distribution 3.1. Governing equations and boundary conditions . . . . . . . . . . . . . . . . . . 3.2. Solution procedure and processing of results . . . . . . . . . . . . . . . . . . . 3.3. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Hydrodynamics and heat and mass transport processes in two-phase flows and at the phase interfaces are encountered in numerous natural phenomena and technical applications in energy conversion, process technology, material processing and microfluidics. The typical two-phase flow configurations include films or rivulet, droplets ⁎ Corresponding author. E-mail address: [email protected].

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0 0 0 0 0 0 0 0 0 0 0 0 0

and bubbles. The capillary forces which are determined by surface or interfacial tension play an important role in hydrodynamics of films, droplets and bubbles, especially if the ratio between the liquid–gas interface area and the volume of liquid is high (e.g. for thin films, small droplets and bubbles). The surface or interfacial tension is generally a function of the fluid composition and temperature. The surface tension of the common liquids decreases with increasing temperature. In the presence of concentration or temperature gradients along the interface between two phases the surface tension gradients induce interfacial stresses and,

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Please cite this article as: Gambaryan-Roisman T, Modulation of Marangoni convection in liquid films, Adv Colloid Interface Sci (2015), http:// dx.doi.org/10.1016/j.cis.2015.02.003

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therefore, bring the liquid into motion [1,2]. This phenomenon is known as Marangoni effect. Thermocapillary flows are Marangoni flows induced by interfacial temperature gradients. They inevitably arise in configurations, in which the temperature gradients are intentionally imposed. One of the examples is a classical configuration of a horizontal planar liquid layer with a free upper surface, which is bounded from two sides by walls having different temperatures [1,3–5]. In this case the nearly constant interface temperature gradient leads to the development of a return flow pattern in a film with a maximal velocity at the liquid–gas interface. This velocity is directed from the hot wall to the cold wall in the vicinity of interface and in the opposite direction within the film, so that at the steady state the total flow rate in each cross-section of the film is equal to zero. Further frequently studied configurations are motivated by the crystal growth techniques. Liquid bridges [4,6,7] are axisymmetric columns of liquid with a free lateral surface and top and bottom in contact with walls of different temperatures. The temperature gradient in liquid bridges is established along the lateral surface in an axial direction. The basic thermocapillary flow in a liquid bridge is characterized by toroidal streamlines. Another configuration is a layer of liquid between two concentric cylindrical walls (annular geometry) kept at different temperatures, where the top side of the annular region is a free surface subject to radial temperature gradient [4,8]. In this configuration the streamlines of the basic flow have a toroidal shape. In all the above configurations the basic flow pattern is unstable starting from a certain critical value of the Marangoni number defined as: Ma ¼

σ T ΔTl ; μa

ð1Þ

where μ is the dynamic viscosity, a is the thermal diffusivity of the liquid, l is the characteristic size, ΔT is the characteristic temperature    is the temperature coefficient difference in the system and σ T ¼ dσ dT of surface tension. The instability of the basic flow leads to development of new flow patterns, including convection cells, waves and oscillatory structures. The Marangoni effect is also important in configurations, in which the basic state is characterized by a constant interface temperature and zero velocity at each point within the liquid. A paradigmatic system for investigation of Marangoni convection is a horizontal liquid film of infinite horizontal extent and a thickness h covering a hot wall of a temperature T w and exposed to a cool gas of a temperature T g . In theoretical investigations it is normally assumed that the heat transfer between the liquid and the gas can be described as thermal boundary condition of the third kind with a constant heat transfer coefficient α. If the experiment is performed in an open air, then the heat transfer coefficient is determined by the free convection. The free convection in the gas can be suppressed by using an isothermal cover placed at a small distance from the liquid–gas interface [4,9]. In this case α is determined by the heat conduction in a gas layer. The evaporation of liquid can also be taken into account by an appropriate definition of α. Heat transport in a film in undisturbed state takes place due to a one-dimensional conduction, and the temperature at the liquid–gas interface is given by   T i ¼ T w − T w −T g

Bi ¼

αh ; k

Bi ; 1 þ Bi

ð2Þ

ð3Þ

where Bi denotes the Biot number and k is the thermal conductivity of the liquid.

The basic state of the quiescent liquid layer is subject to two kinds of thermocapillary instability [3,4]. The first of them is called short-wave instability and is induced by the following mechanism. If a certain point at the liquid–gas interface becomes hotter than the environment due to a temperature fluctuation, the local surface tension decreases. The temperature non-uniformity leads to development of thermocapillary stresses which pull the liquid from the hot spot to the colder regions. This local flow leads to the flow of the hot liquid from the wall to the interface, to compensate the Marangoni-induced outflow of the liquid. As a result, the local temperature further increases, the local surface tension decreases and the thermocapillary stresses become stronger. This process is opposed by the viscosity which reduces the flow velocity and by heat conduction which tends to level out the temperature differences. The random fluctuation leads to the development of persisting flow patterns if the Marangoni effect prevails over viscosity and heat conduction, or if the Marangoni number based on the film thickness as the characteristic length and ΔT = Tw − Ti exceeds a certain critical value. The critical Marangoni number has been first determined theoretically by Pearson [10] by a linear stability analysis under an assumption of a non-deformable liquid–gas interface. The resulting Marangoni number corresponding to neutral stability depends on the dimensionless wave number of disturbance K = 2πh/λ (where λ is the wavelength of the disturbance) and the Biot number: Macr ¼

16K ðK coshK þ Bi sinhK Þ½2K− sinhð2K Þ : 4K 3 coshK þ 3 sinhK− sinhð3K Þ

ð4Þ

For each Biot number the Macr(K) curve is characterized by a single minimum, Mamin. Below this value the quiescent film is stable. At Ma = Mamin the disturbance can grow for one certain wave number. For Ma N Mamin the film is unstable for disturbances in a certain range of wave numbers. For a constant wave number the critical Marangoni number increases with increasing Bi. For Bi = 0 the minimum of the Macr(K) curve corresponds to Mamin = 79.6 reached for K = 1.99 ≈ 2. This means that the wavelength of disturbance which starts to grow as soon as the Marangoni number reaches 79.6 is λ = πh, or has the same order of magnitude as the film thickness. The growth of disturbance leads to development of regular convective cells first observed and reported by Bénard [11]. The size and shape of these cells, as well as onset of instationary convection are determined by the Marangoni number [2,12,13]. The second limiting mode of instability is the so called long-wave instability, which is dominated by the deformability of the liquid–gas interface. A local fluctuation of the interface temperature (hot spot) leads, as discussed above, to the appearance of thermocapillary stresses pulling the liquid in the near-interface region from the hot spot outwards. This outflow leads to the local decrease of the film thickness. From Eqs. (2)–(3) it is seen that the interface temperature decreases with increasing the film thickness. Therefore the local film thinning leads to the further increase of the hot spot temperature and to increasing thermocapillary stresses. The self-sustaining film thinning may lead to the local film rupture. This process is opposed by the hydrostatic pressure which tends to level off the film thickness gradients towards a flat film. The local film thinning is also opposed by the surface tension which tends to reduce the gradients of the interface curvature. The stability of thin film in the presence of Marangoni effect has been studied theoretically in the framework of the long-wave theory [3,14,15]. This theory is applicable for liquid films with low inclination angle of the liquid–gas interface relative to the horizontal, in which the inertial forces and heat convection do not play an important role [16]. Application of the long-wave theory leads to an evolution equation for the film thickness:   i σ 1 h 3  2 2 μht þ ∇ h ∇ σ∇ h− ρgh − T ∇ h ∇T i ¼ 0; 3 2

ð5Þ

Please cite this article as: Gambaryan-Roisman T, Modulation of Marangoni convection in liquid films, Adv Colloid Interface Sci (2015), http:// dx.doi.org/10.1016/j.cis.2015.02.003

T. Gambaryan-Roisman / Advances in Colloid and Interface Science xxx (2015) xxx–xxx

where t denotes the time, g is the gravity, ρ is the liquid density and ∇ is the two-dimensional gradient operator. The interface temperature Ti depends on the local film thickness through dependence of Biot number on the film thickness (see Eqs. (2)–(3)). Linear stability analysis of Eq. (5) predicts that the film is unstable to every disturbance with a wavelength longer than the cut-off wavelength h i−1=2 e e λc ¼ 2πh0 M−Bo if Bo b M;

ð6Þ

e and the Bond number Bo where the modified Marangoni number M are defined as e ¼ M

  3σ T T w −T g 2σ

Bi0 2 ; Bo ¼ gh0 ρ=σ ð1 þ Bi0 Þ2

ð7Þ

and the Biot number based on the average film thickness h0 is Bi0 ¼ αh0 =k:

ð8Þ

e or If Bo N M;   3σ T T w −T g 2ρgh20

3

In several works the film hydrodynamics has been described with the Stokes equation instead of Navier–Stokes equations [25,26]. The Stokes equation disregards the inertial effects and is generally applicable at low Reynolds numbers. It has been shown [25] that the Stokes equation is applicable to the flows at very high Prandtl numbers, where the thermal convection effect is much more important than inertia. Boeck and Thess [26] show that the infinite Prandtl number limit can be applied starting from Pr = 100. Many silicone oils are characterized by Prandtl number in this range. Although in the basic state the above described one-layer system is quiescent and is characterized by constant interface temperature, it is possible to induce controlled Marangoni convection by a slight modification of the system. This controlled convection leads to improved mixing of liquid within the layer and to intensification of heat and mass transport within the layer and can be used for manipulation of small droplets and particles at the liquid–gas interface. In the following section several methods of control or modulation of Marangoni convection in liquid films are presented. The non-uniform wall heating effect is presented in details. The other methods are briefly discussed. The interested reader can find more details on the effects of wall topography and non-uniform wall thermal properties in [27]. 2. Modulation of Marangoni convection: approaches

Bi0 b1; ð1 þ Bi0 Þ2

ð9Þ

the film is fully stabilized by the hydrostatic pressure. Otherwise the disturbances with the wavelength shorter than λc are effectively suppressed by the surface tension, and the disturbances with the wavelength above λc lead to the film rupture [3,4,14]. This is the reason for calling this type of instability the long-wave instability. The long-wave thermocapillary instability, associated film rupture and appearance of dry spots have a deteriorating influence on the function of evaporators or heat transfer apparatuses relying on continuous film flow over hot substrates. Therefore suppression of this instability is an important task for the apparatus design. The two types of instability introduced above are, of course, the limiting cases. In reality, the deformability of the liquid–gas interface and the influence of heat convection on the film behavior are both significant [17]. The interaction and competition between the short-wave and long-wave instability modes have been a subject of numerous studies [9,18]. Van Hook et al. [9] have observed development of short-scale convection cells in fragments of liquid films' leftover after the film rupture induced by the long-wave instability. This is caused by the increase of film fragments' thickness in comparison with the film thickness before the rupture and hence increase of the Marangoni number (Eq. (1)). If the film Marangoni number is below the critical value for the onset of short-wave instability before the rupture and above the critical value after the rupture, the appearance of convective cells can be considered as a consequence of long-wave instability. Golovin et al. [18] have found that the short-scale interface temperature variation may suppress the growth of the long-wave disturbances and therefore exert a stabilizing action on the liquid film. In this case a competition between the long-wave and short-wave instabilities takes place. Theoretical and numerical descriptions of Marangoni instabilities and Marangoni-induced flow are a challenging task, since the position of the liquid–gas interface is not known a priori and should be computed as a part of the solution. One of the most popular methods for simulation of thin film evolution is the long-wave theory [14–16,19–21]. The applicability of the classical long-wave method is limited to the flows with low inclination angle of the liquid–gas interface, in which, in addition, the inertia and heat convection do not play a role. For the full-scale numerical simulation the finite element method in an ArbitraryLagrangian–Eulerian frame of reference [22] as well as the Volumeof-Fluid Method [5,23,24] has been used.

2.1. Non-uniform wall heating Non-uniform or local heating of the wall supporting the liquid film is the most widespread origin of imposed Marangoni flows in liquid films heated at the wall side and cooled at the liquid–gas interface side. Tan et al. [28] modeled a deformation of a liquid film over a wall with a spatially periodic temperature distribution in one direction along the wall (which will be referred to as x-direction). The model has been developed in the framework of the long-wave theory. The evolution equation developed in this work and describing the dependence of film thickness on x contained, in addition to terms appearing in a steady state version of Eq. (5), a term describing the intermolecular forces, including the components due to the van der Waals forces, electrostatic and structural forces [29], which are treated in form of so-called disjoining pressure [30]. This term becomes important for the film thickness below 100 nm and is normally used for description of phenomena in the vicinity of the three-phase contact line [31–35], in particular, the rupture of liquid films [21,36,37]. Tan et al. [28] have found that the wall temperature nonuniformity leads to the deformation of the liquid–gas interface and can even lead to the film rupture. By solving numerically the evolution equation in a range of governing parameters they found that a continuous steady solution of the evolution equation (steady deformed state of the film) exists if 3σ T ΔT w bRcr ; 2ρgh20

ð10Þ

where the critical value Rcr is of an order of unity. Herein ΔTw denotes a difference between the maximal and minimal wall temperature. Comparing this condition with the film stability condition in Eq. (9) it is clear that both are identical except for definition of characteristic temperature difference. The inequalities in Eqs. (9) and (10) express the fact that the film deformation induced by the Marangoni force is restricted by the hydrostatic pressure. Burelbach et al. [38] have experimentally validated the theory developed by Tan et al. [28]. They laid a silicone oil layer over a substrate with periodic temperature distribution arranged using heating and cooling blocks. The film thickness distribution was measured using a mechanical impedance probe. The experiments have been performed in the film thickness range from 125 μm to around 1700 μm, so that the stabilizing effect of hydrostatic pressure played a role. The agreement between the experimental and numerical results was excellent for the thickness around several hundreds of micrometers,

Please cite this article as: Gambaryan-Roisman T, Modulation of Marangoni convection in liquid films, Adv Colloid Interface Sci (2015), http:// dx.doi.org/10.1016/j.cis.2015.02.003

T. Gambaryan-Roisman / Advances in Colloid and Interface Science xxx (2015) xxx–xxx

but was much worse for thicker films where the measured film deformation was far below the predicted one. Among the reasons of this discrepancy are the possible onset of free convection, the increasing role of the heat transport between the liquid and the gas (neglected in the computations by setting Bi = 0 in Eq. (2)) and the short-wave Marangoni instability leading to appearance of convection cells in the system. Let us examine the behavior of interface deformation in linear approximation. Consider a variation of the wall temperature in the form   2πx T w ¼ T w0 þ ΔT w cos ; d

ð11Þ

where d is a spatial period of the wall temperature variation. The linearized form of Eq. (5) valid for small values of interface deformation relative to the average film thickness accepts a solution hðxÞ 3σ T ΔT w ¼ 1− h0 2σ ð1 þ Bi0 Þ

    −1 2πh0 2 2πx e ; þ Bo−M cos 0 d d

ð12Þ

e is computed using Eq. (7) with the average wall temperature where M 0 Two substituting Tw. Although Eq. (12) is only an approximate solution of the evolution equation, it can be used for an analysis of factors influencing the amplitude of deformation. According to Eq. (12), the amplitude of the liquid–gas interface deformation depends on Bi0 nonmonotonously e 0 on Bi0. However, for due to the nonmonotonous dependence of M the range relevant to the experiments reported by Burelbach et al. [38] the last term in the square brackets is insignificant, and the amplitude of the interface deformation decreases with increasing Bi0. Therefore, neglecting the heat transfer between the liquid and gas can lead to overestimation of the film deformation. Sellier and Panda [39] suggested a method for determination of the wall temperature distribution directly from the measurement of the deformation of film covering the wall. The authors started from the steady film evolution equation without the disjoining pressure term, considered an inverse problem and arrived at a closed form expression for the wall temperature distribution. The accuracy of the method, estimated by determination of wall temperature from the film deformation taken from [38] was around 12.5%. Examination of solution in Eq. (12) shows that the amplitude of the film deformation and therefore the sensitivity of the method proposed by Sellier and Panda [39] decrease with decreasing of typical longitudinal scale of temperature variation, and with increasing of average surface tension. Darhuber et al. [40] have shown that Marangoni stresses in liquid films and droplets can be induced by using substrates with embedded programmable heaters. In combination with specially designed wettability patterns on the substrate this system can be used for propulsion of droplets, the intensification of mixing within films and droplets as well as for the splitting of droplets. Mao et al. [41] suggest using non-uniform and time-dependent heating of the wall to pump liquids along the wall. Using the long-wave theory and two-dimensional Volume-of-Fluid direct numerical simulations, the authors predict the flow and temperature field in a liquid film covering a wall with imposed temperature field in a form of traveling wave. Mao et al. [41] have found that the thermocapillary stresses induced by non-uniform time-dependent wall heating lead to development of a traveling wave with a certain phase difference within the film. As a result, a directed motion of the fluid in the direction of the thermal wave propagation is induced. The thermocapillary pumping method suggested by the authors can find application in microfluidics. One of the typical applications of the liquid films is cooling of electronic devices, in which local hot spots are surrounded by unheated areas [42,43]. In the cooling systems the liquid covering the non-uniformly heated walls flows along the wall under the action of gravity [44–46] or shear stress exerted by a parallel gas flow [43,47–50].

The evaporation of liquid into the surrounding gas is often significant [43,48,50,51]. Consider a liquid film flowing under the action of gravity or a parallel gas flow (Fig. 1). The wall, the film and the local heater are assumed to be of infinite extent in the z-direction. The liquid film in the vicinity of heater is hotter than away from the heater. The thermocapillary stresses act from the heater region outwards. At the downstream edge of the heater the direction of thermocapillary flow and the direction of main flow coincide. At the upstream edge of the heater the thermocapillary flow is opposite to the main flow direction. These opposite streams lead to a local thickening of the film and to formation of bump (shown in Fig. 1) and the local film thinning at the downstream edge of the heater. If the temperature gradients induced by local heating are high enough, this bump is unstable to traverse perturbations [52]. This instability leads to appearance of structures periodic in the z-directions, which consist of lanes of thick film separated from each other by valleys of thin film. Depending on the properties of liquid, the flow conditions, the extent of heating, the heat transfer between the liquid and the gas and the wettability of the interface, the film can be ruptured at the valleys, so that the periodic structure leads to formation of rivulets flowing along the x-direction [43]. Numerous works published in the last 20 years are devoted to experimental, theoretical and numerical studies of the shape and stability of the bump, the shape and stability of the resulting transverse periodic structures and the effect of these complex phenomena on heat transport. In the majority of the experiments [42,43,52] and some theoretical and numerical works [48,50] the boundary conditions at the wall are posed for the heat flux, with the qualitative picture of the phenomena described in this paragraph staying valid. In experimental works, besides the classical falling films [42,52], the flow configurations in microchannels and minichannels have been extensively studied in the last decade [43,49]. The focus on micro- and minichannel configurations is motivated by applications in cooling of high power electronic devices. From the same reason, the fluorinert liquids like FC-72 became very widespread test liquids. One of the important goals of experimental investigations was the establishment of the regime map for the film flow (distinguishing between the smooth film flow regimes, regimes with two-dimensional and three-dimensional waves, film breakdown) depending on the hydrodynamic parameters and applied heat flux [43]. The film breakdown is associated with the heat transfer crisis and achievement of Critical Heat Flux (CHT) which sets the upper limit of heat flux which can be removed with a given cooling system. In experimental works the film thickness distribution has been characterized using fluorescence

x Tw = Tw0 + ΔTw Tw = Tw0

bump

Flow direction

4

Fig. 1. Sketch of a liquid film flowing over a wall with a local heater.

Please cite this article as: Gambaryan-Roisman T, Modulation of Marangoni convection in liquid films, Adv Colloid Interface Sci (2015), http:// dx.doi.org/10.1016/j.cis.2015.02.003

T. Gambaryan-Roisman / Advances in Colloid and Interface Science xxx (2015) xxx–xxx

method, capacitive sensors [52], and reflectance phase-shift Schlieren technique [43]. In addition, infrared thermography has been used for characterization of the liquid–gas interface [49]. The typical sizes of heaters used in experiments varied from 4 × 11 mm2 [49] to 150 × 150 mm2 [52]. It has been shown that using the shear-driven film flow much higher local heat fluxes can be applied without the film breakdown and, therefore, much higher heat fluxes can be removed in electronic devices. The majority of theoretical and numerical works aiming at description of hydrodynamics, stability and heat transfer with local heating have been fulfilled in the framework of the long-wave theory [44,46,51,53]. These works are devoted to falling films on vertical and inclined walls at low Reynolds numbers. Scheid et al. [44] considered two-dimensional film evolution for sinusoidal variation of wall temperature along the main flow direction. They found that the interaction between the classical long-wave Marangoni instability and the Marangoni flow induced by non-uniform heating depends on the ratio between the characteristic temperature differences along the wall and across the liquid film. Skotheim et al. [53] analyzed the linear stability of a liquid film subjected to non-uniform heating by an array of heaters and in the limiting case of a single heater. The authors have determined the critical Marangoni number for the onset of transverse instability observed in experiment. Liu and Kabov [46] studied the effect of size, shape and mutual location of heaters on falling film shape and stability. In all these works film evaporation has not been taken into account. Recently, Tiwari et al. [51] performed linear and non-linear stability analysis for evaporating film flowing along a wall with periodic embedded heaters. The evaporation rate has been computed from the kinetic theory under an assumption of evaporation into vapor. It should be noted that the interface temperature difference caused by the departure of thermodynamic equilibrium (as assumed by the authors) is usually very low and can lead to very weak Marangoni effect. Kalliadasis et al. [45] applied an integral boundary layer theory for description of the film shape and for stability analysis of a falling film over a wall with a single heater. The integral boundary layer theory allows a more realistic description of inertial effects in comparison with the long-wave theory. The analysis of Kalliadasis et al. predicts the appearance of bump seen in experiments, as well as the critical Marangoni number for the onset of spanwise instability, which decreases with decreasing Peclet number. Several theoretical and numerical works are devoted to description of film flow over a locally heated wall in the presence of co-current gas flow [47,48,50,54]. Gatapova and Kabov [48] describe theoretically the convective heat transport in a liquid film using a constant Biot number and constant interface shear stress and on the basis of computed temperature distribution on the liquid–gas interface determine the Marangoni-induced film deformation in the form of bump and, under certain conditions, additional ripples downstream. In the same paper the authors describe the effect of liquid evaporation by simultaneously solving the convective heat transport problem in the liquid film and in the gas as well as convective–diffusive vapor transport in the gas phase. They show that in the case of evaporation the assumption of constant Biot number is not justified. Kabova et al. [50] performed a three-dimensional simulation of co-current liquid and gas flow in a channel. The hydrodynamics has been described in the framework of lubrication approximation, whereas convective heat transport in both phases and convectivediffusive vapor transport in the gas phase have been taken into account. The concentration field in the gas phase is dominated by the development of concentration boundary layer. 2.2. Wall topography Using wall topography for the liquid film applications was motivated by heat and mass transfer enhancement [55]. The wall topography affects hydrodynamics and transport processes in a liquid film

5

through a multitude of physical mechanisms, including appearance of recirculation zones [56,57], change of the flow topology and formation of rivulets, accompanied by increasing evaporation rate [33], suppression of the wave formation in falling films [58,59] and prevention of film dryout through the capillary action. These mechanisms have been recently reviewed in [27]. In addition, wall topography induces thermocapillary flows [60–62]. Indeed, a horizontal liquid film on a wall with topographical features is characterized by non-uniform thickness. If the film covers a hot wall and is exposed to a cool gas, the liquid–gas interface temperature is higher at the locations of thin film (over the topography crests) than at the locations of thick film (topography troughs). The induced temperature gradients lead to appearance of thermocapillary stresses which cause the flow of the liquid in the vicinity of the liquid–gas interface from the hot sites (topography crests) to the cold sites (topography troughs). The mass conservation in the liquid film demands that the liquid returns back (to the sites of the topography crests) by an oppositely directed flow in the deeper regions of the film, forming vortices. In addition, the level of the liquid may rise at the cold interface locations and recede at the hot interface locations leading to film deformation. Symmetric wall topography leads to appearance of symmetric vortices which induce the liquid mixing in the film and lead to heat transfer enhancement [23]. Asymmetric topographic features bring an additional effect: they cause a net flow in a definite direction in the liquid film [61]. It can be shown that the magnitude of the induced velocity increases with decreasing of the characteristic lateral dimension of topographical features, whereas the magnitude of the interface deformation increases with increasing of this dimension [27]. The recirculation motion in the vortices is a short-scale phenomenon, whereas the film deformation is a long-scale phenomenon. Marangoni convection in a silicone oil film induced by a row of identical parallel grooves with comparable depth and width (short-scale structure) has been studied experimentally and numerically using the Volume-of-Fluid method [5,23]. The measured flow velocity reached 1.2 mm/s for a wall temperature 40 K above the ambient. The evolution of a liquid film over a long-scale topography of sinusoidal shape has been modeled using the long-wave theory and compared with the results of direct numerical simulations for several sets of parameters [63–65]. The wall topography has been shown to destabilize the liquid film leading to film rupture at a lower Marangoni numbers in comparison with the unstructured walls. However, the film rupture on substrates with topography takes place in a controlled way, so that the rupture points correspond to the topography crests, whereas the liquid is accumulated in the topography troughs. In this way using substrates with topography prevents the development of large dry patches. Several works are devoted to investigating the simultaneous effects of non-uniform wall heating and wall topography on film shape and stability [66,67]. If the Marangoni convection induced by topography overlaps with the flow driven by gravity [34], by centrifugal force [68] or other forces, complex flow patterns may evolve. Their detailed investigation will become a focus of future studies. 2.3. Non-uniform thermal properties of the substrate Volatile or non-volatile liquid films covering heated substrates with spatially varying thermal properties or volatile liquid films covering such substrates which are not necessarily heated exhibit non-uniform temperature of the liquid–gas interface [69–72]. The liquid–gas interface temperature over locations with higher thermal conductivity is in steady state higher than over locations with lower thermal conductivity. This temperature non-uniformity induces thermocapillary flows directed from the locations of high substrate thermal conductivity to the locations of low thermal conductivity. This phenomenon leads to the development of vortices and deformation of the liquid–gas interface, which has been studied using the long-wave theory [69,70]. Similar to the topography-

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T. Gambaryan-Roisman / Advances in Colloid and Interface Science xxx (2015) xxx–xxx

of less than 400 μm from the film surface. This hot point induces, as expected, a toroidal-shaped flow pattern with the flow directed outwards from the hot spot near the liquid–gas interface and with the flow directed inwards at the deeper layers of the film. The authors have measured a flow velocity of around 1.5 mm/s in a 140 μm-thick mineral oil film with an input power of 18 mW applied to the probe. The same authors have presented a device for manipulation of droplets floating on an oil film with the help of a 128-pixel array of individually programmable heaters placed at a distance of 100 to 500 μm over the film surface [74]. These heaters allow a local heating of liquid–gas interface thus inducing Marangoni flows, which, in turn, bring the floating droplets into motion. Wedershoven et al. [75] study the film deformation and rupture caused by an infrared laser irradiation. They measure the film rupture time as a function of laser power and the laser beam radius. The rupture time increases with increasing beam radius at a constant laser power, which can be explained by the reduction of the temperature gradients responsible for Marangoni stresses. At the lowest power of 12.5 mW the suppression of film deformation through the surface tension leads to slowing down of the film thinning and rupture. The control of the Marangoni convection by local and time-dependent radiation heating of the liquid–gas interface is a very promising technique. The optimization of this technique requires detailed investigations of the radiative heat transport between the source and the film surface as well as within the liquid film taking into account the spectral radiative properties of the medium and coupling of heat transfer analysis with description of hydrodynamics. A theoretical basis for such analysis in the long-wave approximation has been developed by Oron [76]. In this work the influence of uniform irradiation on film dynamics and stability has been studied.

Fig. 2. Sketch of liquid film with computational domain.

induced Marangoni convection and to the Marangoni convection induced by the non-uniform wall heating, the short-scale variation of the substrate properties results in weak deformation of film and relatively high recirculation velocities, whereas long-scale variation of substrate properties (with characteristic length scale of the substrate properties variation significantly exceeding the average film thickness) results in substantial deformation of liquid–gas interface [27]. 2.4. Non-uniform heating or cooling of the liquid–gas interface The methods for controlling or modulation of Marangoni convection described in the previous sections are designed to induce a controlled thermal field at the liquid–gas interface. However, this controlled temperature field is established not directly, but by modification of the substrate properties or by modification of thermal boundary conditions at the wall. In this section we briefly review the controlling of Marangoni convection by directly affecting the liquid–gas interface. Basu and Gianchandani [73] created a hot spot at the liquid–gas interface by placing thin resistance-heated cantilever tip at a distance

0.7

θi

0.6

0.5

(a) 0.4

0.3

(d) X

40

(e)

(b)

Umax

Ui

30

(c)

20

10

0

X

X Fig. 3. Simulation results for Bi = 2, Ma* = 1000, D = 4 and Δθ = 0.1. (a) constant temperature contours; (b) streamlines; (c) velocities vector plot and temperature color plot; (d) interface temperature distribution; (e) interface velocity distribution.

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T. Gambaryan-Roisman / Advances in Colloid and Interface Science xxx (2015) xxx–xxx

a

1.0E+03

Umax

1.0E+02

Bi= 2

1.0E+01

Bi= 0.01 1.0E+00

7

sinusoidal wall temperature distribution. The authors have also neglected the heat convection in the fluid, so that the velocity field is a function of temperature field, but the two-way coupling responsible for the development of short-wave instability was not taken into account. The theoretical–numerical two-dimensional model developed in this section is based on solution of creeping flow equations in spatially-periodic domain, as in [79]. Unlike the model in the work of Pendse and Esmaeeli, the model described in this section takes into account the convective heat transport in the film and is able to describe the interaction between the induced Marangoni convection and the intrinsic Marangoni convection driven by the short-wave instability. 3.1. Governing equations and boundary conditions

1.0E-01

Consider a liquid film of a thickness h on a substrate with periodically varying temperature (see Fig. 2). It is assumed that the velocity and temperature fields are stationary. This assumption is justified for liquids with large Prandtl numbers [12], for which also the assumption of creeping flow is justified [25]. The stream function in creeping flow obeys the biharmonic equation:

Ma

b

1.1

θi

0.9

Bi= 0.01

4

∇ ψ ¼ 0:

ð13Þ

The velocity components per definition are determined as partial derivatives of the stream function:

0.7

Bi= 2

0.5



∂ψ ∂ψ ; v¼− : ∂y ∂x

a

0.3

ð14Þ

60

Bi= 2

Ma Fig. 4. Maximal interface velocity (a) and average interface temperature (b) for D = 4 and Δθ = 0.1.

3. Short-scale Marangoni convection in liquid films with periodic wall temperature distribution It can be concluded from the review presented in the previous sections that the long-scale evolution of liquid films induced by the non-uniform wall heating, by the wall topography or by non-uniform thermal properties of the wall has been studied theoretically and numerically much more extensively than the short-scale Marangoni convection. However, the role of convective heat transport associated with short-scale flow field may be significant, especially if the characteristic length of interface temperature variation is comparable with the film thickness. To illustrate this effect, the Marangoni-induced convection on substrates with periodic wall temperature distribution is studied numerically in this section. It is assumed that the convection is a pure short-scale phenomenon in the given parameter range, i.e. that the liquid–gas interface is non-deformable. The theory is developed for liquids with high Prandtl numbers (for example silicone oils), so that the inertia can be neglected and the creeping flow regime in the film can be assumed [26]. Pendse and Esmaeeli [79] published an analytical solution for a Marangoni-induced creeping flow regime in a two-layer system with

Umax

Bi= 0.01 20

Bi= 0.01

b

Ma* = 100

Bi= 2

0

Δθ 1.4 1.2

Ma* = 1000

Bi= 0.01

1

θi

The choice of an appropriate method for controlling the Marangoni convection depends on a specific application. For example, the radiation-induced Marangoni convection can't be realized over a large film area and is of course useless for the cooling applications. Controlling the surface tension-driven flow using different surfactants characterized by different adsorption and desorption kinetics opens additional perspectives for time-dependent modulation of surface tension in multiphase systems [77,78].

Ma* = 1000

40

0.8 0.6

Ma* = 100 Ma* = 1000 Bi= 2

0.4 0.2

Ma* = 100

0

Δθ Fig. 5. Maximal interface velocity and average interface temperature for D = 4 as functions of Δθ.

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T. Gambaryan-Roisman / Advances in Colloid and Interface Science xxx (2015) xxx–xxx

a

b

c

d

e

Fig. 6. Constant temperature contours (upper plot) and streamlines (lower plot) for D = 4. The representation domain is 0 ≤ X ≤ 2, 0 ≤ Y ≤ 1. (a) Bi = 2, Ma⁎ = 100, Δθ = 0.05; (b) Bi = 2, Ma⁎ = 100, Δθ = 0.5; (c) Bi = 0.1, Ma⁎ = 1000, Δθ = 0.05; (d) Bi = 0.01, Ma⁎ = 1000, Δθ = 0.5; (e) Bi = 2, Ma⁎ = 1000, Δθ = 0.05.

The continuity equation is satisfied automatically. The temperature field satisfies the stationary form of the energy equation: 2

2

∂T ∂T ∂ T ∂ T þ u þv ¼a ∂x ∂y ∂x2 ∂y2

! :

ð15Þ

The non-penetration and non-slip boundary conditions at the wall (y = 0) have the form: ∂ψ ¼ 0; ψ ¼ const; ∂y

y ¼ 0:

ð16Þ

The prescribed wall temperature varies sinusoidally along the wall (Eq. (11)).

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T. Gambaryan-Roisman / Advances in Colloid and Interface Science xxx (2015) xxx–xxx

The symmetry boundary conditions for the velocity and temperature fields at x = 0 and x = d/2 are: ∂2 ψ ∂T ¼ 0; ψ ¼ const; ¼ 0; ∂x ∂x2

x ¼ 0; x ¼ d=2:

ð17Þ

The normal velocity component at the liquid–gas interface is equal to zero: ψ ¼ const;

y ¼ h:

ð18Þ

The constant value of the stream function is the same along the wall and along the symmetry planes and can be set equal to zero. The shear stress at the liquid–gas interface is determined by the Marangoni effect:

μ

∂u ∂T ∂2 ψ ∂T ¼ −σ T →μ 2 ¼ −σ T ; ∂y ∂x ∂x ∂y

a

y ¼ h:

ð19Þ

We assume that the heat transport between the liquid and the gas can be described using a boundary condition of a third kind with a constant heat transfer coefficient α, k

  ∂T þ α T−T g ¼ 0; ∂y

y ¼ h;

ð20Þ

where Tg is the temperature of ambient gas. We introduce the dimensionless variables and parameters in the following form: T−T g x y d uh vh ψ ; Y ¼ ; D¼ ; U¼ ; V¼ ; Ψ¼ ; θ¼ ; h h h a a a T w0 −T g   σ T T w0 −T g h 1 þ Bi  : ¼ Ma Ma ¼ Bi μa X¼

ð21Þ

In case of uniform wall temperature the definition of Ma in Eq. (21) coincides with that routinely used in the analysis of short-wave instability. The alternative definition of Marangoni number Ma* is based on the difference between the average wall temperature and the temperature of surrounding gas.

b

1

9

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0 0

0.5

1

1.5

2

2.5

3

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

0

0 0

0.5

1

1.5

2

2.5

3

c1 0.8 0.6 0.4 0.2 0 0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

1 0.8 0.6 0.4 0.2 0 Fig. 7. Constant temperature contours (upper plot) and streamlines (lower plot) for D = 2π, Bi = 2 and Ma⁎ = 1000. (a) Δθ = 0.005; (b) Δθ = 0.05; (c) Δθ = 0.5.

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T. Gambaryan-Roisman / Advances in Colloid and Interface Science xxx (2015) xxx–xxx

5

Application of the first and third boundary conditions from Eq. (26) leads to the relations:

4

 2 d f n   Bn 2  ¼ Ma βn φn ð1Þ; dY Y¼1

ð30aÞ

 dφn  þ Biφn ð1Þ ¼ 0: dY Y¼1

ð30bÞ

Ma*= 1000

Nu

3 2

Ma*= 100

The velocity and temperature field are coupled linearly through the condition in Eq. (30a) and nonlinearly through the energy equation, which after substitution of Eqs. (27) and (29) accepts the form:

1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X∞ dφ l βm Bm f m cosðβm X Þ l¼0 cosðβ l X Þ dY ! X∞ X∞ X∞ df m d2 φ n 2 −β φ − m−1 Bm sinðβm X Þ l¼0 βl φl sinðβl X Þ ¼ n¼0 n n cosðβ n X Þ: dY dY 2



Δθ Fig. 8. Nusselt number for D = 2π and Bi = 2.

X∞

m¼1

ð31Þ Then the governing equations accept the form: 4

After rearrangement this equation can be written as

∇ Ψ ¼ 0;

ð22Þ

∂Ψ ∂θ ∂Ψ ∂θ ∂2 θ ∂2 θ þ : þ ¼ ∂Y ∂X ∂X ∂Y ∂X 2 ∂Y 2

ð23Þ

  df m dφl βl φl þ f m βm þ cos½ðβm −β l ÞX  dY dY ! )   ∞ 2 df dφl d φn 2 − n¼0 cos½ðβm þ βl ÞX  − m βl φl þ f m β m −β φ n n cosðβ n X Þ: dY dY dY 2 −

1 2







m¼1

B l¼0 m

∑ ∑



ð32Þ

The dimensionless boundary conditions are:   ∂Ψ 2πX ; Y ¼ 0; ¼ 0; Ψ ¼ 0; θ ¼ 1 þ Δθ cos D ∂Y

ð24Þ

w where Δθ ¼ T ΔT−T , w0

g

2

∂ Ψ ∂θ ¼ 0; Ψ ¼ 0; ¼ 0; ∂X ∂X 2

X ¼ 0; X ¼ D=2;

∂2 ψ ∂θ  ∂θ ¼ −Ma ; Ψ ¼ 0; þ Biθ ¼ 0; Y ¼ 1: ∂X ∂Y ∂Y 2

ð25Þ

ð26Þ

3.2. Solution procedure and processing of results

X∞

B f ðY Þ sinðβn X Þ; n¼1 n n

θi ¼

1 D

ð27Þ

θi ¼

D 0

θðX; 1ÞdX ¼

Z ∞ 1X φn ð1Þ D n¼0

D 0

cosðβn X ÞdX ¼ φ0 ð1Þ:

1 : 1 þ Bi

where

ð33Þ

ð34Þ

1.4

f n ðY Þ ¼ sinðβn Y Þ−βn Y coshðβn Y Þ sinhðβn Y Þ þ Y ðβn coshβn − sinβn Þ ; sinhβn 2πn : D

1.2

ð28aÞ

1

ð28bÞ

Θ

βn ¼

Z

In the absence of heat convection (pure heat conduction in liquid film)

We look for the solution for the stream function in the form ΨðX; Y Þ ¼

This equation reduces to an infinite system of simultaneous ordinary differential second-order equations for φn, n = 0, 1, … The truncated system (typically up to n = 40) was solved numerically with the finite differences method implemented in a Matlab script. The convergence of computations has been assessed on the basis of computed periodaveraged interface temperature defined in the next paragraph and the maximal interface velocity. The obtained numerical results are used to compute the average interface temperature:

0.8

Ma* = 100

0.6

The solution in Eqs. (27)–(28a) and (28b) satisfies the governing Eq. (22) and all boundary conditions except for the first condition from Eq. (26), which is coupled with the temperature field. The solution for the temperature field is sought as θðX; Y Þ ¼

X∞ n¼0

φn ðY Þ cosðβn X Þ:

ð29Þ

This form of solution satisfies the symmetry boundary conditions at x = 0 and x = d/2.

0.4

Ma* = 1000

0.2 0

Δθ Fig. 9. Parameter Θ (Eq. 41) for D = 4 and Bi = 2.

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The Nusselt number (dimensionless heat transfer coefficient) is defined as qw h qw h ¼ ; Nu ¼ T w −T i k T w0 −T i k

ð35Þ

where qw denotes the average wall heat flux. In a steady state the average wall heat flux is equal to the average heat flux at the liquid– gas interface:   qw ¼ qi ¼ α T i −T g ;

ð36Þ

and, therefore,   qw h α T i −T g h θ ¼

Nu ¼ ¼ Bi i : T w0 −T i k T w0 −T i k 1−θi

ð37Þ

For the case of pure heat conduction, according to the above definition, Nu = 1. We make the first rough estimation of the effect of non-uniform wall heating on long-wave stability of liquid films. Assume that the typical length scale of the film deformation is much larger than the period of wall temperature variation, d, so that the film thickness over the length d is approximately uniform and equal to the period-averaged value h. The period-averaged film thickness and the period-averaged interface x≫d and temperature T i are functions of the long-scale coordinate e the long-scale time τ. It is also assumed that the long-scale velocity field in the liquid film, which accompanies the long-scale film deformation, does not affect the temperature field. This means that the period-averaged interface temperature is completely determined by solution of the problem posed by Eqs. (13), (15)–(20), and the period-averaged velocity is determined by the long-scale velocity associated with the film deformation. Under these assumptions the long-wave evolution Eq. (5) accepts the following form: μhτ þ

 i 1 h 3 σ h 2 i h σhexex − ρgh − T h ∇ T i ex ¼ 0; ex ex ex 3 2

ð38Þ

where the period-averaged interface temperature depends on the x: coordinate e x through the dependence of h on e

∂T T i ex ¼ i hex : ∂h

ð39Þ

Linear stability analysis of Eq. (38) brings to the following expression for the cut-off wavelength (cf. Eq. (6)): " λc ¼ 2πh0

#−1=2 3σ T h0 ∂T i 3σ h ∂T − if Bo b − T 0 i : −Bo 2σ ∂h 2σ ∂h

ð40Þ

Herein h0 denotes the undisturbed film thickness. This means that the first term in the square brackets in Eq. (40) governs the long-wave instability. We introduce a new parameter Θ¼−

3σ T h0 ∂T i ; ~ ∂h 2σ M

ð41Þ

e is given by Eq. (7). This parameter shows if convection stabilizes where M or destabilizes the film flow in comparison to the case of uniform wall temperature in the absence of short-wave instability. If Θ b 1, the convection stabilizes the film, and if Θ N 1, the convection enhances the long-wave instability.

11

3.3. Results and discussion The results of simulations for Bi = 2, Ma* = 1000 (Ma = 666.7), D = 4 and Δθ = 0.1 are presented in Fig. 3 within the computational domain, consisting of a half-period in the x-direction (0 ≤ X ≤ D/2 = 2). In all plots presented in this section the range in the y-direction is 0 ≤ Y ≤ 1 (from the wall to the liquid–gas interface). The isotherms and streamlines are represented in Fig. 3a and b, respectively. The vector velocity plots together with temperature color plot are shown in Fig. 3c, where the maximal temperature is represented in red and minimal temperature in blue. It is clear that the flow pattern consists of one clockwise-oriented vortex within the computation domain. The flow direction at the liquid–gas interface is from X = 0 (highest wall temperature) to X = D/2 (lowest wall temperature). The isotherms are not parallel to each other. This shows that the convective heat transport plays an important role. The temperature distribution at the liquid– gas interface is shown in Fig. 3d. As expected, the temperature decreases with X in the domain. However, the temperature drop at the liquid–gas interface is higher than the temperature drop at the wall (equal to 2Δθ = 0.2). The reason for that is that the convective flow brings the hot liquid from the hottest near-wall location (X = 0) directly to the liquid–gas interface. This solution could not be found under assumption that the heat transport takes place only by conduction (as in [79]). In Fig. 3e the dimensionless velocity distribution along the liquid–gas interface is represented. The velocity maximum is, along with the average interface temperature, an important quantity characterizing the role of Marangoni effect at given conditions. The influence of Marangoni number on the maximal interface velocity (in logarithmic scale) and on the average interface temperature (a semilogarithmic plot) is illustrated in Fig. 4 for D = 4 and Δθ = 0.1. As expected, both the maximal interface velocity and the average interface temperature increase with increasing of Marangoni number. For the case Bi = 2 both the Umax(Ma) and θi ðMaÞ curves are characterized by a sudden change of trend at around Ma N 150. At low Marangoni numbers the liquid–gas interface temperature is constant and close to the value 1/3, which corresponds to the heat conduction-dominated heat transfer. Starting from about Ma = 150, the interface temperature increases with increasing Marangoni number. The rate of growth of θi decreases in the range Ma N 1000. The dependence Umax(Ma) experiences a change of trend in the same range of Marangoni numbers. Namely, the fastest increase of Umax with Ma takes place for 150 b Ma b 600. This can be attributed to the short-wave stability properties of liquid layers heated from below. Indeed, the critical Marangoni number corresponding to the onset of short-wave instability for the wave number corresponding to the period of temperature variation K = 2π/D = 1.57 is equal to Macr = 183. Above this value of Marangoni number the Marangoni convection would be induced in the liquid film even without imposing the wall temperature variation. For Bi = 0.01 (weak heat transfer coefficient at the liquid–gas interface) the critical Marangoni number is Macr = 85. The change of the Umax(Ma) and θi ðMaÞ trends beyond the instability threshold can be observed in Fig. 3. An insight into the competition and interaction between the intrinsic Marangoni convection and the convection induced by variation of wall temperature can be attained by examining the dependencies Umax(Δθ) and θi ðΔθÞ (Fig. 5). Isotherms and streamlines corresponding to several characteristic points on the curves shown in Fig. 5 are plotted in Fig. 6. For Ma ⁎ = 100 and both values of Biot number the maximal flow velocity increases nearly linearly with increasing of Δθ, whereas at the limit of Δθ → 0 (constant wall temperature) the velocity is zero. This is an expected result for the case if the basic state of film with uniform wall temperature is stable (no Marangoni instability), and the flow is induced solely by the spatial variation of wall temperature. In particular, Pendse and Esmaeeli [79] come to the conclusion that the maximal velocity is proportional to the amplitude of temperature variation at the wall. Indeed, for Bi = 2 and Ma⁎ = 100 (Ma = 667) as well as for Bi = 0.01 and Ma⁎ = 100 (Ma = 0.99) the film

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with uniform wall temperature equal to Tw0 would be stable to shortwave disturbances of the wave number K = 2π/D. Even if the amplitude of spatial variation of wall temperature is substantial, the temperature field is dominated by heat conduction (examine the shape of isotherms in Fig. 6a and b), and the dimensionless period-averaged interface temperature stays close to 1/(1 + Bi). For Bi = 0.01 and Ma⁎ = 1000 (Ma = 9.9) the film with uniform wall temperature would be still stable. However, the effect of convective heat transfer can be observed for this set of parameters if the wall temperature distribution is non-uniform. The average interface temperature increases with increasing Δθ. In addition, the shape of isotherms changes substantially between the points Δθ = 0.05 and Δθ = 0.5 (compare Fig. 6c and d). The velocity depends on Δθ strongly and nonlinearly. Note that for Bi = 0.01 the average temperature at the liquid–gas interface in the presence of convection may lie above the average wall temperature, since the interface temperature is determined by the maximal wall temperature. The case Bi = 2, Ma* = 1000 (Ma = 666.7) is qualitatively different from those considered before. This case is characterized by a finite maximal velocity at the limit Δθ → 0, driven by the short-wave instability. This means that the convective heat transfer plays even for uniform wall temperature a dominant role. The isotherms represented in Fig. 6e are significantly distorted in comparison to Fig. 6a, in which the isotherms are nearly parallel to the wall, in spite of the fact that the amplitude of wall temperature variation is in both cases rather low. This distortion is a result of convection mixing the liquid inside a fluid domain. The streamlines in Figs. 3 and 6 correspond to a single vortex per half-period. This is not always the case. Fig. 7 displays the isotherms and streamlines for D = 2π, Bi = 2 and Ma⁎ = 1000 (Ma = 666.7). For Δθ = 0.005 (wall temperature variation amplitude close to zero) two vortices are filling the half-period of wall temperature variation. These two vortices also govern the temperature field (Fig. 7a). Evidently, for Ma = 666.7 the short-wave instability mode with the wave number K = 2π/(D/2) = 2 dominates over the mode K = 2π/D = 1. However, the temperature variation with the amplitude as small as Δθ = 0.05 is enough to change the flow pattern towards one vortex per half-period (Fig. 7b). Further increase of Δθ leads to the distortion of isotherms and to the change of the position of the vortex center (Fig. 7c). The change of the flow pattern affects the overall heat transfer in the liquid film. The Nusselt number as a function of Δθ for D = 2π, Bi = 2 and two values of Marangoni number is illustrated in Fig. 8. For Ma⁎ = 100 (Ma = 66.7) the Nusselt number is close to unity in the whole range of Δθ values. The effect of convective motion on heat transfer is insignificant. For Ma* = 1000 (Ma = 666.7) the convective heat transport dominates over conductive heat transport even for very small amplitudes of wall temperature variation. Moreover, the heat transfer is significantly enhanced by periodic variation of wall temperature. Indeed, for Δθ N 0.05 the Nusselt number increases with Δθ. However, in the small range 0 ≤ Δθ ≤ 0.05 the Nusselt number decreases with increasing Δθ. This behavior can be attributed to the forced change from the two-vortex flow pattern to the one-vortex pattern. Finally, the parameter Θ introduced in Eq. (41) is plotted in Fig. 9 for D = 4 and Bi = 2. For Ma⁎ = 100 (Ma = 66.7), according to this plot, Θ is equal to unity for Δθ → 0 and decreases with increasing Δθ. In this case the spatial variation of the wall temperature leads to increasing film stability to long-wave disturbances. In contrast, for Ma⁎ = 1000 (Ma = 666.7) Θ increases with increasing Δθ, which means that the wall temperature variation destabilizes the film. At the same time it is obvious that for this set of parameters the short-wave Marangoni convection suppresses the long-wave instability, since over a wide range of values of Δθ the parameter Θ is below unity. These first results indicate that influence of short-scale wall temperature variation on long-wave film stability is a rather complex phenomenon. Prediction of this influence requires a rigorous theoretical and numerical analysis and experimental validation.

4. Conclusions Several known methods of controlling or modulation of Marangoni convection in liquid films are reviewed. These methods use different mechanisms for inducing controlled non-uniform temperature field at the liquid–gas interface. The induced temperature gradients have two consequences: recirculation flow and film deformation which can lead to film rupture. The recirculation flow dominates for the short-scale temperature variation, whereas interface deformation dominates for the long-scale temperature variation. Interaction between the controlled Marangoni-driven flow and the directed flow driven by other forces (gravity, shear stress etc.) leads to development of complex flow structures and flow instabilities. A combined theoretical–numerical model of short-scale Marangonidriven flow and heat transfer in liquid films on walls with imposed periodic temperature is developed for liquids with high Prandtl number. It is shown that in the parameter range above the critical Marangoni number for the onset of short-wave instability the convection is important even for infinitely small amplitude of the wall temperature variation. It is also shown that the controlled Marangoni convection can lead to significant heat transfer enhancement. In the future the interaction and competition between the controlled short-scale Marangoni convection and the long-wave Marangoni instability should be studied by a rigorous analysis and the possibility to suppress the long-wave Marangoni instability by inducing controlled temperature gradients at the liquid–gas interface should be checked in a wide range of parameters. Acknowledgment This article has been prepared in the framework of the Marie Curie Initial Training Network “Complex Wetting Phenomena” (CoWet), Grant Agreement no. 607861 and the COST Action MP 1106. References [1] Levich VG. Physicochemical hydrodynamics. Englewood Cliffs, NJ: Prentice-Hall; 1962. [2] Nepomnyashchy AA, Velarde MG, Colinet P. Interfacial phenomena and convection. New York: Chapman and Hall/CRC; 2001. [3] Davis SH. Thermocapillary instabilities. Annu Rev Fluid Mech 1987;19:403–35. [4] Schatz MF, Neitzel GP. Experiments on thermocapillary instabilities. Annu Rev Fluid Mech 2001;33:93–127. [5] Alexeev A, Gambaryan-Roisman T, Stephan P. A numerical model for the thermocapillary flow and heat transfer in a thin liquid film on a microstructured wall. Int J Numer Methods Heat Fluid Flow 2007;17:247–62. [6] Preisser F, Schwabe D, Scharmann A. Steady and oscillatory thermocapillary convection in liquid columns with free cylindrical surface. J Fluid Mech 1983;126:545–67. [7] Wanschura M, Shevtsova VM, Kuhlmann HC, Rath HJ. Convective instability mechanisms in thermocapillary liquid bridges. Phys Fluids 1995;7:912–25. [8] Schwabe D, Möller U, Schneider J, Scharmann A. Instabilities of shallow dynamic thermocapillary liquid layers. Phys Fluids 1992;4:2368–81. [9] Van Hook SJ, Schatz MF, Swift JB, McCormick WD, Swinney HL. Long-wavelength surface-tension-driven Bénard convection: experiment and theory. J Fluid Mech 1997;345:45–78. [10] Pearson JRA. On convection cells induced by surface tension. J Fluid Mech 1958; 489–500. [11] Bénard H. Les tourbillons cellulaires dans une nappe liquid. J Phys Theor Appl 1900; 11:1261–8. [12] Eckert K, Bestehorn M, Thess A. Square cells in surface-tension-driven Bénard convection: experiment and theory. J Fluid Mech 1998;356:155–97. [13] Colinet P, Legros JC, Velarde MG. Nonlinear dynamics of surface-tension-driven instabilities. New York: Wiley; 2001. [14] Davis SH. Rupture of thin liquid films. Waves on fluid interfaces. In: Meyer RE, editor. New York: Academic Press; 1983. p. 291–302. [15] Kopbosynov BK, Pukhnachev VV. Thermocapillary flow in thin liquid films. Fluid Mech-Sov Res 1986;15:95–106. [16] Oron A, Davis SH, Bankoff SG. Long-scale evolution of thin liquid films. Rev Mod Phys 1997;69:931–80. [17] Scriven LE, Sterling CV. On cellular convection driven by surface-tension gradients: effects of mean surface tension and surface viscosity. J Fluid Mech 1964;19:321–40. [18] Golovin AA, Nepomnyashchy AA, Pismen L. Interaction between short-scale Marangoni convection and long-scale deformational instability. Phys Fluids 1994; 6:34–48.

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Please cite this article as: Gambaryan-Roisman T, Modulation of Marangoni convection in liquid films, Adv Colloid Interface Sci (2015), http:// dx.doi.org/10.1016/j.cis.2015.02.003

Modulation of Marangoni convection in liquid films.

Non-isothermal liquid films are subject to short- and long-wave modes of Marangoni instability. The short-wave instability leads to the development of...
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