Nonlinear behavior of an insoluble surfactant partially covering liquid during the transition to equilibrium

The process of relaxation to the mechanical and thermal equilibrium state of an insoluble surfactant film partially covering liquid has been investigated theoretically by the method of direct numerical simulation. Thermometric experimental data, which describe the film dynamics along the upper free boundary of a shallow Hele – Shaw cell, are analyzed and explained. At the initial stage, the temperature along the surface is non-uniform. This distribution generates a large-scale convective motion throughout the volume. The volume of the carrying liquid is relatively small. Therefore, the mutual action of the surfactant film and moving liquid on the dynamics of each other is turned to be significant. Thermo- and concentration-capillary forces on the surface, jointly with the buoyancy one in the volume, initiate the convection but after the heating stops, the non-monotonous propagation of the thermal front can be observed on the surface for some values of governing parameters. The calculations show that the characteristics of the motion are essentially different for temperature and concentration fields. This effect can be explained by the serious distinction of the kinetic coefficients for surfactant transfer and thermal processes.


Introduction
It is known that the free boundary of many liquids can not be pure in fact because an admixture is forced out from the volume on the surface continuously even in the case of small concentration. On the other hand, a monomolecular film of an insoluble surfactant can exist on the external boundary as a result of any natural reason. Often the interface can be originally coated by an impurity as a consequence of a technological process.
The different experimental methods are used in chemical and biological applications to observe the spatial and surface distributions of concentration. Most often, the standard interferometric technique is applied to monitor the field of surfactant and to control locally the pattern of distribution. Nevertheless, the optical methods are expensive, laborious and ineffective for the measurements in the case of extremely small values of concentration.
The experiments [1] demonstrate that the allocation of surfactant molecules with rare packing is very sensitive to a thermocapillary flow, which invariably originates under the action of an external longitudinal temperature gradient. The ability in principle of the surface purification with the help of  [2] for a shallow two-dimensional slot. The special configuration is considered in this work, within the framework of which the surfactant molecules are insufficient to result in a fully covered surface. Thus, the interface is partially clean and subject to non-zero stress, or contaminated by surfactant and no-slip. The theoretical study [2] laid foundation in an understanding of the joint dynamics of a mutually interactive surfactant and liquid under the interface.
Nevertheless, it is important in practice to investigate experimentally the ability of control the surfactant distribution by means of thermocapillary flow for applications. At the same time, it is impossible to affect on a redistribution of pollution for high values of surfactant concentration. Usually, the regular thermocapillary flow is relatively weak, and it can't open a window, free of surfactant, in the case of a thick film. It has been shown in [1], that the thermometric methods can be applied in experiment to visualize dynamics of the surfactant film along interface. This technique permits to register position of the boundary between the clean and impure parts of the surface, including the observation of its dynamics. The steady states and transitional regimes of these interfacial phenomena are interesting from the physical point of view and demonstrate the curious behaviour. Thus, the paper is devoted to the description of a nonlinear transfer of the insoluble surfactant on the gas-liquid interface during the relaxation to mechanical and thermal equilibrium in a Hele -Shaw cell.

Problem statement
The cavity with wide vertical sides in the plane (x, y) has the form of a shallow slot (Fig. 1a). The bottom and all lateral boundaries are rigid, while the upper horizontal surface is free. The clean water is used as a working liquid. Initially, the cavity is heated from above non-uniformly by a contactless optical device, which gives the ability to generate controllable heat flux q, directed vertically downward through the interface (x, z). The thickness 2d of the cavity is much smaller than the length L and the height H. A vector of the heat flux q across the surface has only y-component and its magnitude depends on x-coordinate linearly as in the experiment [3]. Under these thermal conditions, there is a motion of the fluid only in the plane of wide boundaries that leads to the convention for the transversal component of velocity vz = 0. Such approximation permits us to call this convective system a Hele -Shaw cell.  At the initial moment, the uniform film of the insoluble surfactant covers the interface. The oleic acid is considered as a surface-active substance in the course of numerical modeling to reproduce the results of the experiment. Thermocapillary and buoyancy forces produce convective flow in the volume, which compresses the surfactant film in a colder corner of the cavity due to the tangential stresses. Thus, a We use a standard system of differential equations in partial derivatives for the description of nonsteady convective flows in a non-uniformly heated cavity. There are classical equations of thermal convection in Boussinesq approximation [4]. In addition, the distribution of the insoluble surfactant follows the equation for surface concentration [5]: ( ) where v, p, T are the velocity, pressure, and temperature fields in non-dimensional form, Γ is the twodimensional field of surfactant concentration. Index s over the symbols ∇ and ∆ (nabla and Laplace operators correspondingly) means the differentiation on coordinates x and z in the plane of an interface. There are three non-dimensional governing parameters in the set of equations (1) -(3): Here Ra and Pr are the Rayleigh and Prandtl numbers respectively; ScS is the surface analogue of Schmidt number. Following values are taken as units for the length, time, velocity, temperature, pressure, and surface concentration: d, d 2 /ν, χ/d, dA/κ, ρνχ/d 2 , Γ*, where ν is the kinematic viscosity, χ is the thermal diffusivity, κ is the thermal conductivity, A is the amplitude of heat flux through the interface, ρ is the density of fluid, Γ* is the characteristic surfactant concentration. It will be discussed in more detail below.
The simulation criteria (4) include additional parameters; β, DS are the coefficients of thermal expansion and surface diffusion respectively, g is the magnitude of gravity acceleration. The balance of tangential stresses takes into account the action of thermo-and concentration-capillary forces on the upper boundary for y = H: where MaT, MaΓ are the temperature and concentration Marangoni numbers: ( In our examination, the constant Γ * determines the value of surface concentration for which the phase transition from gaseous state to the liquid one occurs, η is the dynamic viscosity, σT, σΓ are the coefficients that determine the dependence of surface tension on a temperature and surfactant concentration. The dependence of heat flux on the time along the upper boundary for y = H is modeled by the formula 0 < t < t1: where coefficient a′ is defined as a = dA κ ′ ′ ; t1 -the time moment, when the heating stops. The first condition determines the linear distribution of heat, transferred across the interface per unit of time on initial stage. The second one begins to work after the heating cutoff when the convective system gives back the heat into the environment. The emission of heat continues until the temperature becomes uniform, for which T → T0 (T0 is the environment temperature, a' is the empirical heat-transfer coefficient). Parameter a' determines the intensity of heat emission into surrounding space. In the course of our calculations, the non-dimensional coefficient a' was equal to 0.03. During numerical simulation, the hydrodynamic system comes at first into the steady motion under the influence of heating which is constant in dependence on time. Such stage is characterized by the presence of the region free of surfactant on the interface and the volumetric one vortex flow exists in the plane of wide boundaries. Then the heating is removed sharply and the system relaxes to thermal and mechanical equilibrium state. The motion of thermal front along the interface is presented in fig. 1b.
The steady states were accurately investigated experimentally and theoretically for realistic values of governing parameters in the work [1]. Good agreement of experimental data and calculation results was received in this work. On the other hand, the non-steady relaxation process of this convective system after the heating cutoff also demonstrates surprising behavior. Namely, the evolution to thermal and mechanical equilibrium is accompanied by the non-monotonous dynamics of the temperature field along the interface.
The problem (1) -(3) with boundary conditions (5), (7), (8) was solved numerically with the help of the mathematical package "Comsol Multiphysics". This software is a cross-platform package for the simulation of various physical processes, which has many applications for the numerical solution of the problems in the area of continuous media mechanics. The solution of standard problem during the simulation of thermal convection and interfacial phenomena has few stages, including the creation of the calculation grid, assignment of the initial and boundary conditions, implementation of the calculation procedure, and visualization of the obtained data. The non-uniform rectangular mesh was generated over the calculations that had 60:300 cells with the crowding near the up left corner. In the process of numerical simulation, we used Multifrontal Massively Parallel sparse direct Solver (MUMPS) and performed parallel calculations on 12 cores. The values of governing parameters were equal to L = 10, H = 5, Pr = 7, Ra = 13·10 5 , ScS = 800, MaT = 2·10 7 , MaΓ = 4000. Non-dimensional criteria were chosen in compliance with the following values of parameters: d = 0.002 m, β = 1.8·10 -4 1/K, ν = 10 -6 m 2 /s, χ = 14.4·10 -8 m 2 /s, DS = 10 -8 m 2 /s, η = 10 -4 Pa·s, κ = 0.6 Wt/(m·K), g = 9.8 m/s 2 , A' = 10 Wt/(m 2 ·K), A = 5000 Wt/m 2 , σT = 0.01 Pa·m/K, σΓ = 0.0008 Pa·m.

Results of calculation
In the course of numerical simulation, the conditions close to experiment are modeled. After the beginning of heating, the thermocapillary force produces Marangoni flow, which pushes the surfactant to the cold area of the surface. At the same time, concentration-capillary mechanism occurs because of increasing concentration inhomogeneity and stabilizes the flow, acting in an opposite direction. During the setting of a steady state two regions are formed on the interface. There are the free zone and the area covered by surfactant. These regions are divided by the stationary stagnation point. The coordinate of stagnation point can be obtained both from a profile of temperature and field of surface concentration. We suppose that it is necessary to distinguish the coordinates of stagnation points xT and xΓ, gotten from the analysis of temperature and concentration fields, respectively. In a steady state, these coordinates coincide but they become different for transitional regimes.  Thus, let us distinguish the "thermal" and "solutal" stagnation points xT, xΓ. The "thermal" stagnation point is identified using the kink on the linear temperature profile. Its initial position is taken from temperature dependence on longitudinal coordinate xT(t1) for steady state (curve 1, fig. 3a). The "solutal" stagnation point is determined on the edge of concentration field xΓ(t1) (curve 1, fig. 3b). The calculations show that the coordinates xT, xΓ of "thermal" and "solutal" stagnation point coincide in the steady state. As the time reaches the value t1 = 2000 s, the heating turns off and the system seeks to return into the former state with homogeneous distribution of surfactant.
At this stage, the diffusion and concentration-capillary mechanism dominate and the size of open window begins to decrease with the time. Meanwhile, because the cooling of interface could not be instant, the temperature gradient diminishes gradually along the up boundary and it is not enough to maintain earlier developed thermocapillary flow ( fig. 4a).   (fig. 4b). Finally, the cold flow reaches the left corner of the cell and free region is closed according to analysis of both temperature and concentration profiles. Meanwhile, the calculations show that there is an evident difference between characteristic values of relaxation time for the concentration field and the temperature one. If inclined profile of temperature completely disappears according to experimental observation one can think, that the surfactant completely covers the liquid. In fact, the real edge of the surfactant film can be far from the lateral narrow wall and free region of interface still exists.
It is necessary to note, that the position of "solutal" stagnation point is practically unaffected by the competition of conductive and convective mechanisms of heat transfer in a volume.

Conclusion
Direct numerical simulation has been fulfilled to explain a nonlinear dynamics of an insoluble surfactant film along the upper free boundary of a liquid. The surface-active substance moves under an influence of thermo-and concentration capillary forces. At first, the surfactant coats the interface partially and it is aggregated in the cold corner of a cavity. The process of relaxation to the mechanical and thermal equilibrium, after a heating off, demonstrates unexpected behaviour. On the one hand, the observation of the temperature field permits to visualize the evolution of the impurity film, but the direct thermometric technique can not be applied directly for determination of the stagnation point position. There is the cardinal difference between the dynamics of temperature field and concentration one during their transfer along the surface. It has been demonstrated that the nonlinear behaviour of the stagnation point motion, which is detected on temperature field, is the result of the simultaneous action of interfacial phenomena and thermal convection in the volume.