Numerical modelling of thermocapillary deformation in a locally heated thin horizontal volatile liquid layer

The thermocapillary flow in a thin horizontal layer of viscous incompressible liquid with free surface is considered. The deformable liquid layer is locally heated. The problem of thermocapillary deformation of the locally heated horizontal liquid layer has been solved numerically for two-dimensional unsteady case. The lubrication approximation theory is used. Capillary pressure, viscosity and gravity are taken into account. Evaporating rate is supposed to be proportional to the temperature difference between the liquid surface and ambient. Heat transfer in the substrate is also simulated. The deformation of the free surface has been calculated for different values of the heating power and thickness of the liquid layer. Initially the liquid layer has flat surface and uniform temperature. The model predicts the thermocapillary deformation of the liquid surface and the formation of the thin residual layer of the liquid. 1 Problem statement Heat transfer in thin liquid layers with local heating is an important challenge for thermal stabilization technique of electronic equipment [1-3]. The problem consider a thin horizontal volatile liquid layer on the substrate with local heater, Figure 1. Liquid is viscous and incompressible. Heater is thin and uniform. Two-dimensional unsteady problem is considered. Initially the liquid layer has flat surface and uniform constant temperature T0. At the time t0=0 the heater is activated and starts heating the substrate and liquid. Shear stress occurs on the liquid surface caused by the heterogeneity of its temperature. Thermocapillary flow and deformation of the liquid surface is developing. The mechanism of convective heat transfer in the liquid and deformation of surface in the heat problem is also taken into account. Gravity, surface tension, thermocapillary effect, viscosity are included in the evolution equation for the liquid layer thickness. Figure 1. Scheme of the process: liquid layer on the substrate with local heater. Liquid Substrate Heater ↓ g Ta – ambient temperature DOI: 10.1051/ , 00003 (2016) MATEC Web of Conferences matecconf/2016 84 8400003 International Symposium IPHT 2016 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). 2 Mathematical formulation 2.1 Basic equations 2.1.1 Motion equation of the liquid in the approximation of the lubrication theory The mathematical formulation of the problem includes the Navier-Stokes equations, continuity and thermal conductivity [4]:


Problem statement
Heat transfer in thin liquid layers with local heating is an important challenge for thermal stabilization technique of electronic equipment [1][2][3]. The problem consider a thin horizontal volatile liquid layer on the substrate with local heater, Figure 1. Liquid is viscous and incompressible. Heater is thin and uniform. Two-dimensional unsteady problem is considered. Initially the liquid layer has flat surface and uniform constant temperature T0. At the time t0=0 the heater is activated and starts heating the substrate and liquid. Shear stress occurs on the liquid surface caused by the heterogeneity of its temperature. Thermocapillary flow and deformation of the liquid surface is developing. The mechanism of convective heat transfer in the liquid and deformation of surface in the heat problem is also taken into account. Gravity, surface tension, thermocapillary effect, viscosity are included in the evolution equation for the liquid layer thickness. The mathematical formulation of the problem includes the Navier-Stokes equations, continuity and thermal conductivity [4]: The dimensional variables that are used into equations: u -velocity, F -vector of external forces, ȡ -density, p -capillary pressure, Ȟ -kinematic viscosity, T -temperature, a -thermal diffusivity.

The evolution equation for liquid film thickness
The evolution equation for liquid film thickness in the case of horizontal substrate, in thin layer approximation [4][5][6], is as follows: Where, t h -velocity of the surface, α -heat transfer coefficient for evaporation, lv r -latent heat of vaporization.

The energy equation
The law of temperature distribution in the liquid layer is determined by the energy equation: Where ( ) 0, outside the heater , , 0, on the heater -bulk density of the heat sources, the function ( ) , , Q t x z has a constant nonzero value in the area of the heater which is known from the experimental conditions, p a C λ ρ = ⋅ -thermal diffusivity; λ -coefficient of thermal conductivity; p C -specific heat; ρ -density of the liquid.

The boundary conditions
Since it is made the replacement of the variables for the rectification of the liquid surface, then move on to the conjugation conditions at the liquid-substrate interface.

The initial conditions
Initially the liquid layer has flat surface and uniform temperature:

Numerical modelling
The evolution equation of the liquid layer thickness (2) is approximated at grid's nodes with finite volume method [6]. System of nonlinear algebraic equations obtained by the approximation is solved with Newton's method. The Jacobians are calculated using numerical linearization [6].
Geometric parameters and parameters of the liquid, substrate and heater are specified.    T  T  Q  a  T  T  C   T  T  Q  a  T  T  C τ ρ τ ρ Initial condition: The resulting system of linear algebraic equations is solved at each step with the tridiagonal matrix algorithm. Calculations are performed successively. The time step for the evolution equation is done after the time step for the energy equation.

Numerical results
The deformation of the free surface has been calculated for different values of the heating power and thickness of the liquid layer. The model predicts the formation of the thin residual layer of the liquid and film breakdown at sufficiently intensive heating. Calculated deformations of the liquid layer with the time are shown in, Fig. 3. The thin residual layer is formed in the center, over the heater. It predicts the formation of dry spots. A local decreasing of the liquid surface temperature is shown in Fig. 4, curve 4. It is caused by the intense evaporation and decreasing of the thermal resistance of the liquid layer.

Conclusions
The numerical calculations with given model of thin horizontal liquid layer with local heating have shown that local heating of a horizontal liquid layer causes deformation of the liquid surface and thermocapillary flow. Significant reduction of liquid film thickness occurs in the greatest value of the temperature gradient zone. The model predicts the formation of the thin residual layer of the liquid. A local decreasing of the liquid surface temperature is caused by the intense evaporation and the decreasing of the thermal resistance of the liquid layer.