Numerical modelling of heat transfer in the layer of viscous incompressible liquid with free boundaries

The dynamics of a viscous incompressible liquid layer and the temperature distribution in it are investigated numerically in threedimensional case. The planar layer with free boundaries under condition of zero gravity is studied on the basis of the special class of exact solutions of the Navier-Stokes equations. The thermocapillary forces and additional tangential stresses on the boundaries caused by the environment are taken into account. The influence of additional tangential stresses on the layer dynamics and heat distribution is studied.


Introduction
The problems of a nonstationary liquid flow in domains with free boundaries are rather difficult to study.Such problems may include the consideration of additional shear stresses caused by the external environment [1,2].Investigations of the flows in areas with free boundaries carried out with the use of exact solutions of special type are presented in [3,4].The numerical modelling of deformation of the viscous liquid layer with free boundaries and temperature distribution in it is given in [5][6][7].
In the present paper, the modelling of liquid flows which take place under the influence of thermocapillary forces and additional shear stresses on the free boundaries of infinite horizontal liquid layer is performed with the help of special class of exact solutions of the Navier-Stokes equations.The velocity field and temperature distribution in a layer with moving boundaries are determined by numerical algorithms.The results of numerical study of the dynamics of free boundaries and the temperature distribution within the layer depending on the direction of additional shear stresses are presented.

Statement of the problem and numerical algorithm
Consider the infinite plane-parallel layer of incompressible viscous thermally conducting fluid Ω = {(x, y, z): - < x < +, - < y < +, -Z(t) < z < Z(t)} under condition of zero gravity.Unknown function satisfies the Navier-Stokes and heat transfer equations: Here v = (u, v, w) is the velocity vector, p is the pressure, T is the temperature, Re is the Reynolds number (Re = υ* l/), Pr is the Prandtl number (Pr = ν/χ), ν and χ are the kinematic viscosity and thermal diffusivity coefficients, l is the characteristic length of the flow domain, υ* is the characteristic velocity, t * = l/υ* is the characteristic time.
Temperature and shear stresses τ1(x, t) and τ2(y, t) on free boundaries are the given functions of time and longitudinal coordinate: where A(t), B(t), Θ(t), τ(t) are the arbitrary functions that depend on time.
The kinematic and dynamic conditions on the free boundaries should be as follows: Here n = (0, 0, ±1) is the unit vector of external normal, s1 = (1, 0, 0) and s2 = (0, 1, 0) are the unit tangential vectors, D(v) is the tensor of deformation rate, , g P is the external pressure, ,   is the relations of the densities and kinematic viscosity coefficients of gas and liquid, Ma is the Marangoni number (Ma = σTT * l/(ρνχ)), T * is the characteristic temperature, 2 * * p   is the characteristic pressure, ρ is the liquid density, vg is the gas velocity, σT is the temperature coefficient of surface tension (σ = σ0 − σT(T − T0), σT > 0).It is assumed that the action of normal stresses from the external environment may be neglected.
Let the solution that determines the layer dynamics have the form [3,4]: .
The functions f(z, t), g(z, t) and the position of the free boundary Z(t) are defined as the solutions of the integro-differential equations [3,4].
Also the problem has been supplemented by conditions at infinity and initial position of the layer.
Functions f(z, t), g(z, t) and Z(t) are determined numerically using the "predictorcorrector" algorithm of second-order accuracy [5].
The values of temperature distribution in the layer can be determined in the parallelepiped Ω = {(x, y, z): -N < x < N, -L < y < L, -Z < z < Z}.The "lateral walls" x = ± N and y = ± L of the parallelepiped (so-called the working boundaries) are artificially introduced.The boundary conditions at the "lateral walls" are the result of the heat transfer equation and the conditions at infinity.Such conditions may be called the "smooth" conditions (see.[1,2]).The relations on the boundaries x = ± N and y = ± L of a parallelepiped can be specified as Txx = 0 and Tyy = 0, respectivelly.These conditions are obtained as a result of estimated execution expressions T → T∞ (T∞ = const) at infinity and additional relations on the assumption Tx → 0 on x → ±, Ty → 0 on y → ±.
EPJ Web of Conferences 159, 00047 (2017) DOI: 10.1051/epjconf/201715900047 AVTFG2016 The temperature function is determined numerically.Finite-difference scheme of second order approximation for the solution of the heat transfer equation is based on the stabilizing corrections method [8,9]:   Here Ki are the finite-difference analogues of the corresponding (каких именно) differential operators: K1  (1/(Re Pr))∂ 2 /∂z 2 , K2  (1/(Re Pr))∂ 2 /∂x 2 , K3  (1/(Re Pr))∂ 2 /∂y 2 ; Sk is the finite-difference analog of the convective term: S  -uTx -υTy -wTz; T k (x, y, z) = T(x, y, z, t k ), ∆t is the time step.The finite-difference scheme (1) is implemented using the grid (xn, yl, zm) which is movable along the vertical coordinate due to the mobility boundaries z = ±Z(t).Thus the grid along the coordinate z has the form: is the new position of the free boundary on the time layer k + 1.The Newton interpolation formula is used to transition to the new spatial grid at the new time step [7].The sweep method is applied to implementation the scheme (1) at each time sublayer.

Numerical results
In the present paper, the results of numerical calculations of the dynamics of the liquid layer and the temperature distribution in it are performed.The modelling of the liquid layer spreading in cases where the action of the thermocapillary forces can be intensified or reduced by the additional tangential stresses is presented.The position of the free liquid boundary at the initial time is defined as Z0 = 0.5, Θ is assumed to be 10.Functions A(t), B(t) and ( ) t The less intense layer spreading is observed in the case when the gas-related tangential forces have the opposite direction in comparison with the capillary forces (0 = -0.1,see fig. 1).The case when the additional tangential forces intensify the action of the capillary forces occurs for 0 = 0.1 (see fig. 2).
The research was supported by the Ministry of Education and Science of Russia (Agreement No 14.613.21.0011, project identifier RFMEFI61314X0011).