The new partially invariant solution of two-dimential motions of heated viscous liquid equations is considered. For factor-system arised the initial boundary value problem is formulated. This problem is inverse one and describing of common motion of two immiscible liquids in a plane channel under the action of thermocapillary forces. As Marangoni number is small (so-called creeping flow) the problem becomes the linear one. Some a priori estimates are obtained and input data conditions when solution tends to stationary one are found. In Laplace transforms the exact solution is obtained as quadratures and some numerical results of velocities behavior in layers are presented.
It is well known that in a non-uniformly heated liquid a motion can arise. In some
applications of liquid flows, a joint motion of two or more fluids with surfaces
takes place. If the liquids are not soluble in each other, they form a more or
less visual interfaces. The petroleum-water system is a typical example of this
situation. At the present time modelling of multiphase flows taking into account
different physical and chemical factors is needed for designing of cooling systems
and power plants, in biomedicine, for studying the growth of crystals and films,
in aerospace industry [
Nowadays, there are exact solutions of the Marangoni convection [
Thermocapillary convection problem for two incompressible liquids separated
by a closed interface in a container was investigated in [
Statement of the Problem The 2D motion of a viscid incompressible heat-conducting liquid in the absence of mass forces is described by the system of equations (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) 1 1 1 + 1 1 + 2 1 + = ( 1
+ 1 ), 2 + 1 2 + 2 2 + = ( 2
+ 2 ), 1 + 2 = 0, + 1 + 2 = ( + ).
Here 1(, , ) and 2(, ,
) are the components of the velocity vector, (, , ) is the pressure, (, , ) is the temperature, > 0 is the density, > 0 is the kinematic viscosity and >
0 is the thermal conductivity of the liquid.
The quantities > 0, > 0 and >
0 are constant.
The system of equation (2.1)–(2.4) admits a four-dimential Lie subalgebra solution of rank 2 and defect 3 should be sought for in the form 4 = ⟨ , 1 + , , ⟩. Its invariants are , , 2 and a partially invariant 1 = 1(, , ), 2 = (, ), = (, , ), = (, , ).
Inserting the exact form of the solution into the equations (2.1)–(2.3) yields 1 = (, ) + (, ), + = 0, + + 2 = ( ) + , 1 = (, ) − ( ) 2 2 lution of the form with some function ( ) that is arbitrary so far.
Regarding the temperature field, we assume that equation (2.4) has the so = (, ) 2 + (, ) + (, ). As we see below, (2.6) is in good accord with conditions on the interface.
The stationary solution of the Navier-Stokes equations in the form (2.5) for
= 0 for pure viscous fluid was found for the first time by [
It is known that the temperature dependence of the surface tension coefficient
is the one of the most important factors leading to the dynamic variety of the
interfacial surface. In the papers [
We assume for simplicity that (, ) ≡ 0, (, ) ≡ 0. The latter condition means that the temperature field has an extremum at = 0, more exactly, a maximum for (, ) < 0 and a minimum for (, ) > 0.
Let us apply the solution of the form (2.5), (2.6) to described joint motion of two immiscible liquids in the flat layer 0 < < ℎ considering that the wall = 0 and = ℎ are solid and the line : = (, ) is their common interface, see Fig. 1.
Introduction the index = 1, 2 for the liquids and using (2.5) and (2.6), we come to the conclusion that the unknowns satisfy the equations + + 2 =
+ ( ), + = 0, 1 ( ) 2
2
= (, ) −
, =
− − ,
+ 2 + = , + =
+ 2
in domain 0 < < (, ) for = 1 and in domain (, ) < < ℎ for = 2.
At the interface = (, ) the conditions hold [
(2.12) In (2.12) (u) is the strain-rate tensor, ( 1) is the surface tension coefficient, is the mean curvature of the interface, whereas ∇ = ∇ − n(n · ∇) on the right-hand side designates the surface gradient. For most of real liquid media the dependence ( 1) is approximated well by the linear function ( 1) = 0 − 1 , where 0 > 0 and > 0.
They are assumed constant and determent by experimental methods. Projecting condition (2.12) to the tangent direction = (1 + 2)−1/2(1, ), and using (2.13), (2.6) we obtain [ 2( 2 − 2) − 1( 1 − 1)] +
(1 − 2)( 2 2 − 1 1 ) = − ( 1 + 1 ) = − [2 1 + ( 1 2 + 1 )].
2 The projection (2.12) to the normal n yields 1 1 − 2 2 +
+ 2[ 2 (u2) − 1 (u1)]n · n [ 2 2( ) − 1 1( )] 2
2 = [ 0 − ( 1 2 + 1)] (1 + 2)3/2 .
The boundary conditions on the solid walls have the form 1(0, ) = 0, 1(0, ) = 0, 2(ℎ, ) = 0, 2(ℎ, ) = 0, with some given functions 0( ) and 0( ).
The initial conditions for the velocities are zero because of we study the properties of the solution of the problem simulating the motion only under the 0( ), (, 0) = 0( ), (, 0) = 0( ). action of thermocapillary forces (, 0) = 0, (, 0) = 0. Besides, (, 0) =
Note several specific features of the formulated problem. This is a nonlinear and inverse one since the functions ( ) are unknowns also. It is easy to understand if we exclude the functions (, ) from the equations of mass conservation. Then the problem reduces to the conjugate problem for the functions (, ), (, ) and (, ). The problem for (, ) given (, ) and (, ) can be separated. The functions (, ) can be recovered by quadrature from the second equation (2.8) up to a function of time. The last condition in (2.10) and the fourth from (2.16) are the additional conditions on ( ), = 1, 2.
Let us introduce the characteristic scales of length and time as well as functions , , , and , namely, the quantities 0, 02/ 1, 0 0/ 1, 0 02/ 1, 0, 0 0/ 1,
0/( 1 0), where 0 = const > 0 is the average value of thickness of the first layer of the liquid at = 0, 0 = m>a0x | 20( ) − 10( )| > 0, or 0 = max max | 0( )| > 0, if 20( ) = 10( ). In the dimensionless variables, some factor appears at the nonlinear terms in (2.7), the Marangoni number The same applies to the kinematic condition (2.11)
M = 0 03/( 1 1). ¯ ¯ + ¯M ¯(¯(,¯ ¯), ¯)¯¯ = M¯1(¯(,¯ ¯), ¯).
Assume that the M ≪ 1. The latter holds either in the thin layers or large viscosities. Then the nonlinear terms in the equations can be neglected and the latter become linear. In particular, the kinematic condition (2.20) has the form ¯¯ = 0, i.e. ¯ = ¯( ). Let us turn to (2.15). After transition to the dimensionless variables on the right-hand side the Weber number We = 0/( 0 02) appears instead of 0. In the real conditions We ≫ 1 for the most of liquid media; for example, for the water–air system We ∼ 106. ¯ =
Therefore, for these Weber numbers, (2.14) assume the form ¯¯¯ = 0, i.e. + 0. We assume later that
= 0 and the interface is the plane = 0 < ℎ parallel to the solid walls = 0 and = ℎ ; in what follows, the index 0 for 0 will be omitted. 3
A priori Estimates Let us present the so-obtained linear problem in its entirely in dimensional form =
+ ( ), (, 0) = 0, 1(0, ) = 0, 2(ℎ, ) = 0, 1(, ) = 2(, ), (2.19) (2.20) (3.1) (3.2) (3.3) (3.4) where 0 < < for = 1 and < < ℎ for = 2. The first equality in (3.6) follows from (2.10) whereas the last in the no-slip condition 2(ℎ, ) = 0.
Let us write the problem for the functions (, )
= , (, 0) = 0( ), (3.1)–(3.5), it is necessary firstly to infer the estimates for the solutions of initialboundary value problem (3.7)–(3.10). We perform the change of variables ︁∫ 0
2 2 (, ) − 1 1 (, ) = −2 1(, ), 1(, ) = 0, 2(, )
= 0, ︁∫ ℎ 1(, ) = ¯1(, ) + 2(, ) = ¯2(, ) + 10( )( − )2
2 20( )(
− )2 ,
The functions ¯ (, ) in their domains satisfy the equations ¯1 = 1¯1 ¯2 = 2¯2 + + 2 1 10( ) 2
− 2 2 20( ) (ℎ − )2 − ′10( )( − )2
2 ′20( )(
− )2 (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) (3.14) where the prime denotes differentiation with respect to time. Boundary conditions (3.9) for ¯1 and ¯2 become homogeneous, whereas (3.10) preserve it form. Initial conditions (3.8) for ¯1 and ¯2 change ¯1(, 0) = 10( ) − ¯2(, 0) = 20( ) − 10(0)( − )2 20(0)( − )2 ≡ ¯10( ), ≡ ¯20( ).
Let us multiply (3.1), (3.2) by 1 1¯1 and 2 2¯2 1, 2 and integrate over the segments [0, ], [, ℎ ] taking into account (3.8) and (3.9). Then add up the result. We infer that + 1 ¯12 + 2 ¯22 ( ) = 1 1 ∫︁ 2 0 where are the coefficients of the specific heat capacity. Along with (3.15) there is another identity
From (3.15) and (3.17) we obtain the inform estimates in ︁∫ 0 ︁∫ 0 ︁∫ ℎ ︁∫ ︁∫ ℎ 0
where
︂( 8 | (, )| 6 2 ( ) ( )
( ) = 1 ¯120( ) + 2
¯220( ) As to functions (, ), ( ) the following estimates hold | 1(, )| 6 2 ︂[ ( ) (︂ 1 ( ) + 4 2 21(, )︂)] 1/4
, ︂( 8 | 2(, )| 6 2 ( ) 2( ) 5 1 , | 1( )| 6 2 ︂[ 1( ) (︂ 1 3( ) + 4 2 ( ′1(, ))2 )︂] 1/4 + 122 1 (︂ 8 ( ) 5 1 1 | 2( )| 6 ︂( 8 1( )
,
The functions 1( ), 1( ), 2( ) and 3( ) have the same structures as ( ), ( ). 4 Stationary Flow The problem (3.1)–(3.10) has the stationary solution ( ), ( ), 2( ) =
ℎ 1( ) = (1 − )ℎ (3 2/ℎ 2
− 2/ℎ ) 1 =
2 2[ + (1 − )] (3 2/ℎ 2
− 2(2 + )/ℎ 2(1 − ) 2[ + (1 − )] [ (+20 −(1−10 ))] ℎ + 10, + 1 + 2 ) , 2 = + (1 − ) 1
︁[ ( 20 − 10)ℎ + 10 + (1 − ) 20︁] , 1 = − 3 (1 − ) 2[ + (1 − )] , 2 = −(1 − )ℎ 2[ + (1 − )] , 3 1( ) = −2 2[ + (1 − )] ℎ 3 − ℎ 2 , ︂( 3
2 )︂ ℎ ︂)
2 2( ) = −2(1 − ) 2[ + (1 − )] ℎ 3 − 3
︂( 2 −(2 + ) ℎ 2 − 2 + (1 + 2 ) ︂[( 3
︂) ︂(
︂)] ℎ − .
Introducing the differences (, ) = ( ) − (, ), (, ) = ( ) − (, ) ︂] , (3.23) (3.24) (3.25) (4.1) (4.2) (4.3) (4.4) ( ) and under the conditions of convergence of the integrals and carrying out the calculations analogous to those in Section 2, we can prove that the solution of the nonstationary problem reaches the steady regime ( ), ∞ on physical parameters of liquid and layers thicknesses. ‖ ( ) −
‖ 6 More exactly, ‖ (, ) − ( )
‖ 6 − 1 , ‖ (, ) − ( )‖ 6 − 2 , − 3 with the positive constant , , , 1, 2, 3 depending 5
Nonstationary Motion and Numerical Results To describe the nonstationary motion of two viscous thermally conducting liquids the Laplace transform will be applied to problem (3.1)–(3.10). As a result we come to boundary value problem for images ˆ (, ) of functions (, ) and images ˆ (, ) and ˆ ( ) of functions (, ), ( ) (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) (5.7) (5.8)
In condition (5.2) and equation (5.4) ˆ 0( ), ˆ ( ) are images of functions 0( ), ( ) respectively. The solutions of problem (5.1)–(5.8) can be written as ˆ ˆ − = − 0( )
, ˆ1(0, ) = ˆ10( ), ˆ2(ℎ, ) = ˆ20( ), ˆ1(, ) = ˆ2(, ), 1ˆ1 (, ) = 2ˆ2 (, ), ˆ
− ˆ = − ˆ ( )
, ˆ1(0, ) = 0, ˆ2(ℎ, ) = 0,
ˆ1(, ) = ˆ2(, ), 2 ˆ2 (, ) − 1 ˆ1 (, ) = −2 ˆ1(, ), ˆ1(, ) = 0, ˆ2(, )
= 0. ︁∫ ℎ ︁∫ 0
︂√
ˆ (, ) = 1 sh + 2 ch ︂√ − √ 1 ∫︁
︂[ ˆ (, ) = −2 ˆ1(, ) 1 sh
+ 2 ch ︂√ 0( ) sh ︂√ ︂√
+ ( ) ︂] ( − ) ,
(5.9) , (5.10) where ˆ ( ) = −2 ˆ1(, ) ( ).
The values 1, 2, 1, 2 and ˆ ( ) determined from the boundary conditions (5.2), (5.3), (5.5)–(5.8). Due to the cumbersome the type of these values is not given here.
Let us assume that lim 0( ) = 0, = 1, 2, using the formulas (5.9), (5.10) →∞ and presenting for the values 1, 2, 1, 2 and ˆ ( ) we can prove the limit equalities lim (, ) = ( ), →∞ lim (, ) = ( ), →∞ lim ( ) = , →∞ where ( ), ( ), are determined by formulas (4.1), (4.2).
Let us apply the numerical method of inversion of Laplace transformation to obtained formulas (5.9), (5.10). The graphs only for the velocities are given because the have a real physical meanings. All numerical calculations were made for the system of liquid silicon–water. Thickness of the layers is the same and equal to 1 mm. The corresponding values of the defining parameters are given in Table 1. ¯ w
0 −0.040
0.5 0.02 0.01 v¯ 0 −0.01 −0.020 0.08 0.04 ¯ w
0 −0.040 0.02 0.01 −0.020 0.5 1 ξ 1.5 2 Fig. 5. Evolution of functions ¯ for 10( ) = 1 + − cos(10 ). Total line is the stationary profiles, −− – = 1, − · − – = 4, · · · – = 2 6
The two-dimensional horizontal layer is a matter of great importance in connection with the theory of convective stability applications in the design of cooling systems, in studying the growth of crystals and films, or in the aerospace industry. We have presented a theoretical and numerical study of a creeping flow of two immiscible viscous heat conducting liquids in thin layers. The flow arises due to heat exchange with the localized parabolic heating of the borders and through the thermocapillary forces on the interface. The following results are obtained: (1) the exact solution describing the stationary thermocapillary convective flow is found; (2) a priori estimates of the initial boundary value problem are established and sufficient conditions on input data when solution tends to stationary one are obtained; (3) the solution of the non-stationary problem in the form of final analytical formulas in the Laplace representation is found and some numerical results of velocities behaviour in layers are presented. Acknowledgments. This research was supported by the Russian Foundation for Basic Research (14-01-00067).