=Paper=
{{Paper
|id=Vol-1839/MIT2016-p23
|storemode=property
|title= On the inverse problem of a creeping motion in thin layers
|pdfUrl=https://ceur-ws.org/Vol-1839/MIT2016-p23.pdf
|volume=Vol-1839
|authors=Victor Andreev
}}
== On the inverse problem of a creeping motion in thin layers==
https://ceur-ws.org/Vol-1839/MIT2016-p23.pdf
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
On the Inverse Problem of a Creeping Motion
in Thin Layers
Victor Andreev
Institute Computational Modelling SB RAS, Siberian Federal University,
Krasnoyarsk, Russia
andr@icm.krasn.ru
Abstract. The new partially invariant solution of two-dimential mo-
tions 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 lin-
ear 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.
Keywords: Thermocapillarity, a priori estimates, conjugate initial-boundary
value problem, asymptotic behaviour, numerical simulation
1 Introduction
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 [1-4].
Nowadays, there are exact solutions of the Marangoni convection [5-7]. One
of the first solutions was obtained in [8]. This is the Poiseuille stationary flow
of two immiscible liquids in an inclined channel. As a rule, all such flows were
considered steady and unidirectional. The stability of such flows was investigated
in [9, 10]. As for non-stationary thermocapillary flows, studying of them began
recently [11, 12].
Thermocapillary convection problem for two incompressible liquids separated
by a closed interface in a container was investigated in [13]. Local (in time)
unique solvability of the problem was obtained in Holder classes of functions.
The problem of thermalcapillary 3D motion of a drop was studied in [14]. More-
over, its unique solvability in Holder spaces with a power-like weight at infinity
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was established. Velocity vector field decreases at infinity in the same way as
the initial data and mass forces, the temperature diverges to the constant which
is the limit of the initial temperature at infinity. The present work is devoted
to studying of solutions of a conjugate boundary value problem arising as a re-
sult of linearization of the Navier-Stokes system supplemented with temperature
equation. The description of the 2D creeping joint motion of two viscous heat
conducting fluids in flat layers is also provided here. The motion arises due to
thermocapillary forces imposed along two interfaces, after which the unsteady
Marangoni convection begins. Such kind of convection can dominate in flows
under microgravity conditions or in motions of thin liquid films.
2 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
1
π’1π‘ + π’1 π’1π₯ + π’2 π’1π¦ + ππ₯ = π(π’1π₯π₯ + π’1π¦π¦ ), (2.1)
π
1
π’2π‘ + π’1 π’2π₯ + π’2 π’2π¦ + ππ¦ = π(π’2π₯π₯ + π’2π¦π¦ ), (2.2)
π
π’1π₯ + π’2π¦ = 0, (2.3)
ππ‘ + π’1 ππ₯ + π’2 ππ¦ = π(ππ₯π₯ + ππ¦π¦ ). (2.4)
Here π’1 (π₯, π¦, π‘) and π’2 (π₯, π¦, π‘) are the components of the velocity vector,
π(π₯, π¦, π‘) is the pressure, π(π₯, π¦, π‘) is the temperature, π > 0 is the density, π > 0
is the kinematic viscosity and π > 0 is the thermal conductivity of the liquid.
The quantities π > 0, π > 0 and π > 0 are constant.
The system of equation (2.1)β(2.4) admits a four-dimential Lie subalgebra
πΊ4 = β¨ππ₯ , ππ’1 + π‘ππ₯ , ππ , ππ β©. Its invariants are π‘, π¦, π’2 and a partially invariant
solution of rank 2 and defect 3 should be sought for in the form
π’1 = π’1 (π₯, π¦, π‘), π’2 = π£(π¦, π‘), π = π(π₯, π¦, π‘), π = π(π₯, π¦, π‘).
Inserting the exact form of the solution into the equations (2.1)β(2.3) yields
π’1 = π€(π¦, π‘)π₯ + π(π¦, π‘), π€ + π£π¦ = 0,
1 π (π‘)π₯2
π€π‘ + π£π€π¦ + π€2 = π (π‘) + ππ€π¦π¦ , π = π(π¦, π‘) β , (2.5)
π 2
ππ¦ = ππ£π¦π¦ β π£π‘ β π£π£π¦ , ππ‘ + π£ππ¦ + π€π = 0
with some function π (π‘) that is arbitrary so far.
Regarding the temperature field, we assume that equation (2.4) has the so-
lution of the form
π = π(π¦, π‘)π₯2 + π(π¦, π‘)π₯ + π(π¦, π‘). (2.6)
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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
π = 0 for pure viscous fluid was found for the first time by [15]. It describes the
liquid impingement from infinity on the plane π¦ = 0 under the no slip condition
on it. In the paper [16], this solution for the flow between two plates or for the
flow in a cylindrical tube (axisymmetric analogue of solution (2.5)) was applied.
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 [17, 18] the stationary solutions in form (2.5),
(2.6) was found at π(π¦, π‘) β‘ 0, π = const for a flat layer with a free boundary
π¦ = π = const and a solid wall π¦ = 0. The non-uniqueness of solution depending
on the physical parameters of the problem was revealed. A similar problem in
the case of half space was investigated in [19].
We assume for simplicity that π(π¦, π‘) β‘ 0, π(π¦, π‘) β‘ 0. The latter condition
means that the temperature field has an extremum at π₯ = 0, more exactly, a
maximum for π(π¦, π‘) < 0 and a minimum for π(π¦, π‘) > 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 < π¦ < β considering that the wall
π¦ = 0 and π¦ = β are solid and the line π€ : π¦ = π(π₯, π‘) is their common interface,
see Fig. 1.
Fig. 1. Geometry of the Marangoni convection problem
Introduction the index π = 1, 2 for the liquids and using (2.5) and (2.6), we
come to the conclusion that the unknowns satisfy the equations
π€ππ‘ + π£π π€ππ¦ + π€π2 = ππ π€ππ¦π¦ + ππ (π‘), π€π + π£ππ¦ = 0, (2.7)
1 ππ (π‘)π₯2
ππ = ππ (π¦, π‘) β , πππ¦ = ππ π£ππ¦π¦ β π£ππ‘ β π£π π£ππ¦ , (2.8)
ππ 2
πππ‘ + 2π€π ππ + π£π πππ¦ = ππ πππ¦π¦ , πππ‘ + π£π πππ¦ = ππ πππ¦π¦ + 2ππ ππ (2.9)
in domain 0 < π¦ < π(π₯, π‘) for π = 1 and in domain π(π₯, π‘) < π¦ < β for π = 2.
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At the interface π¦ = π(π₯, π‘) the conditions hold [1]
π€1 (π(π₯, π‘), π‘) = π€2 (π(π₯, π‘), π‘), π£1 (π(π₯, π‘), π‘) = π£2 (π(π₯, π‘), π‘), (2.10)
ππ‘ + π₯π€1 (π(π₯, π‘), π‘)ππ₯ = π£1 (π(π₯, π‘), π‘), (2.11)
ππ1 ππ2
π1 (π(π₯, π‘), π‘) = π2 (π(π₯, π‘), π‘), π1 = π2 ,
ππ ππ
ππ1 ππ2
π1 (π(π₯, π‘), π‘) = π2 (π(π₯, π‘), π‘), π1 = π2 ,
ππ ππ
π1 > 0, π2 > 0 are the heat conductivity coefficients and n = (1 + ππ₯2 )β1/2 (βππ₯ , 1)
is the normal to the line π¦ = π(π₯, π‘).
The dynamic condition for π€ has a vector form [1]
(π1 β π2 )n + 2[π2 π·(u2 ) β π1 π·(u1 )]n = 2ππΎn + βπ€ π, ππ = ππ ππ . (2.12)
In (2.12) π·(u) is the strain-rate tensor, π(π1 ) is the surface tension coefficient,
πΎ is the mean curvature of the interface, whereas βπ€ = β β n(n Β· β) on the
right-hand side designates the surface gradient. For most of real liquid media
the dependence π(π1 ) is approximated well by the linear function
π(π1 ) = π 0 β π
π1 , (2.13)
where π 0 > 0 and π
> 0. They are assumed constant and determent by
experimental methods. Projecting condition (2.12) to the tangent direction
π = (1 + ππ₯2 )β1/2 (1, ππ₯ ), and using (2.13), (2.6) we obtain
π₯
ππ₯ [π2 (π£2π¦ β π€2 ) β π1 (π£1π¦ β π€1 )] + (1 β ππ₯2 )(π2 π€2π¦ β π1 π€1π¦ )
2
= βπ
(π1π₯ + ππ₯ π1π¦ ) = βπ
[2π1 π₯ + ππ₯ (π1π¦ π₯2 + π1π¦ )]. (2.14)
The projection (2.12) to the normal n yields
[π2 π2 (π‘) β π1 π1 (π‘)]π₯2
π1 π1 β π2 π2 + + 2[π2 π·(u2 ) β π1 π·(u1 )]n Β· n
2
ππ₯π₯
= [π 0 β π
(π1 π₯2 + π1 )] . (2.15)
(1 + ππ₯2 )3/2
The boundary conditions on the solid walls have the form
π€1 (0, π‘) = 0, π£1 (0, π‘) = 0, π€2 (β, π‘) = 0, π£2 (β, π‘) = 0, (2.16)
π1 (0, π‘) = π10 (π‘), π2 (β, π‘) = π20 (π‘), (2.17)
π1 (0, π‘) = π10 (π‘), π2 (β, π‘) = π20 (π‘), (2.18)
with some given functions ππ0 (π‘) and ππ0 (π‘).
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
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action of thermocapillary forces π€π (π¦, 0) = 0, π£π (π¦, 0) = 0. Besides, π(π₯, 0) =
π0 (π₯), ππ (π¦, 0) = π0π (π¦), ππ (π¦, 0) = π0π (π¦).
Note several specific features of the formulated problem. This is a nonlinear
and inverse one since the functions ππ (π‘) are unknowns also. It is easy to under-
stand if we exclude the functions π£π (π¦, π‘) from the equations of mass conservation.
Then the problem reduces to the conjugate problem for the functions π€π (π¦, π‘),
ππ (π¦, π‘) and π(π₯, π‘). The problem for ππ (π¦, π‘) given π£π (π¦, π‘) and ππ (π¦, π‘) can be sep-
arated. The functions ππ (π¦, π‘) 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 ππ (π‘), π = 1, 2.
Let us introduce the characteristic scales of length and time as well as func-
tions π€π , π£π , ππ , ππ and ππ , namely, the quantities π0 , π02 /π1 , π
π0 π0 /π1 , π
π0 π02 /π1 ,
π0 , π
π0 π0 /π1 , π
π0 /(π1 π0 ), where π0 = const > 0 is the average value of thickness
of the first layer of the liquid at π‘ = 0, π0 = max |π20 (π‘) β π10 (π‘)| > 0, or
π‘>0
π0 = max max |ππ0 (π¦)| > 0, if π20 (π‘) = π10 (π‘). In the dimensionless variables,
π π¦
some factor appears at the nonlinear terms in (2.7), the Marangoni number
M = π
π0 π03 /(π1 π1 ). (2.19)
The same applies to the kinematic condition (2.11)
Β―ππ‘Β― + π₯ Β― Β―π(Β―
Β―Mπ€( π₯, π‘Β―), π‘Β―)Β―ππ₯Β― = MΒ―
π£1 (Β―π(Β―
π₯, π‘Β―), π‘Β―). (2.20)
Assume that the M βͺ 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
Β―ππ‘Β― = 0, i.e. Β―π = Β―π(π₯). Let us turn to (2.15). After transition to the dimensionless
variables on the right-hand side the Weber number We = π 0 /(π
π0 π02 ) appears
instead of π 0 . In the real conditions We β« 1 for the most of liquid media; for
example, for the waterβair system We βΌ 106 .
Therefore, for these Weber numbers, (2.14) assume the form Β―ππ₯Β―π₯Β― = 0, i.e.
Β―π = πΌπ₯ + π0 . We assume later that πΌ = 0 and the interface is the plane π¦ = π0 < β
parallel to the solid walls π¦ = 0 and π¦ = β; in what follows, the index 0 for π0
will be omitted.
3 A priori Estimates
Let us present the so-obtained linear problem in its entirely in dimensional form
π€ππ‘ = ππ π€ππ¦π¦ + ππ (π‘), (3.1)
π€π (π¦, 0) = 0, (3.2)
π€1 (0, π‘) = 0, π€2 (β, π‘) = 0, (3.3)
π€1 (π, π‘) = π€2 (π, π‘), (3.4)
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π2 π€2π¦ (π, π‘) β π1 π€1π¦ (π, π‘) = β2π
π1 (π, π‘), (3.5)
β«οΈ π β«οΈβ
π€1 (π§, π‘) ππ§ = 0, π€2 (π§, π‘) ππ§ = 0, (3.6)
0 π
where 0 < π¦ < π for π = 1 and π < π¦ < β for π = 2. The first equality in (3.6)
follows from (2.10) whereas the last in the no-slip condition π£2 (β, π‘) = 0.
Let us write the problem for the functions ππ (π¦, π‘)
πππ‘ = ππ πππ¦π¦ , (3.7)
ππ (π¦, 0) = π0π (π¦), (3.8)
π1 (0, π‘) = π10 (π‘), π2 (β, π‘) = π20 (π‘), (3.9)
π1 (π, π‘) = π2 (π, π‘), π1 π1π¦ (π, π‘) = π2 π2π¦ (π, π‘). (3.10)
In order to obtain a priori estimates for π€π (π¦, π‘), ππ (π‘) of the solution of
(3.1)β(3.5), it is necessary firstly to infer the estimates for the solutions of initial-
boundary value problem (3.7)β(3.10). We perform the change of variables
π10 (π‘)(π¦ β π)2
π1 (π¦, π‘) = π
Β―1 (π¦, π‘) + , 0 6 π¦ 6 π0 β‘ π,
π2
(3.11)
π20 (π‘)(π¦ β π)2
π2 (π¦, π‘) = π
Β―2 (π¦, π‘) + , π 6 π¦ 6 β.
(β β π)2
The functions π
Β―π (π¦, π‘) in their domains satisfy the equations
2π1 π10 (π‘) πβ²10 (π‘)(π¦ β π)2
π
Β―1π‘ = π1 π
Β―1π¦π¦ + β β‘ π1 π
Β―1π¦π¦ + π1 (π¦, π‘), (3.12)
π2 π2
2π2 π20 (π‘) πβ²20 (π‘)(π¦ β π)2
π
Β―2π‘ = π2 π
Β―2π¦π¦ + β β‘ π2 π
Β―2π¦π¦ + π2 (π¦, π‘), (3.13)
(β β π)2 (β β π)2
where the prime denotes differentiation with respect to time. Boundary condi-
tions (3.9) for π
Β―1 and π
Β―2 become homogeneous, whereas (3.10) preserve it form.
Initial conditions (3.8) for π
Β―1 and π
Β―2 change
π10 (0)(π¦ β π)2
Β―1 (π¦, 0) = π01 (π¦) β
π Β―01 (π¦),
β‘π
π2
(3.14)
2
π20 (0)(π¦ β π)
Β―2 (π¦, 0) = π02 (π¦) β
π 2
Β―02 (π¦).
β‘π
π
Let us multiply (3.1), (3.2) by π1 π1 πΒ―1 and π2 π2 π
Β―2 π1 , π2 and integrate over
the segments [0, π], [π, β] taking into account (3.8) and (3.9). Then add up the
result. We infer that
β«οΈ π β«οΈβ β«οΈ π β«οΈβ
ππ΄(π‘)
+ π1 Β―21π¦ ππ¦ + π2
π Β―22π¦ ππ¦ = π1 π1
π π1 π
Β―1 ππ¦ + π2 π2 π2 π
Β―2 ππ¦, (3.15)
ππ‘
0 π 0 π
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β«οΈ π β«οΈβ
π 1 π1 π 2 π2
π΄(π‘) = Β―21 ππ¦ +
π Β―22 ππ¦,
π (3.16)
2 2
0 π
where ππ are the coefficients of the specific heat capacity. Along with (3.15) there
is another identity
β«οΈ π β«οΈβ [οΈ β«οΈ π β«οΈβ ]οΈ
1 π
π1 π1 Β―21π‘ ππ¦ + π2 π2
π Β―22π‘ ππ¦ +
π π1 π 2
Β―1π¦ ππ¦ + π2 π 2
Β―2π¦ ππ¦
2 ππ‘
0 π 0 π
β«οΈ π β«οΈβ
= π1 π1 π1 π
Β―1π‘ ππ¦ + π2 π2 π2 π
Β―2π‘ ππ¦. (3.17)
0 π
From (3.15) and (3.17) we obtain the inform estimates in π¦
(οΈ )οΈ1/4
8ππ
|ππ (π¦, π‘)| 6 πΉ (π‘)π΄(π‘) + |ππ0 (π‘)|, (3.18)
ππ2
where
β«οΈ π β«οΈβ [οΈ β«οΈπ‘ β«οΈπ‘ ]οΈ
2π1 4π1 π
πΉ (π‘) = π1 Β―210 (π¦) ππ¦+π2
π Β―220 (π¦) ππ¦+
π π210 (π ) ππ + (πβ²10 (π ))2 ππ
π1 π3 5
0 π 0 0
[οΈ β«οΈπ‘ β«οΈπ‘ ]οΈ
2π2 4π2 2 ββπ β² 2
+ π20 (π ) ππ + (π20 (π )) ππ β‘ πΉ (π‘), (3.19)
π2 (β β π)3 5
0 0
[οΈ βοΈ (οΈ β«οΈπ‘ βοΈ β«οΈπ‘ )οΈ
βοΈ π1 2π π
π΄(π‘) 6 πβ2πΏπ‘ π΄(0) + β1 πΏπ
π |π10 (π )| ππ + π πΏπ
|πβ²10 (π )| ππ
π1 π3 5
0 0
βοΈ (οΈ β«οΈπ‘ βοΈ β«οΈπ‘ )οΈ]οΈ2
π2 2π2 ββπ
+ βοΈ πΏπ
π |π20 (π )| ππ + π πΏπ
|πβ²20 (π )| ππ . (3.20)
π2 (β β π)3 5
0 0
As to functions π€π (π¦, π‘), ππ (π‘) the following estimates hold
[οΈ (οΈ )οΈ]οΈ1/4
πΈ(π‘) 4π
2 ππ21 (π, π‘)
|π€1 (π¦, π‘)| 6 2 πΉ (π‘) + , (3.21)
π1 5π1
)οΈ1/4 (οΈ
8
|π€2 (π¦, π‘)| 6 πΈ(π‘)πΉ2 (π‘) , (3.22)
π2
[οΈ (οΈ )οΈ]οΈ1/4 (οΈ )οΈ1/4
πΈ1 (π‘) 4π
2 π(πβ²1 (π, π‘))2 12π1 8πΈ(π‘)
|π1 (π‘)| 6 2 πΉ3 (π‘) + + 2 πΉ2 (π‘) ,
π1 5π1 π π1
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(οΈ )οΈ1/4 (οΈ )οΈ1/4
8πΈ1 (π‘) 12π2 8πΈ(π‘)
|π2 (π‘)| 6 πΉ3 (π‘) + πΉ 2 (π‘) , (3.23)
π2 (β β π)2 π2
where
β«οΈπ‘
πΈ(π‘) 6 πβ4πΏ1 π‘ π»(π )π4πΏ1 π ππ, (3.24)
0
[οΈ(οΈ )οΈ1/2 ]οΈ
2π
8π1 2
π»(π‘) = πΉ (π‘)π΄(π‘) + π10 (π‘) . (3.25)
π π12
The functions πΈ1 (π‘), πΉ1 (π‘), πΉ2 (π‘) and πΉ3 (π‘) have the same structures as πΈ(π‘),
πΉ (π‘).
4 Stationary Flow
The problem (3.1)β(3.10) has the stationary solution π€ππ (π¦), ππ π (π¦), πππ
π
(1 β πΎ)π΄β(3π¦ 2 /β2 β 2πΎπ¦/β)
π€1π (π¦) = ,
2πΎπ2 [πΎ + π(1 β πΎ)]
π
πΎπ΄β(3π¦ 2 /β2 β 2(2 + πΎ)π¦/β + 1 + 2πΎ)
π€2π (π¦) = ,
2(1 β πΎ)π2 [πΎ + π(1 β πΎ)]
(ππ 20 β ππ 10 ) π¦
ππ 1 = + π10 , (4.1)
[πΎ + π(1 β πΎ)] β
1 [οΈ π¦ ]οΈ
ππ 2 = π(ππ 20 β ππ 10 ) + πππ 10 + πΎ(1 β π)ππ 20 ,
πΎ + π(1 β πΎ) β
3π
π(1 β πΎ)π΄ 3π
πΎπ΄
π1π = β , π2π = β ,
πΎβπ2 [πΎ + π(1 β πΎ)] (1 β πΎ)βπ2 [πΎ + π(1 β πΎ)]
ππ 1 (0) = ππ 10 , ππ 2 (β) = ππ 20 , π = π1 /π2 , π = π1 /π2 , πΎ = π/β < 1, π = π1 /π2 ,
(ππ 20 β ππ 10 )πΎ
π΄= ; (4.2)
πΎ + π(1 β πΎ)
(οΈ 3 )οΈ
π π
(1 β πΎ)π΄β π¦ πΎπ¦ 2
π£1 (π¦) = β β 2 ,
2πΎπ2 [πΎ + π(1 β πΎ)] β3 β
[οΈ(οΈ )οΈ
π π
πΎπ΄β2 π¦3 3 (4.3)
π£2 (π¦) = β βπΎ
2(1 β πΎ)π2 [πΎ + π(1 β πΎ)] β3
(οΈ 2 )οΈ (οΈ )οΈ]οΈ
π¦ π¦
β(2 + πΎ) 2 β πΎ 2 + (1 + 2πΎ) βπΎ .
β β
Introducing the differences
ππ (π¦, π‘) = ππ π (π¦) β ππ (π¦, π‘), ππ (π¦, π‘) = π€ππ (π¦) β π€π (π¦, π‘) (4.4)
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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 π€ππ (π¦),
ππ π (π¦) and πππ under the conditions of convergence of the integrals
β«οΈβ β«οΈβ β«οΈβ
π πΏπ
|ππ π0 β ππ0 (π )| ππ, π πΏπ
|πβ²π0 (π )| ππ, ππΏπ |πβ²β²π0 (π )| ππ. (4.5)
0 0 0
More exactly, βπ€π (π¦, π‘) β π€ππ (π¦)β 6 ππ πβπΏ1 π‘ , βππ (π¦, π‘) β ππ π (π¦)β 6 ππ πβπΏ2 π‘ ,
βππ (π‘) β πππ β 6 π πβπΏ3 π‘ with the positive constant ππ , ππ , π , πΏ1 , πΏ2 , πΏ3 depending
on physical parameters of liquid and layers thicknesses.
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)β(3.10). As a result we
come to boundary value problem for images π Λπ (π¦, π) of functions ππ (π¦, π‘)
πΛ
π π0π (π¦)
Λπ¦π¦ β
π =β , (5.1)
ππ ππ
π
Λ1 (0, π) = π
Λ10 (π), π
Λ2 (β, π) = π
Λ20 (π), (5.2)
π
Λ1 (π, π) = π
Λ2 (π, π), π1 π
Λ1π¦ (π, π) = π2 π
Λ2π¦ (π, π), (5.3)
Λπ (π¦, π) and πΛπ (π) of functions π€π (π¦, π‘), ππ (π‘)
and images π€
π πΛπ (π)
Λππ¦π¦ β
π€ Λπ = β
π€ , (5.4)
ππ ππ
π€
Λ1 (0, π) = 0, π€
Λ2 (β, π‘) = 0, (5.5)
π€
Λ1 (π, π) = π€
Λ2 (π, π), (5.6)
Λ2π¦ (π, π) β π1 π€
π2 π€ Λ1π¦ (π, π) = β2π
Λ
π1 (π, π), (5.7)
β«οΈ π β«οΈβ
π€
Λ1 (π¦, π) ππ¦ = 0, π€
Λ2 (π¦, π) ππ¦ = 0. (5.8)
0 π
In condition (5.2) and equation (5.4) π Λπ0 (π), πΛπ (π) are images of functions
ππ0 (π‘), ππ (π‘) respectively. The solutions of problem (5.1)β(5.8) can be written as
βοΈ βοΈ β«οΈπ¦ βοΈ
π π 1 π
Λπ (π¦, π) = πΆπ1 sh
π π¦+πΆπ2 ch π¦β β π0π (π§) sh (π¦βπ§) ππ§, (5.9)
ππ ππ πππ ππ
π¦π
[οΈ βοΈ βοΈ ]οΈ
π π πΏπ (π)
π€ π1 (π, π) π·π1 sh
Λπ (π¦, π) = β2π
Λ π¦ + π·π2 ch π¦+ , (5.10)
ππ ππ π
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where πΛπ (π) = β2π
Λ π1 (π, π)πΏπ (π).
The values πΆπ , πΆπ2 , π·π1 , π·π2 and πΛπ (π) determined from the boundary condi-
1
tions (5.2), (5.3), (5.5)β(5.8). Due to the cumbersome the type of these values is
not given here.
Let us assume that lim ππ0 (π‘) = ππ π0 , π = 1, 2, using the formulas (5.9), (5.10)
π‘ββ
and presenting for the values πΆπ1 , πΆπ2 , π·π1 , π·π2 and πΛπ (π) we can prove the limit
equalities
lim ππ (π¦, π‘) = ππ π (π¦), lim π€π (π¦, π‘) = π€ππ (π¦),
π‘ββ π‘ββ
lim ππ (π‘) = πππ ,
π‘ββ
where ππ π (π¦), π€ππ (π¦), πππ 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 siliconβwater. Thickness of the layers is the same and
equal to 1 mm. The corresponding values of the defining parameters are given
in Table 1.
Table 1. Physical properties of liquids
Item liquid silicon water
3
π, kg/m 956 998
β6 2
π Γ 10 , m /s 10.2 1.004
π, kg Β· m/s3 Β· K 0.133 0.597
β6 2
π Γ 10 , m /s 0.0675 0.143
Γ¦ Γ 10β5 , kg/s2 Β· K 6.4 15.14
Figure 2β5 show the profiles of the dimensionless functions
π€Β―π (π, π ) = π€π (π¦, π‘)π2 /(π
π΄) (π = π¦/π, π = π1 π‘/π2 are the dimensional variables)
and transverse velocity π£Β―π (π, π ) = π£π (π¦, π‘)π2 /(π
π΄β) with π20 (π‘) = 0. In particu-
lar, the functions π€ Β―π are negative, so reverse flows arise here. Figure 2, 3 show
the results of calculations when π10 (π ) = sin π , π20 (π ) = 0. That is the limit of
π10 (π ) at π β β does not exist and the velocity field does not converge to a
stationary one.
Figure 4, 5 show an evolution of the convergence of functions π€ Β―π and trans-
verse velocities π£Β―π to stationary regime for the case π10 (π ) = 1 + πβπ cos(10π ),
π20 (π ) = 0. These results are good agreement with the a priori estimates were
obtained in Section 4.
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0.08
wΜ 0.04
0
β0.04
0 0.5 1 1.5 2
ΞΎ
Fig. 2. Evolution of functions π€Β―π for π10 (π ) = sin π . Total line is the stationary profiles,
ββ β π = 4, β Β· β β π = 5, Β· Β· Β· β π = 7
0.02
0.01
0
vΜ
β0.01
β0.02
0 0.5 1 1.5 2
ΞΎ
Fig. 3. Evolution of functions π£Β―π for π10 (π ) = sin π . Total line is the stationary profiles,
ββ β π = 3, β Β· β β π = 6, Β· Β· Β· β π = 8
0.08
0.04
wΜ
0
β0.04
0 0.5 1 1.5 2
ΞΎ
Fig. 4. Evolution of functions π€ Β―π for π10 (π ) = 1 + πβπ cos(10π ). Total line is the
stationary profiles, ββ β π = 1, Β· Β· Β· β π = 4
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0.02
0.01
0
vΜ
β0.01
β0.02
0 0.5 1 1.5 2
ΞΎ
Fig. 5. Evolution of functions π£Β―π for π10 (π ) = 1 + πβπ cos(10π ). Total line is the sta-
tionary profiles, ββ β π = 1, β Β· β β π = 4, Β· Β· Β· β π = 2
6 Conclusion
The two-dimensional horizontal layer is a matter of great importance in connec-
tion 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 in-
dustry. 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 con-
vective 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).
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