ISSN: 2277-9655
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[Venherskyi* et al., 5(11): November, 2016]
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IJESRT
INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH
TECHNOLOGY
INVESTIGATION OF THE PROPERTIES OF SOLUTION OF VARIATIONAL
PROBLEM COUPLED FLOW OF SURFACE AND GROUNDWATERS
Petro Venherskyi *, Heorhiy Shynkarenko*,**
Department of applied mathematics and informatics, Franko national university of Lviv, Ukraine
**
Department of mathematics and applied informatics, Opole University of Technology, Poland
*
DOI: 10.5281/zenodo.167102
ABSTRACT
The mathematical model of surface and ground flows of water are observed. Formulated initial - boundary and
variational problems. We give conditions of the interaction of these two streams on the common border. And
formulated a theorem on uniqueness and limited solution by variational problem of coupled stream.
KEYWORDS: Surface flow, groundwater flow, watershed, coupled flows, conditions of interface, convergence
of numerical scheme, conditions of uniqueness solutions of the problem.
INTRODUCTION
To simplify the description of the motion of water flows on the watershed area is held vertical decomposition of
the problem - the whole area is divided into layers: the surface layer of the atmosphere, land surface, unsaturated
zone, saturated zone, zone traffic pressure and so on. In each layer to describe the movement of moisture used
models of different dimensions and their solutions are connected by boundary conditions [4-8].
We select in solid medium (liquid) moving surface layer (Figure 1) of such a structure
 F  t  :   x1 , x2 , x3   R 3 ,   x   x3    x, t  x   x1 , x2     t  .
Denote projection of its lower
and upper
  t  :   x1 , x2 , x3   R 3
 F  t  :   x1 , x2 , x3   R 3
x3   x  , x   x1 , x2     t 
x3   x,t  x   x1 , x2     t 
bases on the plane 0x1 x2 . The rest of the surface layer
Г F  t  :   x1 , x2 , x3   R 3 ,   x   x3    x, t  x  ( x1 , x2 )    t 
will be called the lateral surface layer F  t  .
Similarly denote part of fluid that moves in the soil, so
 P  t  :   x1 , x2 , x3   R 3 , h  x   x3    x  , x   x1 , x2     t 
the projection of the lower part will be written as
 P  t  :   x1 , x2 , x3   R 3
x3  h  x  , x   x1 , x2     t  .
Then, a layer of ground water
Г P  t  :   x1 , x2 , x3   R 3 , h  x   x3    x  x  Γ P  t 
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ISSN: 2277-9655
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Figure 1. General view of the model of flows and their cross-section.
MATERIALS AND METHODS
1. Initial boundary value problem of interaction of water flows.
We formulate initial boundary problem of motion of surface and groundwater flows on the surface
watershed considering boundary and initial conditions [1-3].
 Find unknown quantities u, p,   such that

satisfy the following system of equations :
 
3
3
 ik


 ui   
 ui uk    f i  
 0,


 t
k 1 xk
k 1 xk

 ij   pF  ij   ij ,
  2  e , i, j  1, 2, 3
ij
 ij

u j 
1  ui
eij  

,
2
xi 

 x j


3  u 
 
k

 0,
in
F  (0, T ],

xk
k 1
 t

3
   
 m  
k
  in P  (0; T ],



 x 


t

x
j 1
j
j



where {ui(x,t)}3i=1
(1)
(2)
(3)
and pF  pF ( x, t ) – the sought velocity vector of fluid and hydrostatic pressure,
respectively; F   g f i ( x)i 1 – the mass forces; =(x,t)>0 – density of the mass water flow; =(x )>0 –
3
 
coefficient of viscosity; eij
3
i , j 1
 
,  ij
3
i , j 1
– tensors velocities of deformation and stress of the liquid at the
point x in time t; ij – Kronecker symbol; k  k ( x, t ) - coefficient of filtration; m  m( x, t ) - the coefficient of
specific water loss;    ( x, t ) - known function of sources of water influx;
  x3 
pp
- piezometric pressure;
g
flow (flow separation);
q  k     ( x, t ) - the velocity vector of fluid in the ground;

q

(4)
(5)
,  - volume porosity;
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nF  nP - vectors normal to the boundary area F and P in accordance;
  F P , F P  {}, F P  , F  F   F ; P   P   P .
Boundary conditions:
ui  0 on  F , i = 1, 2,3,
(6)
 n   , on  F ,

0 
0 
(7)
в F  (0, T],
(8)
 u1
 u2
t
x1
x2
where R - velocity of falling rain drops, u10, u20 - horizontal components of velocity on the free surface
 ( x.t ) (  F );
u3  R 
 nP   на  P ;
1  2  0 на  P ,
(9)
(10)
3   I на  P ,
(11)
where I – known function that describes the velocity of flow of fluid through the surface
P .
Initial conditions:
u t  0  u0 ,
in .
p |t  0  p0 ,
(12)
 |t  0  0 .
Contact flow conditions on a common boundary Г:
 nn (u, pF )  p p ,
 n  0 ,
(13)
un  n .
2. Variational formulation of the problem of interaction of water flows.
We introduce the following bilinear forms:
3
3
ui
3
qi ds, Cv ( w, q)   2  e( w) : e( q) ds,
M v (r; w, q)    rwi qi ds, Nv ( w; u, q)     wk
xk
v i 1
v
v k 1 i 1
Av (w, q)    w divqds, Yv (w, q)    w qnd , Bv ( p, w)    3 p .wds.
v
v
v i 1
Introduce spaces:
H F :   ( H 1 ( ))3 |   0 on  ,
F
H p :   H 1 (p ) |  0 on  , W : H F  H P ,  j : W  R, j  1,3
___
3
 ,     fii ds   (n pa    )d 
1
i1

,

( x, t )  g
0
 ,    un  d   ,  
dp     gd
2
3



 p
,
p
.
F
F
F
Denote
   g , m 
m

,
Then write the following variational problem:
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Find
u, p, V  Q W ,
M F (  ; u ,  )  N F (u; u,  )  AF ( p,  )  CF (u,  )
(14)
Y (u,  )  1 ,  ,   V
BF (u, )  Y ( , u)  2 , ,   Q
(15)
M P (m; , )  AP ( , )  Y ( , )  3 , ,   W
(16)
with initial conditions
M F (u(0)  u0 ,  )  0,
(17)
BF ( p(0) - p0 , )  0;
(18)
M P ( (0)  0 , )  0.
(19)
Calculate considering initial conditions (17) - (19) and boundary conditions (6) - (11), and values of variables u
and p with relations (14) and (15). Then of conditions of coupling flows (interface conditions) (13) and
boundary condition (9) the value of the variable  is calculated from (16).
3. The uniqueness and limited solution of variational problems of coupled flow.
We formulate the following theorem.
Theorem 1. Let variational problem (14) - (19), data which satisfy the conditions of regularity
u0  H F ,0  H P , t1 , t2  0, T 
(20)
has a solution (u(t ),  (t)) .
Then the solution (u(t ),  (t)) is the single solution of the problem (1) - (13)
u  L (0, T ; H F ),   L2 (0, T ; H P ) .
2
(21)
Moreover, the solution (u(t ),  (t)) depends continuously on the data of (1) - (13) and under these
conditions will correct a priori estimation
t

2
2
2
2
1
u
(
t
)


(
t
)

( u (t ) W   (t ) W )d   


2
0

(22)
t
2
2
2
2
1 
C  u (0)   (0)    ( u (t ) W   (t ) W )d  
 0
2 
with constant C>0, the value of which does not depend on the values of our interest.
Proof. Considering the conditions of the theorem (21), we have
l1  L2 (0, T ; H F ), l2  L2 (0, T ; H P ) ,
resulting of (14), (15) will correct estimates
t
  1 , u d  
0
t
1
( 1 ( )
2 0
t
t
1
0  2 , u d   2 0 (  2 ( )
2
 u ( ) ) d
(23)
 u ( ) ) d
(24)
2
2
2
Since the initial condition (12) for the function u(t), we have
2
u (0)  u 0
2
.
H
(25)
Given the inequality of Poincaré-Fridrihsa [9], has a place next estimation
u W  C u H  , u  H  ( ),
F
F
F
(26)
where
H ( F )  {u  H ( F ) | u  0 на Г   F },
H ( F )  space inverted to H ( F ).
Similarly, from (16) to function  (t ) we write
2
 (0)   0
2
H
.
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(27)
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Further, in view of introduced norms in paragraph 4 [1], there is C  const  0 that
W C 
P
H P
,   H  (P ),
where
H ( P )  {  H ( p ) |   0 on  P },
H ()  space
inverted tо
then
H ().
 (0) H  C  (0) H   C 0
P
P
(28)
H P
,   H 1 (P ),
(29)
From relations (24) - (27) and (29) come to a priori assessment (22).
On basis of this evaluation by considerations of opposite the uniqueness solution of problem (1) - (13) is proved.
CONCLUSION
Based on the laws of conservation of considered the basic equations and boundary and initial
conditions describing the coupled motion flows of surface and of ground water with unknown values of velocity
and piezometric pressure. Variational problems formulated compatible flow and obtained the contact conditions
on a common boundary based on the laws of motion a continuous medium. We prove uniqueness and limited
variation problem solution coupled flow.
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