CC BY
33
УДК 517.9
Group Analysis of Equations of Hydrostatic Model of Viscous Fluid
Alexander A. Rodionov*
Institute of Computational Modeling SB RAS, Akademgorodok, 50/44, 660036 Russia
Daria A. Krasnova^
Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 24.01.2019, received in revised form 10.02.2019, accepted 20.02.2019 The group properties of three-dimensional equations of hydrostatic model of viscous fluid are studied. Several exact solutions are presented. The free surface of fluid and pressure on this surface are determined.
Keywords: viscous fluid, hydrostatic model, group analysis, exact solution. DOI: 10.17516/1997-1397-2019-12-3-342-350.
The group analysis of differential equations is a powerful tool for studying non-linear equations and boundary value problems. This method was introduced by Sophus Lie in his works in the 19th century. The interest to the group analysis was revived by L. V. Ovsyannikov, who pointed out a method for describing the properties of differential equations [1, 2].
One of the main problems in the group analysis of differential equations is to find the permissible group of transformations of the system of equations on the set of solutions of these equations.
Lie group properties of differential equations were studied by L. V. Ovsyannikov [2], N. H. Ibragimov [3], V. V. Pukhnachov and by their followers S.V.Habirov, Y. N. Pavlovsky, A. A. Buchnev, O. V. Bytev, V. K. Andreev. At present the group properties of equations of fluid mechanics are studied by V.K.Andreyev, O. V.Kaptsov, V. V. Pukhnachev, A. A. Rodionov [4].
The Navier-Stokes equations are a system of differential equations that describe the motion of a viscous fluid. The aim of this study is to perform a group analysis of the hydrostatic model of three-dimensional Navier-Stokes equations and to find exact solutions of this model.
1. Basic equations and problem statement
The three-dimensional Navier-Stokes equations for the motion of a viscous incompressible fluid are
* aarod@icm.krasn.ru tkrasnova-d@mail.ru © Siberian Federal University. All rights reserved
(4)
Ut + UUx + VUy + wuz + pPx = v(nxx + Uyy + uzz),
1
Vt + UVx + VVy + WVz + pPy = v(Vxx + Vyy + Vzz ), (!)
1
Wt + UWx + VWy + WWz + pPz = V(Wxx + Wyy + Wzz) - 9, Ux + Vy + Wz = 0.
Here u, v, w are the components of the velocity vector along the x, y, z directions, p is pressure, t is time, g = const > 0 is the acceleration of gravity in the z direction, v is the dynamic viscosity coefficient, p = const is the fluid density (it is assumed that p =1).
Let us assume that the pressure in the fluid linearly depends on the depth
Pz = -9. (2)
This assumption is often used in oceanology [5]. Then
p(x,y,z,t) = -gz + q(x,y,t), (3)
q(x,y,t) is a new function. In this case system (1) is rewritten as
Ut + UUx + VUy + WUz + qx = v(Uxx + Uyy + Uzz), Vt + UVx + VVy + WVz + qy = V (Vxx + Vyy + Vzz), Wt + UWx + VWy + WWz = v(Wxx + Wyy + Wzz), Ux + Vy + Wz = 0, qz = 0.
Let r : z = n(x, y, t) be the equation for the free boundary of a fluid on which the following kinematic and dynamic conditions are satisfied
nt + u(x, y, n(x, y, t), t)r/x + u(x, y, n(x, y, t), t)r/y = w(x, y, n(x, y, t),t), (5)
(pa - p)lt + 2vD -it = 2aH it, (6)
where pa(x, y, t) is the atmospheric pressure, p = -gr(x, y, t) + q(x, y, t), 1 is the normal to the free surface, H is the mean curvature that depends on the position of the point on the surface, a = const is the surface tension coefficient, D = D(u,v,w) is the deformation rate tensor [4].
If the solution of system (4)is known then the equation of free surface can be found from relation (5), and pa(x,y,t) can be found from (6).
We apply group analysis [2] to equations of system (4). It is required to find the Lie algebra of admissible operators for this system and to construct exact solutions.
A similar study of the hydrostatic model of an ideal fluid was carried out in [6].
2. Group properties of the equations
Let us consider the group properties of equations (4). Let us introduce the following index designations: u1 = u, u2 = v, u3 = w, u4 = q, x1 = x, x2 = y, x3 = z, x4 = t. In these new designations equations (4) supplemented by requirement (4) assume the following form:
U1 + u1 U1 + u2u2 + u3u3 + uf - V(ru\i + u22 + U^) = 0,
U2 + u1 U2 + u2u2 + u3u3 + u2 - v(u^1 + u22 + Uo3) = 0,
u4 + u1u1 + u2u3 + u3u3 - v(u11 + u22 + u33) = 0, u1 + u2 + u3 = 0, U3 = 0.
The lower index denotes differentiation.
The admissible operator for system (7) has the following form
X=u) dx+nk(x u) dut,
Here summation is performed over i,k = 1, 2,3,4. Let us extend the operator to the first derivatives
X = X + Zk 9 Zk = dnk + l dnk kl^
X = X + Qttt , Q = TT- + Ui^-J - U.I— + Ui —
(djj + t 8g_\ \dx} + i dul )
and to the second derivatives
d y d y d 2 d 2 d X _ X +Q_ au* = X +Cl1 +Z22 + Zl1 dUU-- +Z22 dUU22 + ■■■■■
where
za _ dza + idza + i dzi_ a foe + i
Zi_ _ dx_ + U dU + U_k dulk Uik\ dxi + U_ Oxj) '
where summation is performed over l, k.
Let us note that values u\2, u\3, u23, uy2, uy3, u23, u32, u33, u23 are absent in system (7). We use the invariance criterion [2]. Acting via operator X onto equations (7), the governing
equations are obtained. Let us consider equation (7) and substitute U4 , U2 , U3 , U1 for remaining variables. Splitting governing equations with respect to independent variables, we obtain coordinates of operator X
£ 1 = C4x 1 + f (x4), £2 = C4x2 + C5x 1 + h(x4),
£3 = C2 + C3x4 + C4x3, £3 = C1 +2C4x4, n1 = -C4u1 - C5u2 + f'(x4), n2 = -C4u2 + C5U1 + h'(x4), n3 = C3 + 2C4x41 n4 = -2C4U4 - x1f ''(x4) - x2h''(x4) + p(x3),
where C1,...,C5 are constants, f (x4), h(x4), x4) are arbitrary functions.
It is proven that the Lie algebra for the system of equations (4) is formed by the operators
X1 = dt, X2 = 8Z, X3 = tdz + 8W, X4 = xdx + ydy + zdZ + 2tdt - udu - vdv - wdw - 2qdq,
X5 = xdy - ydx + udv - vdu, (8)
X6 = f (t)dx + f '(t)du - xf ''(t)dq, Xr = h(t)dy + h' (t)dv - yh''(t)dq, X8 = v(t)6q.
The first operator is responsible for the transfer in time t, the second and third operators are responsible for the transfer and the Galileo transformation along the z axis, the fourth operator is responsible for the tensile transformations, the fifth operator is responsible for the rotation around the zaxis. The sixth, seventh and eighth operators contain arbitrary functions f (t), h(t), p(t) that depend on time. They define the infinite-dimensional part of the Lie algebra of admissible operators.
For the first time a group analysis of equations of system (1) was carried out by V. O. Bytev [7]. The difference of the obtained result from operators (7) is that two operators responsible
for the rotation around the x and y axes are absent in (7), and the infinite-dimensional operator along the z axis similar to Xe,X7, operators is represented as two finite-dimensional operators X2, X3.
If the two-dimensional hydrostatic model is considered then variable y and velocity V in equations (4) should be excluded. Then for equations
Ut + UUx + WUz + qx = v(Uxx + Uzz), Wt + UWx + WWz = v(Wxx + Wzz ), (9)
Ux + Wz =0, qz =0
the algebra of admissible operators is determined by the following operators
X1 = dt, X2 = dz, X3 = tdz + dw,
X4 = xdx + zdz + 2tdt - udu - wdw - 2qdq, (10)
X5 = f (t)dx + f'(t)du - xf''(t)dq, Xe = <f(t)dq.
3. Exact solutions
Example 1. Let us find a solution to equations (4) with operators from basis (8)
/ d d d d \ (X3 = ^ + dW5 X7 = % + dV, (h(t)= t)).
The invariants of these operators are
J = (x,t; u;V - t; w - z; qj.
(12)
Therefore, the invariant solution of equations should have the form
(u,v,w,q) = (U (x,t); y + V (x,t); t + W (x,t); Q(x,t)j, (11)
where U, V, W, Q are functions of two variables.
Let us substitute functions (11) into the system and obtain the factor-system:
Ut + UUx + Qx = vUxx, Vt + UVx + tv = vVxx, Wt + UWx + tw = vWxx, Ux + 2 =0.
It follows from the last equation of factor-system (12) that
2t
U (x,t) = -— + h1 (t), (13)
where h1 (t) is an arbitrary function. It follows from the first equation in (12) that
2x ' 3x2
Q(x,t) = — h1(t) - xh1(t) --r- + h2(t), t t2
with an arbitrary function h2(t). It can be seen from the second equation of system (12) that
2x V
Vt = vVxx +( — - h1 (t)j Vx - j. (14)
To solve equation (14) we use the reference book by A. D.Polyanin [8] (Section (1.8.6), p. 129) and consider the following equation
wt = awxx + [xm(t) + n(t)]wx + k(t)w.
In the case of equation (14) we have
21
a = v, m(t) = j, n(t) = -h1(t), k(t) = - j.
Let us introduce the following designations (A, B,C = const)
Bt2
F(t) = B exp^J m(t)dt
C B2t5
t = F 2(t)dt + A + A,
S = xF(t) + J g(t)F(t)dt + C = xBt2 - B J t2hi(t)dt + C.
Using new variables (x,t) ^ (S,t), we obtain
V(x,t) = M(S,t) exp
k(t)dt
-M (S,T ),
where M(S,t) is a new function.
Let us assume that B = 1, A = C = 0, then F(t) = t2,
Vt = -1M (S, t) + 1
t5
t =5, S = xt2 - t2h1(t)dt, M (S, t )
Vx
1
dS ,r 0T
M m + Mt m
' dS _ 3T M^— + MT — dx dx
t2
+ Ms(2x - th\(t)) + MTt3,
Mst; Vxx = t Mss.
Substitution of
M (S,T ] + Ms (2x - th1(t))+ Mt t3 = vt3 Mss +Çt + hi(t)^j Ms t - M (S,T )
t2
in (14) gives us the following equation
Mt = vMss.
The heat equation with constant coefficients is obtained.
t2
(15)
Let us note that if variable S is taken in the form S = xt2 - J t2h1(t)dt then it can be treated
1
as a Lagrangian variable since S = x at t = 1.
Let us consider the simplest solution for (15) [5]: M(S) = aS + ¡3. Then
V(x,t) = 'j^xt2 - Jt2hi(t)d^j +
3 t '
where a, 3 = const.
t
From the third equation of factor-system (12) we obtain
f2x \ W
Wt _ + i t - hi(t) ) Wx - —.
(16)
The solution of equation (16) is found by analogy with equation (14), that is,
W (x,t)_ t(xt2 -J t2hy(t)d?j + t,
where e, t _ const.
As a result, the exact solution of equations (4) is obtained in the form
u(x, t) _ hy(t)--tx,
v(x,y,t)_t y + a(xt2 - J t2hy(t)dt) + 3
w(x,z,t)_i z + e(xt2 -J t2hy(t)dt) + t
. . 2xhy(t) ,, 3x2 , ^ q(x,t) _ -1--xhy (t) - + h2(t),
where hy(t),h2(t) are arbitrary functions; a,3,e,t are constants.
Kinematic condition (5) on the free boundary r : z _ n(x, y, t) for a given solution has the form
(17)
dt
dn + (hy(t) - +(l y + a(xt2 - I t2hy(t)dt) + 3
dn dy
+ e(xt2 -J t2hy(t)dt^ + t
One can see that n(x,y,t) _ t$(Jy, J2) - eJy - t, where J2) is an arbitrary function of
arguments
Jy _ xt2 -Jt2hy(t)dt, J2 _ 1 (y + aJy + 3).
From dynamic condition (6) (pa -p\r)tt + 2vD ■ it _ 2aHlt one can determine the atmospheric pressure at the free boundary
Pa _ P\r + -
1 + V9 + t4(a2 + e2)
+ 2aH.
Taking into account solution (17), the deformation rate tensor and the pressure on the fluid surface are
1 , -4 at2 et2
D _2t
at2 et2
Hr = -9n(x,y,t) + q(x,y,t) = -g[t®(J1, J2) - £J - ft] + 2xh^ - xh^t) - ^x + h2(t). Example 2. Let us consider equation (14):
MT = vMss,
and use solution for this equation [8]
M (S, t ) _ a(S2 + 2vt ) + 3.
Then we obtain
, N a(S2 + 2vt )+ 3 a (( 2 f 2, \2 2vt5) 3 , ,
V(x,t) = y t = j{(xt2 - t2h1(t)dtJ +—)+ J. (18)
In a similar way we can find W(x,t):
W (x,t) = J [(xt2 -J t2h1(t)dt) 2 + ^ + J.
As a result, one more exact solution of equations (4) is obtained
u(x,t) = hi(t)--
1
v(x,y,t) = j 1
w(x, z,t) = j
y + a^(xt2 -J t2h1(t)dtj 2 + + 3
z + £ ^xt2 -J t2h1(t)dtsj + + V
(19)
q(x,t) = 2xh^J - xhi(t) - ^T + h2(t),
where h1 (t),h2(t) are arbitrary functions; a,3,£,p are constants. Thus, the set of solutions of equations (4) can be constructed from the set of solutions of equation (14). Example 3. Let us consider operators
d „ d „ d
{X2 = dt ; X5 = dz ; X = ^H1).
The invariants of these operators are J = {y; u; v; w; q}. Therefore, the invariant solution of equations should have the following form
(u,v,w,q) = (U(y); V(y); W(y); Q(y)). (20)
The factor-system for the stationary solution has the form
VUy = vUyy, VVy + Qy = vVyy, VWy = vWyy, Vy = 0. (21)
One can see from (21) that
C, C,
V = Ci, Q = C2, U (y)= Di + D2e~ y, W (y) = Hi + H2e~ y. Finally, we obtain the exact solution of equations (14)
u(y) = Di + D2e CVy, v(y) = Ci, w(y) = Hi + H2e Vy, q(y) = C2, (22)
where Ci,C2,Di,D2, Hi,H2 are constants.
Kinematic condition (5) on the free boundary z = n(x, y, t) for solution (22) has the form
I + (Di + D2eVy) di + Ci dy = Hi + H2eVy.
The solution to this equation is
1 ( v c, \
n(x,y,t) = — ( Hiy + H2 — e v y 1 +$(Ji,J2),
where §(Jy,J2) is an arbitrary function of arguments
v C.
Ji _ y - Cyt, J2 _ Dyy + D2—e-y - Cyx.
Cl
In the dynamic condition (6) on the free boundary z _ n(x, y, t) the deformation rate tensor has zero values on the diagonal for solution (22). Therefore, it is easy to determine that the external atmospheric pressure at the free boundary is
Pa _ -gn(x, y,t) + C2 + C— eCyjDf+H2 + 2aH.
Example 4. Let us give an example when the factor-system gives a contradiction. Let us consider operators
d d d d d d . . \ (X4 _ t— + —; X6 _ t— + —, ay(t) _ tXr _ t— + —, a2(t) _ t). \ dz dw dx ou dy dv /
{x y z
t; u - -J; v - t; w - -J ; ^. Therefore, an invariant
y z
—; w--;
tt
solution of equations should have the following form
(u, v, w, q)_ (x + U (t); y + V (t); t + W (t); Q(t)), (23)
where U, V, W, Q are functions of one variable. Then the factor-system has the form
Ut - ^ + z + U*)\ = 0. Vt -1 + + V)\ = 0, (24)
w - I + (f + = 1 + 1 + 1 =0'
The last relation in (24) is contradictory. Therefore, a solution of form (23) does not exist.
References
[1] L.V.Ovsjannikov, Group Properties of Differential Equations, English translation by G. W. Bluman, Novosibirsk, USSR Academy of Science, Siberian Section, 1962.
[2] L.V.Ovsjannikov, Group Analysis of Differential Equations, English translation by W.F.Ames, Academic, New York, 1982.
[3] N.H.Ibragimov, Transformation Group in Mathematical Physics, Transformation Groups Applied to Mathematical Physics, Reidel, Dordrecht, 1985.
[4] V.K.Andreev, O.V.Kaptsov, V.V.Pukhnachev, A.A.Rodionov, Applications of Group-Theoretical Methods in Hydrodynamics, U.S.A, Kluer Academic Publishers, 1998 (in Russian).
[5] K.F.Bowden, Physical Oceanography of Coastal Waters, Ellis Horwood Limited Publishers, New York, Halsted Press, 1983.
[6] A.A.Rodionov, Hydrostatic Model for an Ideal Fluid: Group Properties of Equations and their Solutions, Journal of Siberian Federal University. Mathematics and Physics, 8(2015), no. 3, 320-326 (in Russian).
[7] V.O. Bytev, Group properties of the Navier-Stokes equations, Numerical methods of continuum mechanics. Novosibirsk, 3(1972), no. 3, 13-17 (in Russian).
[8] A.D.Polyanin, Reference book. Linear Equations of mathematical physics, Moscow, Nauka, 1983 (in Russian).
Л____гл гл гл
Групповой анализ уравнении гидростатическом модели вязкой жидкости
Александр А. Родионов
Институт вычислительного моделирования СО РАН Академгородок, 50/44, Красноярск, 660036
Россия
Дарья А. Краснова
Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041
Россия
Рассматриваются групповые свойства уравнений трехмерной гидростатической модели вязкой жидкости. Представлено несколько примеров точных решений. Определяются свободная поверхность жидкости и давление на ней.
Ключевые слова: групповой анализ, гидростатическая модель, вязкая жидкость, точные решения.
