CC BY
7
Journal of Siberian Federal University. Mathematics & Physics 2015, 8(1), 3—6
УДК 532.51
The 2D Motion of Perfect Fluid with a Free Surface
Victor K. Andreev*
Institute of Computational Modelling SB RAS Akademgorodok, 50/44, Krasnoyarsk, 660036 Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 10.10.2014, received in revised form 10.11.2014, accepted 20.12.2014 The 3D continuous subalgebra is used to searching new partially invariant solution of incompressible perfect fluid equations. It can be interpreted as a non-stationary motion of a plane layer with one free surface. The velocity field and pressure are determined in analytical form by using Lagrangian coordinates.
Keywords: perfect fluid, partially invariant solution, non-stationary motion, free surfaces.
Governing flow equations and main results
The Euler equations for 2D motions of a perfect fluid are recorded by
Ut + UUX + VUy + 1 px =0, Ux + Vy =0,
P 1 (1)
Vt + UVx + VVy + - Py = 0,
where p is the constant fluid density, u and v are the velocity components in the x and y directions, respectively, p is the pressure. The group of point transformations admitted by the system (1) is computed in [1]. Corresponding this group basic continuous Lie algebra includes the three parametrical subalgebra (dx,tdu + dx, dp). It has the invariants t, y, v and partly invariant solution of (1) rang two and defect two necessary to seek in the form u = u(x,y,t), v = v(y,t), p = p(x, y,t). From continuity equation ux + vy =0 we obtain the relations
u(x,y,t) = Ui(y,t)x + U2(y,t), Ui(y,t)+ Vy (y, t) = 0. (2)
Impulse equations (1) are equivalent to the following
1 x2 U1t + VU1y + U1 = f(t^ pp = 1(y,t) - f (t)y , (3)
ly = -Vt - VVy, U2t + VU2y + U1U2 = 0
with arbitrary function f (t).
Let us introduce the Lagrangian coordinates (n, t) by the solving Cauchy problem
dt
— = v(y,t), y\t=0 = n. (4)
*andr@icm.krasn.ru © Siberian Federal University. All rights reserved
We introduce the following denotations
u 1 (n,t) = u1(y(n,t),t), uu2 (n,t) = u(y(n,t),t), v (n,t) = v(y(n,t),t), where y(n,t) is a solution of (4). Then the first equations (3) can be reduced to Riccati equation
o o - , ,
«1t + «1 = f (t).
It has general solution
«1 = dt ' ln
Here g(t) is the solution of the Cauchy problem
g(t) 1 + «10 (n)
g2(t)
dt
g" — f (t)g = 0, g(0) = 1, g'(0) = 0,
and u10(n) is the initial value of function u1(y,t).
The another functions can be found by the formulae
(5)
(6)
y(n,t)=-à L
1 + Uio(n)
- 2 ( t)
dt
-l
dn;
(n,t) = — / UU1 (n,t)exp — / UU1 (n,t)
00
«2 (n, t) = «20(n) exp
dt
dn;
l (n,t) = l1 (t) — Vt (n, t) exp
0
m 1 (n, t) dt
— / «1 (n,t) dt
0
dn
(7)
(8) (9)
(10)
with arbitrary function 11(t). So, all unknowns can be determined in analytical form.
Now we show that this solution can be interpreted as an unsteady motion in a strip with one free boundary, see Fig. 1.
Fig. 1 Geometry of the motion
Really, at the initial time liquid fills the strip of thickness y = h0 = const. The line y = 0 is a
f y
rigid wall. Initial velocity field has the form u0(x,y) = u10(y)x + u20(y), v0(y) = — / u10(y) dy,
0
v0(0) = 0. The upper line y = h0 is a free boundary and at the initial time the pressure p(h0,0) coincides with outer pressure pout = p10 + p00x2/2. For all t > 0 the strip motion is described
t
1
0
t
n
1
0
0
n
by the formulae are found above, where pout = Pi(t) — p0(t)x2/2 must be given, so f (t) = p0(t), f (0) = p00. The evolution of the free boundary is defined as
1 rho h(i) = sW i
1 + uio(n)
1
s2(t)
dt
(11)
Let us consider two simple cases of the solution (5)-(11), when u10 = a = const or u10 = bn, b = const. For the first case the exact solution can be written in Eulerian coordinates as
d 1 u(x, y, t) = dt ln G(t)x + U2o(G(t)y),
d
v(y,t) = -— In G(t)y,
(12)
1 d2 y2 x2
- p(x, y, t) = li (t) + ^ ln G(t) ^ - f (t) y ,
where
G(t) = g(t) 1 + a The equation of the free boundary is
y = h(t)
s2(t)
dt
G(t)
(13)
(14)
If we take g(t) = cos wt, f (t) = —w2 (g(t) = ch wt, f (t) = w2), w = const, then the solution exists up to the time t* = n/2w (exists for all time). The solution has to be periodic one if g(t) = 2 — cos wt, f (t) = w2 cos wt(2 — cos wt)-1.
For the second case the formulae have a more complicated shapes and we give here only equation of the free boundary, namely,
y = h(t)
1
bg(t) ¡0 g-2(t) dt
ln
1 + bho i g-2(t) dt
./o
(15)
Remark 1. In well-known [2] solutions are sought of the shape ^(x,y,t) = F(y,t)x + G(y,t) for stream function (u1 = , v1 = —The unknowns satisfy the eq's
Fty + (Fy)2 — FFyy = i-!^ Gty + Fy Gy — FGyy = f2(t), G = J U dy — hF + hy, htt — fi(t)h = f2(t).
Some particular solution are presented in handbook, see [2, table 13.9, p. 944]. But in this paper we have found exact solution in analytical form.
The problem has a stationary solution. Indeed, the function v(y) satisfies the eq'n vvyy —v^ = —f0 = const with general solution
sin
a) v = v ici
b) v = ±v/fo(C2 ± y)
vICÏ (C2 ± y)
c)
d)
ch
/ fo ,
v = V Cish
V^ (C2 ± y) V^ (C2 ± y)
fo > 0, Ci < 0;
fo > 0, Ci =0;
fo < 0, Ci > 0;
fo > 0, Ci > 0.
l
t
o
t
1
o
h
o
v=
However, the only case a) has a physical meaning. Really, let us take h0 = n/\J|C1|, then we obtain formulae
rr ny hoVTo . ny — V Jo x cos -— , v =- sin-
h-0 ' n h0
1 , f0 ^ 2 , ho • 2 ny" -p = I0 — -77 x + —
p 2 * n2
which describe the flow in a strip 0 < y < h0, |x| < to, with rigid walls y = 0, h0. The solution obtained is the periodical with respect variable y.
1 p = 10 - — fx2 + sin2 —^ , 10 = const > 0, p 2 V n2 ho y
Conclusion
The partially invariant solution of the perfect fluid equations is investigates. This new solution describes the unsteady motion with a free surface. As was shown by examples the solution may has collapse in finite time or to be periodical one. Not that this phenomenon depends on pressure gradient f (t). There has been previous works devoted exact solutions of perfect fluid motions with a free surfaces [3].
This work is supported by the SB RAS (project 44)-
References
[1] A.A.Buchnev, A Lie group admissible for the equations of an ideal incompressible fluid, Dinamika Sploshnoy Sredy, Novosibirsk, (1971), no. 7, 212-214 (in Russian).
[2] A.D.Polyanin, V.F.Zaitsev, Handbook of nonlinear partial differential equations, CRC Press. Taylor and Francis Group, 2012, 942-949.
[3] V.K.Andreev, O.V.Kaptsov, V.V.Pukhnachev, A.A.Rodionov, Application of GroupTheo-retical Methods in Hydrodynamics, Kluver Acad. Publ., Dordrecht, Boston, London, 2010.
Двумерное движение идеальной жидкости со свободной поверхностью
Виктор К. Андреев
Непрерывная трёхмерная подалгебра используется для нахождения нового частично инвариантного решения уравнений идеальной несжимаемой жидкости. Оно интерпретируется как нестационарное движение плоского слоя со свободной поверхностью. При этом поля скоростей и давлений определяются (с помощью переменных Лагранжа) в аналитическом виде.
Ключевые слова: идеальная жидкость, частично инвариантное решение, нестационарное движение, свободная поверхность.
