Unsteady 2D motions a viscous fluid described by partially invariant solutions to the Navier-Stokes equations Текст научной статьи по специальности «Математика»

УДК 532.51

Unsteady 2D Motions a Viscous Fluid Described by Partially Invariant Solutions to the Navier-Stokes Equations

Victor K. Andreev*

Institute of Computational Modelling RAS SB Akademgorodok, 50/44, Krasnoyarsk, 660036 Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 10.02.2015, received in revised form 03.03.2015, accepted 30.03.2015 3D continuous subalgebra is used to searching partially invariant solution of viscous incompressible fluid equations. It can be interpreted as a 2D motion of one or two immiscible fluids in plane channel. The arising initial boundary value problem for factor-system is an inverse one. Unsteady problem for creeping motions is solved by separating of variables method for one fluid or Laplace transformation method for two fluids.

Keywords: partially invariant solution, viscous fluid, free boundary problem, interface.

Introduction

The Navier-Stokes equations for 2D motions of a viscous fluid are recorded by

1

ut + uux + Vuy + px V(uxx + uyy J:

1

Vt + UVx + VVy + pPy = V(Vxx + Vyy) — g, ux + Vy °

(0.1)

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 and g is the gravity acceleration, v is the fluid viscosity. The group of point transformations admitted by the system (0.1) is computed in [1, 2]. Corresponding this group basic continuous Lie algebra includes the three parametrical subalgebra (8x, du + tdx, dp}. It has the invariants t, y, v and partly invariant solution of (0.1) 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) = w(y,t)x + ui (y,t), w(y,t) + Vy (y,t) = 0.

Navier-Stokes equations (0.1) are equivalent to the system

1

„2

wt + Vwy + w = f (t) + Vwyy, -p = l(y,t) - f (t)^r - gy,

p 2

y

y

- / w(z, t) dz, ly = vVyy — Vt — VVy

J 0

(0.2)

uit + Vuiy + wui — Vuiyy.

In what follows we assume that ui(y,t) = 0.

+ andr@ icm. krasn. ru

© Siberian Federal University. All rights reserved

1. Flow in layer with two rigid walls

In this section the solution (0.2) under consideration shall be interpreted as 2D motion viscous liquid fills the layer 0 < y < h with a rigid walls y = 0, y = h = const. Let us attach the initial and boundary conditions

rh

w(y, 0) = wo(y), wo(0) = wo(h)=0, / wo(z) dz = 0;

■) o

,h

w(0,t) = w(h, t) = 0, / w(z,t) dz = 0.

o

Thus, the function w(y,t) is the solution of integro-differential equation

г у

wt — wy w(z, t) dz + w2 = vwyy + f (t)

o

(1.1)

(1.2)

(1.3)

with initial and boundary conditions (1.1), (1.2).

Here and further suppose the Reynolds number Re = max |wo(y)|h2/v C 1. In such case

ye[o,i]

we can neglect the nonlinear terms in equation (1.3) and the following initial boundary value problem is arised

wt = vwyy + f (t), y e (0, h), t> 0; (1.4)

w(y, 0) = wo(y); (1.5)

w(0,t)=0, w(h, t) = 0, I w(y,t) dy = 0. (1.6)

o

Integrating equation (1.4) we obtain function f (t)

f (t) = h (wy(0,t) — wy(h, t)), f (0) = ^ (woy(0) — woy(h)). Hence, we deduce the so-called loaded equation

wt = vwyy + wy (1,t) — wy (0,t).

But we determine new function W(y,t) = wy (y,t). It satisfies the problem

Wt = vWyy, y e (0,1), t > 0;

W (y, 0) = woy;

yW(y, t) dy = 0;

(1.7)

(1.8)

(1.9)

(1.10)

W(y, t) dy = 0.

o

This problem is not classical one for the heat equation (1.4). The problem (1.8)-(1.11) has the exact solution

, . f 4A|vt\ .

W(y, t) = ak exM-----I sin

fc=i

Ak(2y - h) h

where Ak is kth positive root of the equation

tg Ak = Ak, Ak ^ (2k + 1)n/2, k ^ ro.

(1.11)

o

h

A constants ak are the Fourier series coefficients of known function w0y, i.e.

Afc (2y — h)

woy = ak sin

k = 1

h

Hence, from (1.7)

OO / A \ 2 I *\

f (t) = h [W(0, t) - W(h, t)] = —h ak exp f-----h2A j sin Ak =

k = 1 ' 2

2v \ Ak / 4Ak vt

—exp — ^

k=i

Functions w(y,t) and velocity component v(y, t) can be found by the formulae

A k(2y — h)

, , h ^—л ak ( 4Ak vt . .

ЛУ, t) = ^ Z-j ^ exp -------Tv I < cos Ak — cos

h2

h

7(y,t) = 2 XIAkexP' —

4Ak vt\ f h

Ak (2y — h) , , . л sin | ----;----- | + sin Ak

— y cos A k D

h2 2Ak

k— 1

2. Flow in layer with one rigid wall and free boundary

In the same assumptions like section 1 the function w(y,t) is governed by the equation (1.4). The initial data and boundary conditions are (1.5), (1.6), but it is necessary to change second condition in (1.6) on wy(h, t) = 0. For the function f (t) one obtains

f (t) = h wy(0,t)

(2.1)

As concerning function W(y,t) = wy (y,t) it satisfies the equation (1.8) with initial data (1.9) and boundary conditions

W(h,t)=0, f\h — y)W(y,t) dy = 0.

o

Using separation of variables technique the problem can be solved to obtain W:

Ak (h — y)

W (y,t) = 2D bk Й+А exp, —

k=1

Ak

A|vt\ .

“h^)siI

h

Therefore

,.(y,t) = 2 £ bk It^exA Ak vt

k=1

h2

Ak(h — y) h

- cos Ak ,

and from (2.1), (0.2) we get

f (t) = h^E b.^^ expf — Akvt

k=1

Ak

h2

cos

/ 0vV (1 + A|) ( A\vt

= 1 2z^ bk A3 exp '

h2

. h y cos Ak + — Ak

, Ak (У - h) 1 . • Л sin | -----;---- I + sm Ak

k=1 ' k

A constants bk are the Fourier series coefficients of function w0y, i.e.

w0y = h 13 b

(1 + Ak) „5

k=1

A2

sin

Ak (h - y)

Ak

3. Layered motion of two immiscible fluids

Let us consider a system of two immiscible fluids separated by the interface y = h1. The parameters of the fluid moving in the band 0 < y < h1, x e R are indicated by the subscript "1", and the parameters of the fluid moving in the band h1 < y < h, x e R are indicated by the subscript "2". In the plane motion considered here, the functions wj(y,t) and fj(t), j = 1, 2, are the solutions of the equations

wjt = vj wjyy + fj (t), (31)

related by the conditions on the interface [3]

/ w1(z, t) dz = 0;

J 0

w1(h1 ,t) = w2(h1 ,t), ^1w1y(h1 ,t) - ^2w2y(h1,t) = 0;

the no-slip conditions on the solid boundaries of the flow domain

w1(0,t)=0, w2(h,t) = 0;

rh

and initial data

h

/ w2(z, t) dz = 0,

Jhi

w1 (y, 0) = w1o(y), 0 < y < h1,

w2(y, 0) = w2o(y), h1 < y < h.

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

Remark 1. The initial value problem (3.1)-(3.5) has not a solution expanded into a Fourier series.

A priori estimates. Using equalities (3.2), (3.4) and integrating equation (3.1) we obtain the relations

f1 (t) = [w1y(0,t) - w1y(h1,t)], f1(0) = U [w10y(0) - w10y(h1)]

V1

h

f2(t)

V2

h1

[w2y(h1,t) - w2y(h, t)], f2(0)

V2

h - h1 y y h - h-1

There exists the energetic identity for the problem (3.1)-(3.5)

[w20y(h1) - w20y(h)].

(3.7)

d

dtE(t)+ M1

fhl 2 fh 2

/ wXy dy + ^2 / w2y dy = 0, /0 Jhi

where

1 f hi 1 fh

E(t) = X P1 / w2 dy + x P2 / w2 dy.

2 0 2 hi

(3.8)

(3.9)

oo

k

h

Lemma 1. The following inequality holds

/ W dy + w2 dy < M M1 / wu dy + М2 / wjL dy ,

IQ Jh! \ J Jhi

where M is the solution of the variational problem

rh

M = sup

Vi,V2GV

ph i ph

/ V dy + / v2 dy

J0___________Jhi___________

p hi p h

M1 / vly dy + М2 / dy

0 hi

Here V C W2(0, h 1) x W2(h 1; h) and conditions (3.3), (3.4) for v 1, v2 are satisfied.

Z2 z0,

The proof is given in [4]. Due to this lemma we get M = (h — h1)2/p1z^, where z0 is the

minimal positive root of the equation

sin(a1z) cos(a2z) + a2 sin(a2z) cos(a1z) = 0.

Here a1 = h1/(h — h1), a2 = (p1/p2)1/2. From (3.8), (3.9) we get inequality

§ + *e « 0, , = A™ (.L, A. ),

dt M \p 1 P2)

hence

E(t) ^ E(0)e

-lit

(3.10)

1 /*hi 1 /*h

with E(0) = - pM w1o(y) dy + x P2 / w2o(y) dy.

2 J0 2 Jhi

1

2 0 2 hi

Moreovere, there is another identity for the problem (3.1)-(3.6)

0

hi /»h -1 о / /* hi /* h

,2 л I л.,2 , 1 _d I I л.,2 лл> , ,, I л„2

2 dt

PW w2t dy + P2 / w2t dy + om ( М1 / w2y dy + М2 w2Ly dy

hi

and then following estimates hold

C hi

0

w2y dy < —

W0 M1

0

Г 2 d ^ W0

w2y dy < --- :

/hi М2

hi

(3.11)

where

hi 2 h 2 W0 = M1 / w10 dy + М2 / w20 dy.

0 hi

From (3.4), (3.10), (3.11) we have the estimates

К-(y,t)| <

/8E(0)W^1/4 V vj

-it/2

(3.12)

Therefore, the motion of fluids are slowed down by the viscous friction according to inequalities (3.12).

Now, let us go over to estimate the function f-(t) defined by (3.7). Firstly, the new unknowns Vj(y,t) = w-t(y,t) are satisfies the problem (3.1)-(3.6) with fjt instead of f- (t) and initial data at t = 0 equal to

Vm(y) = ^гшюуу(y) + ^ [w10y(0) — wWy(h.1)],

V20(y) = V2w20yy (y) +

h1

V2

h — h1

[w20y (h1) — w20y (h)].

h

h

h

h

0

e

Hence, we get estimates like (3.12)

(3.13)

here

1 rhi 1 rh

Ei(0) = 2 Pi yo Vi2o Ы + 2 P2 jh V22o(y) dУ,

y* hi y* h

Wo = Mi Vi20y Ы dy + М2 V22oy Ы

o hi

If we multiply equation (3.1) by y(h1 — y) (j = 1) or (y — h1)(h — y) (j = 2) and integrate, then we obtain equalities

h3 Г hi

A fi(t)= y(hi — y)wit(y,t) dy — vihiwi(hi,t),

6

(h — hi)3

6

f2(t)= (y — hi)(h — y)w2t(y,t) dy — V2(h — hi)w2(h,t).

hi

Using inequalities (3.12), (3.13) we get estimates

Ifj(t)l < Cje-st/2,

with constants are

Ci = 6

8E i(0)W0i

vi

i\ i/4

+

6vi (8E(0)Wo\i/4

hi, V

vi

C =6{8Ei(0Wi ^i/4 + 6V2 (8E(0)Wo y/4

V2 J (h — hi)2 \ V2

4. Solution in the Laplace representation

Let us apply the Laplace transform to problem (3.1)-(3.5)

wj (y,p)= wj (y,t)e~pidt.

Jo

As a result, we obtain a boundary-value problem for the ODE:

f j (P) wjo(y)

vj

j = 1, 2;

vj vj

wi(0,p)=0, w2(h,p) = 0;

wi(hi,p) = w2(hi,p), Miwiy (hi ,p) — M2w2y (hi,p) = 0; y* hi y* h

/ wi(y,p) d = 0, / w2(y,p) dy = 0

0 hi

wj (y,p) = Cji sh ^\l~y^J + C ch ^\JV~.y^J —

----= / wjo(z) sh(. /А (y — z)) dz +

Vppj Jyj V V vj )

with the exact solution

(3.14)

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

f j (p)

(4.6)

p

where y 1 =0, y2 = h

U

C1

-| -1

ch ( 4 / — hi ) — 1

vi

\ flip)

\ P

pp

sh a — hi — * — hi

+

+ - I

V1 J 0

hi

J ww(z) sh ^^ (y — z) ) dz

i{'

Cl = 4-<|Gch( 4/-^h) +

C1 = !( M

2 ДI P

sh ( ,/ —h) - sh F/ —hi

— G sh J h

Cl = — FM, Д = 1 — ch( I— (h — hi) P VV V2

G=(yh f,(p)+if

Vpv2 v2 Jh1

I W2o(z) sh ^ — (y — z) ) dz

dy.

Taking into account formulae (4.6), (4.7) from (4.5), we get

fi =

P

aia^ — a2 аз

(Gia4 — G2a2), f2 =

P

ai a4 — a2a3

(G2ai — Gia3),

here

ai =

dy ,

a4

(4.7)

(4.8)

(4.9)

Gi =

Vpvi

J Wio(z) sh ^(hi — z)^ dz + -—= J W20(z) sh ^(hi — z)^ dz—

—Vi s4\/ Vihi

n-i rhi

cMi / — hi ) — 1

i‘[I wio(z> sh( vv>dz

rv

d sh( vU—hi)/«,

phi

dy—

J W2o(z) shf JV. (y — zU dz

dy-,

G2

df Lwio(z) ch (A(hi—z))dz+-fe JIH w2o(z) ch (U(hi—z))dz—

"ж2 ch(\ /V)hi

T-1 Г hi Г

ch ( 1 / — hi ) — 1

+ £Д d,(v V2(h — hi>

L' [fwi0 (z) sh (U(y—z))dz

J W2o(z) sh ^ (y — z^j dz

hi h

dy+

dy.

v

v

1

Simple, but cumbersome calculations with the use of asymptotic representations for functions sh x and ch x show that

lim wj(y,t) = limpWj(y,p) = 0, lim f(t) = limpf(p) = 0.

t—— ^ p——U t—p——u

Lust results obtained are good agrement with the a priori estimates (3.13) and (3.14).

Conclusions

The partly invariant solution of Navier-Stokes equations is investigated. This solution may describes the plane unsteady motions of a viscous fluid in a strip with two rigid walls, the fluid motion with one rigid wall and free boundary or the motion of a two immiscible fluids with interface in a strip bounded rigid walls. The motion arised due to initial velocity field. It was shown that this problem can be reduced for creeping motions to the linear initial boundary inverse problem for parabolic equations. Two problem were solved by Fourier method. At that time, the interface problem is solved by using some properties of the Laplace transformation. For any cases the motions are retarded by viscous friction.

References

[1] V.O.Bytev, Group-theoretic properties of the Navier-Stokes equations, Chislennye metody sploshnoi mehaniki, 3(1972), no. 3, 13-17 (in Russian).

[2] V.K.Andreev, O.V.Kaptsov, V.V.Pukhnachev, A.A.Rodionov, Application of Group-Theoretical Methods in Hydrodynamics, Kluver Acad. Publ., Dordrecht, Boston, London, 2010.

[3] V.K.Andreev, Yu.A.Gaponenko, O.N.Goncharova, V.V.Pukhnachev, Mathematical Models of Convection, Walter de Gruyter GmbH and Co KG, Berlin/Boston, 2012.

[4] V.K.Andreev, On Inequalities of the Friedrichs type for Combined Domains, J. Siberian Federal Univ., Math. and Physics, 2(2009), no. 2, 146-157.

Частично инвариантные решения уравнений Навье—Стокса, описывающие нестационарные двумерные движения вязкой жидкости

Виктор К. Андреев

Трехмерная непрерывная подалгебра используется для нахождения частично инвариантного решения уравнений вязкой несжимаемой жидкости. Оно интерпретируется как двумерное движение одной или двух несмешивающихся жидкостей в плоском канале. Возникающая начальнокраевая задача для фактор-системы является обратной. Нестационарная задача для ползущих движений решена методом разделения переменной для одной жидкости и методом преобразования Лапласа для двух жидкостей.

Ключевые слова: частично инвариантное решение, вязкая жидкость, задача со свободной границей, поверхность 'раздела.