Uniqueness of a solution of an ice plate oscillation problem in a channel Текст научной статьи по специальности «Физика»

УДК 517.95 + 532.582

Uniqueness of a Solution of an Ice Plate Oscillation Problem in a Channel

Konstantin A. Shishmarev* Alexander A. Papin^

Altai State University Lenina, 61, Barnaul, 656049 Russia

Received 09.01.2018, received in revised form 10.02.2018, accepted 20.04.2018 In this paper an initial-boundary value problem for two mathematical models of elastic and viscoelastic oscillations of a thin ice plate in an infinite channel under the action of external load is considered in terms of the linear theory of hydroelasticity. The viscosity of ice is treated in the context of the Kelvin -Voigt model. The joint system of equations for the ice plate and an ideal fluid is considered. Boundary conditions are conditions of clamped edges for the ice plate at the walls of the channel, condition of impermeability for the flow velocity potential and the damping conditions for the oscillations at infinity. The uniqueness theorem for the classical solution of the initial-boundary value problem is proved.

Keywords: Euler equations, viscoelastic oscillations, ice plate, external load, uniqueness. DOI: 10.17516/1997-1397-2018-11-4-449-458.

1. Formulation of the problem

Oscillations of an ice plate in a channel under the action of external load applied to the ice are considered. The channel has a rectangular cross section with finite depth H, (—H < z < 0), and width 2L, (—L < y < L), and it is infinitely long, —to < x < to, (x, y, z) is the Cartesian coordinate system. Fluid in the channel is inviscid and incompressible with density pi. The thickness of the ice plate hi and rigidity D, D = Eh3/[12(1 — v2)], are constant, where E is the Young modulus and v is the Poisson ratio. The ice plate is clamped at the channel walls, y = ±L. We denote unbounded domains occupied with the ice plate and the liquid by n c R2 and Q c R3, respectively

n = {—to < x < to, —L<y<L}, Q = {—to < x < to, —L<y<L, —H < z < 0}. Boundaries of these domains are r = dQ = r U r2 U r3 and G = dn = Gi U G2, where r = ri U r2 = {—to <x < to, y = ±L, —H <z < 0} U {—to <x < to, —L<y<L, z = —H}, r2 = {—to < x < to, —L <y < L, z = 0}, r3 = {x = ±to, —L<y<L, —H < z < 0}, G1 = {—to < x < to, y = ±L}, G2 = {x = ±to, —L < y < L}.

Let us introduce QT = Q x [0,T], nT = n x [0,T], where t e [0,T] is time. The problem is formulated in terms of the vertical displacement of the ice plate w(x,y,t) and the velocity potential of the flow x, y, z, t) in the context of the theory of linear hydroelasticity [1].

* shishmarev.k@mail.ru t papin@math.asu.ru © Siberian Federal University. All rights reserved

An irrotational flow of an ideal fluid in QT with the velocity potential p(x, y, z, t) is considered. The potential p satisfies the Laplace equation in the flow domain

A^(x,y, z,t) = 0, (x,y,z,t) e QT. (1)

The boundary conditions of impermeability in r are

=0 (y = ±L), yz = 0 (z = -H), (2)

and the linearised kinematic condition and Bernoulli integral in r2 are

Pz(x,y, 0,t) = wt(x,y,t), p(x,y, 0,t) = -pwt(x,y, 0,t) - pigw(x,y,t), (3)

where p(x, y, 0, t) is the hydrodynamic pressure on the ice-fluid interface, g is the gravitational acceleration. We assume that fluid oscillations are damped out far away from the load. This boundary condition in r3 is

p(x,y, z,t), Px(x,y, z,t) ^ 0 (|x|^to). (4)

The ice displacement w(x,y,t) satisfies the equation of a thin viscoelastic plate in the ice plate domain

D{1 + Tdt) V'w + Mwtt = P(x,y,t)+ p(x,y, 0,t), (x,y,t) e Ut. (5)

The initial conditions are

w(x,y, 0) = w1(x,y), wt(x,y, 0) = w2(x,y), (x,y) e n. The clamped conditions in Gi are

w = 0, wy =0 (y = ±L), (6)

and the damping conditions far away from the load in G2 are

w = 0, wx =0 (|x|^w). (7)

The initial-boundary value problem (1)-(7) describes oscillations of an ice plate clamped to channel walls under the action of external load P(x, y, t). Here t = n/E is the retardation time;

d4 d4 d4

n is the viscosity of ice; V4 = ——r + 2 + ——r; M = ph is the mass ratio; pi is the

ox4 ox2oy2 ox4

density of ice; p(x,y, 0,t) is the velocity potential at the lowest boundary of the ice plate; the external load is a smooth localized function P(x,y,t).

Let a pair of functions w(x, y, t) and p(x, y, z, t) be a solution of the system of equations (1)-(7) if these functions are defined in QT and nT and they have the following properties: (a) functions w(x,y,t) and p(x,y,z,t) satisfy system of equations (1)-(7), initial and boundary conditions and they are continuous in nT and QT, respectively; (b) functions w(x,y,t), wx(x,y,t), p(x,y,z,t) and px(x,y, z,t) vanish when x ^ to; (c) all derivatives up to third order of w(x,y,t) exist and they are continuous in nT and bounded when x ^ to. The main result of this paper is the following theorem.

Theorem 1. Problem (1)-(7) has unique classical solution.

The system of equations (1)-(7) describes forced oscillations of an ice plate clamped to two walls of a channel (parameters of the ice plate and channel may vary). Joint systems of unsteady equations of a thin viscoelastic or elastic plate and fluid were investigated numerically and analytically for various initial-boundary conditions (see, for example, [1-4] etc). There

is an excellent review of the problems with a thin unbounded ice [1]. Unsteady problems of the ice motion under the action of external load described by equations (1)-(5) in unbounded domain were analytically and numerically investigated [3,4]. Free oscillations of an ice cover described by equations (1)-(4) without taking into account damping and conditions (4) and (7) were studied [5]. Problems of oscillations of an ice plate clamped to a vertical cylinder in a close mathematical formulation were studied [6,7]. Problems with one wall, in particular, in non-linear formulation were considered [8]. The solution of the problem with two walls for given external load was studied numerically by the normal mode method [9].

Two basic approaches were used to study problems. The first approach is the numerical study of a steady-state solution of the problem in the coordinate system moving together with the load. In this coordinate system the ice displacement and the velocity potential do not depend on time, and solution is obtained using numerical methods. Second approach is to solve unsteady problem, where the ice displacement is represented by the series of integrals using the Fourier transform. These integrals depend on time. Parameters of the ice displacement are determined using asymptotic methods as limiting values of forced unsteady hydroelastic waves which are developed with time. Both approaches do not answer the question of existence of the solution. However the ice displacement and strain distributions caused by the moving load are calculated. Comparing calculated strain distribution with critical value allow one to predict the area where the ice is broken [8,10].

Solvability of initial-boundary value problems for the Euler equations and unsteady equation of a thin plate were studied in detail separately [11-16]. Initial-boundary value problems for unsteady Euler equations of a potential flow of a homogeneous liquid with a free surface were considered in detail [11]. Solvability of initial-boundary value problems for the Euler equations with a free surface was considered [12]. Uniqueness of the solution of the flow problem with a given vortex for the unsteady Euler equations of an ideal incompressible fluid was proved [13]. Equation (5) determines oscillations of a thin plate within the Kirchhoff-Love hypothesis of the linear theory of elasticity [17]. This equation is known as the Euler-Bernoulli beam equation in one dimensional case with t = 0. Existence of a classical solution of boundary value problems for the elastic beam equation with a nonlinear right-hand side and fixed edges of the beam was proved [18]. Solvability of the initial-boundary value problem for oscillations of a viscoelastic beam with inhomogeneous boundary conditions was studied using variational methods [19]. Solvability of problems with rigid inclusions and cracks in thin elastic plates within the Kirchhoff-Love hypotheses or Timoshenko hypotheses was studied in detail (see, in example, [14]). Existence of a solution of the bending problem of an elastic plate with rigid inclusion was proved [15]. Unique solvability of the problem of joining of two elastic homogeneous beams, one of which is the Euler-Bernoulli beam and the other is the Timoshenko beam, was proved [16]. In this paper we study the uniqueness of the solution of the joint problem of a plate bending and motion of an ideal fluid. In order to prove the Theorem 1 we used approaches developed before [13].

2. Uniqueness of viscoelastic oscillations

Let us assume that two different non-trivial solutions wi,y>i and w2,y2 of system (1)-(7) exist. Functions w = — w2 and y = — satisfy the following initial-boundary value problem

Ap(x,y,z,t) =0, (x,y,z,t) e QT, (9)

pz = wt (z = 0), py = 0 (y = ±L), pz =0 (z = -H), (10)

w = 0, wy =0 (y = ±L), (11)

p(x,y,z,t), px(x,y,z,t), w(x,y,t), wx (x, y, t) — 0 (|x| —^ to). (12)

w(x, y, 0) = 0, wt(x, y, 0) = 0. (13)

We are to prove that the solution of problem (8)-(13) is nothing but w = 0 and p = 0. The solution (p,w) of problem (8)-(13) has the following properties

I w(x,y,t) dn = 0, t e [0,T], (14)

J n

/ Vp^^y^,^2 dQ= p(x,y, 0,t)wt(x,y,t) dn, t e [0,T], (15)

Jn Jn

/ Vp(x,y,z, 0)|2 dn = 0, p(x,y,z, 0)=0. (16)

Jn

Integrating equation (9) in QR = {-R < x < R, —L < y < L, —H < z < 0} and using the Divergence theorem, we obtain

Apdiln = py\y dq + Pz |z=_ h dY + Vx\xr=_-r dY = °.

nr ri r r?ru:r2r r3r

Here index R in r denotes corresponding boundaries of QR. Using conditions (10)-(12), in the limit R ^ to the last equation provides

i pz(x,y, 0,t) dn = ( wt(x,y,t) dn = — i w(x,y,t) dn = 0.

Jn Jn dt Jn

Using condition (13), from this equation we arrive at (14). Now we multiply equation (9) by p and then integrate the result in QR. After applying the Divergence theorem, we obtain

/ p Ap—Qr = - \Vp\2 —Qr + p (Vp ■ n) —Tr = jqr jqR jvr

= - J\vp\2 dQR + J (ppyWyl-L dY + J (ppz)\Z-°_H dY + J (ppx)\XrZRR dY = 0. mR rl

r r2r^r2r r3r

Using conditions (10)-(12), in the limit R ^ to we obtain from the last relation

- i \Vp\2 dQ+ i p^p(x,y, 0,t) dG = - i \Vp\2 dQ+ i p(x,y, 0,t)wt dG = 0.

jn jg dz jn j g

Condition (15) is obviously obtained from this relation. After taking t = 0 in (15), we have / \Vp(x,y,z, 0)\2 dQ= p(x,y, 0,0)wt(x,y, 0) dn = 0.

Jn Jn

Using (13), from this equation we arrive at (16).

Equation (9) provides d

dt(DT V4w + D V4 wdr + Mwt + pl y(x,y, 0,t)+ pl g wdr^j =0. (17)

The expression in brackets in (17) is equal to zero for t = 0 due to conditions (13) and (16). Taking this into account, we obtain

DtV4w + D i V4wdT + Mwt + ply + plg i wdT = 0. (18)

Jo Jo

Multiplying equation (18) by wt(x,y,t) and then integrating the result in nR = {—R < x< R, —L < y < L}, we get

r tit t \ r

4 4 2

Dt wtV w dnR + D / V4wdT wt dnR + M wf dnR +

Jnn Jnn\ JO J Jnn

+pi ywt dnr + pig I / wdT I wt dnr = 0. (19)

Jnn Jnn \Jo J

Using (15), the forth therm in the left hand side of (19) is

i i y(x,y, 0,t)wt(x,y,t) dnR = i \Vy(x,y, 0,t)|2 dQ. J Jnn Jn

Let us transform all other terms in (19). 1. The first term in the left hand side of (19) is

i wxxxxwt dnR = - d i w2xx dnr + i wxxxwt\Z=-R dy — i wxxwxt\Z=-R dy,

■Jnn 2 dt Jnn J-L J-L

f wyyyywt dnr = 1 d f wly dnr + f wyyywt\yy=-L dx — f wyywyt\yy=-L dx,

Jnn 2 dt J nn J-R J-R

j wxxyywt dnR =2d j wly dnR + j wxyywt\x=-R dy — j wxywxt\Vy=L-L dx.

t

2. We introduce function u = J wdT, ut = w, utt = wt. Then the second term in the left hand

o

side of (19) is

- i u2xxdnR — i uLtdnR + i uxxxutt\x=-Rdy — i uxx uxtt \ x=-Rd inn 2 dt Jnn Jnn J-L J-L

i uxxx$ittdnR = - i u2xxdnn - i ulxtdn.R +f uxxxutt\x=RRdy - i uxxuxtt\ax=RRdy, JnR 2 dt Jnn Jnn J-L J-L

f uyyyyuttdnR = - —22 f ulydnR - f ulytdnR + f uyyyutt\Vy=RLdx - f uyUytt\УyZRLdx, Jnn 2 — Jnn Jnn J-R J-R

2 dP^ uyy dnR — in uyyt/mR + / u'yyy utt\y=-Ldx — j R uy{W'ytt\y= L

i uxxyy utt dH-R =- -¿of u2xy dnR —i u2xytdnR +[ uxyy utt\x=-Rdy —i uxy uxtt\yy=LLdx. ■jnn 2 dt Jnn jnn J-l J-R

t

3. Using function u = f wdT, the last term in the left hand side of (19) is

o

L WdT)

1 d2

wdT J wt —Hr =

Inn \Jo

wt dnR =--r- I u2 dnR — ut dnR.

t R 2 dt2Ln R Ln t R

Summarizing all relations for the integrals, we obtain

1 d

Mi w2 —Mr + pi i pwt dQ + Dt 1 —(i w2x —nR +/ w2yy —Mr + 2 f w2Xy d^) +

Jnr Jn 2 dt \Jnr JnR Jnr J

+ D1 —K! uXx dnR + in uyy dnR + 2 In uXydn^ + Pl 1 dtX Jn u dnR =

= ir +ir + d( f uXxt d^R + / uXyt dnR + 2 i u2 t dn J + Pl f u2 —Rr, (20)

\Jnr Jnr Jnr J Jnr

where

ir = D^J wxxwxt\X=rr dy - J wxxxwt\XX=rr dy - 2 J wxyywt\X=RR dy^ +

+ D ^J uxxuxtt\X=RR dy - J uxxxutt\X=RR —y - 2 J uxyyutt\XZRR d^ , ir = dt^ j R wyy wyt\yyZ-L dx - j R wyyy wt\Vy=-L dx + 2 j R wxy wxt\yyZ-L x +

+ Dij R uyyuytt\y=-L dx - j r uyyyutt\yyZ-L dx + 2 j R uxyuxtt\yZ—L x .

In the limit R — to in (20) for y = ±L we have

w = wy = wt = wyt = 0,

and for \x\ —> ^o

(w,wx,wt,wxt) — 0.

Additionally, for t = 0 we have

w = wt = u = wx = wy = wxy = ux = uy = uxy = 0.

Thus the boundary integrals in (20) are equal to 0, ir =0 and ^ = 0. We introduce

Y (t) = i (wXx + w2y + 2wXy ) dn, Z (t) = f (uXx + u2yy + 2uXy ) dn. Jn Jn

Using (15), relation (20) becomes

/[ 1 d 1 d2

w2tm + Pl jjVp\2dQ + -Dt-Y(t) + -Ddt2Z(t)+

+ 1 pl^i u2d,n = DY(t) + pi j u2—n. (21)

2 dt2 Jn Jn

Using relation

t

2

(x,y,t) = 2 w(x,y,T )wT (x,y,T )—t

_

and Holder inequality, we obtain

J w2(x,y,t)dn < 2 J^J w2(x,y,T )d^ (^J w2 (x,y,T )dn^ dt.

It follows from the last relation that

et

J w2(x,y,t)dn ^ T J (^J wt(x,y,rdr.

(22)

Let us introduce

Y (t) = M J (J w2t dn^ dr + 1 DrY (t), Ci =ma^ 2 , Tm) '

Using (21) and (22), we obtain the following inequality

d~, , d2 ( 1 „„, , 1

dtY(t)+ ~2DZW + 1 Pi i u2dn) < CiY (t).

Then we have

Jt(YeClÎ)+ e(jDZ(t) + 2Pi i u2dn)

dH 1DZ (t) + 1

e~Clt + e~Clt *(2DZ(t) + 2Pi I u2dn)) +

+Cie-Ci t H 2 dz (t)+2 pi iu2dM) <

(23)

Upon integrating (23) with respect to t from 0 to t\ and taking into account that

d

Y (0)=0, dZ (t)

dt

we obtain

d11

0, I u2dn

, I u

t=o dt Jn

= 0

t=o

Ye-Cltl + e-Gltl iX2DZ(ti) + 2Pi i u"dn) +

+ C\e-Cltl^2DZ(ti) + 2P; J^ u2d^ + C2 J 1 e-Clt 1 DZ(t) + 1 pi ^ u2d^ < 0. (24)

The first, third and forth terms in the left hand side of the last inequality are nonnegative. Thus

dk{ 2DZ (ti) + 2 p; Lu2dn) <0.

We conclude from the last inequality that Z(t) = 0, u(x,y,t) = 0. Considering (23), (22) and (21), it is easy to show that w = 0, V^ = 0, ^ = 0. Theorem 1 is proved.

3. Uniqueness of elastic oscillations

Viscous damping is not well understood. Because of this, damping coefficient r = 0 is neglected in many studies. We are to prove the analogue of the Theorem 1 for the case r = 0. In this case system of equation (8)-(13) becomes

DV4w + Mwtt = -pwt(x,y, 0, t) - pigw, (x,y,t) G nT, (25)

A<(x,y,z,t) = 0, (x,y,z,t) e ttT, (26)

<z = wt (z = 0), py = 0 (y = ±L), <z = 0 (z = —H), (27)

w = 0, wy =0 (y = ±L), (28)

<p(x,y,z,t), px(x,y,z,t), w(x,y,t), wx (x,y,t) ^ 0 (|x|^to), (29)

w(x,y, 0) = 0, wt(x,y, 0) = 0. (30)

Let us that the solution (<,w) of problem (25)-(30) satisfies conditions (14)-(16). The equation (25) gives

DV4w + Mwtt + p<t + pi gwt = 0. (31)

Multiplying equation (31) by wt(x,y,t) and then integrating the result over nR = {—R < x< R, —L < y < L}, we obtain

D (V4w) wt + M wttwt dRR + pi <twt dnfl + pig wwt dRR = 0. (32) JnR JnR JnR JnR

Taking into account Section 2 and integrating by parts the terms of equality (32), we arrive at the identity

M d f f ^ D d [ 2

t jtLwdnR+pi Lr <wMr + 2 dt Jnr wxx+

+ w2y +2w2XydnR + f w2dnR = Ir, (33)

2 dt J nr

where the sum of boundary integrals Ir is defined in the same way as the sum of boundary integrals ir and Ir in equation (20). Using (27)-(29), in the limit R ^ to in (33) we obtain

dt(MM Ju wX dn+ 2 ^ iy<2 dn + D Jn Y(x,y,t) dn+ ppf Jn w2 dn^=0. (34) For t = 0 we have

w = wt = wxx = wxy = wyy = 0. (35)

The expression in brackets in (34) for t = 0 is equal to zero due to conditions (16) and (35). Taking this into account, we obtain

M J wX dn+ 2 J iy<l2 dQ + D J Y (x,y,t) dn+ ppg j w2 dn = 0.

n n n n

All terms in the left hand side of the last relation are nonnegative. Then we can conclude that

wt = 0, V<(x, y, z, t) = 0, Y(x, y, t) = 0, w(x, y,t) =0 and, consequently, <(x, y, z, t) = 0.

References

[1] V.Squire, R.Hosking, A.Kerr, Langhorne Moving loads on ice, Kluwer Academic Publishers, 1996.

[2] I.V.Sturova, Unsteady three-dimensional sources in deep water with an elastic cover and their applications, J. Fluid Mech., 730(2013), 392-418.

3] V.D.Zhestkaya, M.R.Dzhabrailov, Numerical solution of the problem of motion along an ice cover with a crack, PMTF, 49(2008), no. 3, 151-156 (in Russian).

4] A.A.Matiushina, A.V.Pogorelova, V.M.Kozin, Effect of Impact Load on the Ice Cover During the Landing of an Airplane, International Journal of Offshore and Polar Engineering, 26(2016), no. 1, 6-12.

5] A.A.Korobkin, T.I.Khabakhpasheva, A.A.Papin, Waves propagating along a channel with ice cover, European Journal of Mechanics - B/Fluids, 47(2014), no. 3, 166-175.

6] P.Brocklehurst, A.A.Korobkin, E.I.Parau, Hydroelastic wave diffraction by a vertical cylinder, Philos. Trans. Roy. Soc. London, Ser. A: Math. Phys. Eng. Sci., 369(2011), 2833-2851.

7] S.Malenica, A.A.Korobkin, Water wave diffraction by vertical circular cylinder in partially frozen sea, In Proc. 18th Internations Workshop on Water Waves and Floating Bodies, Le Croisic, France, 2003.

8] P.Brocklehurst, Hydroelastic waves and their interaction with fixed structures, PhD thesis, University of East Anglia, UK, 2012.

9] K.Shishmarev, T.Khabakhpasheva, A.Korobkin, The response of ice cover to a load moving along a frozen channel, Applied Ocean Research, 59(2016), 313-326.

10] V.M.Kozin, V.D.Zhestkaya, A.V.Pogorelova, S.D.Chizhumov, M.P.Dzhabailov, V.S.Moro-zov, A.N.Kustov, Applied problems of the dynamics of ice cover, Akad. Estestvennyh Nauk, Moscow, 2008 (in Russian).

11] L.V.Ovsyannikov, General equations and examples, The problem of unsteady motion of fluid with a free boundary, Novosibirsk, Nauka, 1967 (in Russian).

12] V.I.Sedenko, Solvability of initial-boundary value problems for Euler's equations for flows of an ideal incompressible nonhomogeneous fluid and an ideal barotropic fluid bounded by free surfaces, Mat. sb., 185(1995), no. 11, 57-78.

13] W.A.Weigant, A.A.Papin, On the uniqueness of the solution of the flow problem with a given vortex, Mathematical notes, 96(2014), no. 6, 820-826.

14] A.M.Khludnev, Problems of the theory of elasticity in nonsmooth domains, Moscow, Fizmat, 2010 (in Russian).

15] A.M.Khludnev, On the bending of an elastic plate with an exfoliated thin rigid inclusion, Sib. journal. industr. mat., 14(2011), no. 1, 114-126 (in Russian).

16] N.V.Neustroeva, N.P.Lazarev, Junction problem for Euler-Bernoulli and Timoshenko elastic beams, Sib. Elektron. Mat. Izv., 13(2016), no. 1, 26-37.

17] S.P.Timoshenko, J.Goodier, Theory of Elasticity, Moscow, Nauka, 1975 (in Russian).

18] H.Lu, L.Sun, J.Sun, Existence of positive solutions to a non-positive elastic beam equation with both ends fixed, Boundary Value Problems, (2012), no. 56, https://doi.org/10.1186 /1687-2770-2012-56.

[19] M.Basson, M.de Villiers, N.F.J.van Rensburg, Solvability of a Model for the Vibration of a Beam with a Damping Tip Body, Journal of Applied Mathematics, (2014), Article ID 298092, http://dx.doi.org/10.1155/2014/298092.

Единственность решения задачи о колебаниях ледового покрова в канале

Константин А. Шишмарев Александр А. Папин

Алтайский государственный университет Ленина, 61, Барнаул, 656015 Россия

В 'работе, в рамках линейной теории гидроупругости, рассматривается начально-краевая задача для математических моделей упругих и вязкоупругих колебаний тонкой ледовой пластины в бесконечном канале, вызванных внешней нагрузкой. Вязкоупругие свойства льда моделируются на основе реологического закона Кельвина-Фойгта. Совместная система уравнений динамики пластины и идеальной жидкости замыкается условиями жесткого защемления для пластины на стенках канала, условиями непротекания для потенциала скорости течения и условиями затухания колебаний на бесконечности. Доказана теорема единственности классического решения поставленной начально-краевой задачи.

Ключевые слова: уравнения Эйлера, вязкоупругие колебания, ледовый покров, внешняя нагрузка, единственность.