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16Russian Journal of Nonlinear Dynamics, 2020, vol. 16, no. 2, pp. 275-290. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd200204
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 74J30
Modeling of Nonlinear Waves in Two Coaxial Physically Nonlinear Shells with a Viscous Incompressible Fluid Between Them, Taking into Account the Inertia of its Motion
L. I. Mogilevich, S. V. Ivanov, Yu. A. Blinkov
This article investigates longitudinal deformation waves in physically nonlinear coaxial elastic shells containing a viscous incompressible fluid between them. The rigid nonlinearity of the shells is considered. The presence of a viscous incompressible fluid between the shells, as well as the influence of the inertia of the fluid motion on the amplitude and velocity of the wave, are taken into account.
A numerical study of the model constructed in the course of this work is carried out by using a difference scheme for the equation similar to the Crank-Nicolson scheme for the heat equation.
In the case of identical initial conditions in both shells, the deformation waves in them do not change either the amplitude or the velocity. In the case of setting different initial conditions in the coaxial shells, the amplitude of the solitary wave in the first shell decreases from the value specified at the initial instant of time, and in the second, the amplitude grows from zero until they equalize, that is, energy is transferred.
The movement occurs in a negative direction. This means that the velocity of deformation wave is subsonic.
Keywords: nonlinear waves, elastic cylindrical shells, viscous incompressible fluid, Crank-Nicolson difference scheme
Received November 30, 2019 Accepted January 30, 2020
This work was supported by the RFBR grant No. 19-01-00014a.
Lev I. Mogilevich mogilevichli@gmail.com
Yuri Gagarin State Technical University of Saratov ul. Politechnicheskaya 77, Saratov, 410054 Russia
Sergey V. Ivanov evilgraywolf@gmail. com Yuriy A. Blinkov blinkovua@gmail.com
Saratov State University
ul. Astrakhanskaya 83, Saratov, 410012 Russia
1. Introduction
The study of the wave process in elastic shells is widely used in the new structures in modern engineering industry. The propagation of deformation waves in nonlinear shells was considered in [10-13]. In [14-16] the interaction of the shell with a viscous incompressible fluid was considered without taking into account wave phenomena.
In the case of a related problem the equations of fluid dynamics and the ones of the elastic body dynamics are solved simultaneously, taking into account the boundary conditions. This approach is used for studying the oscillations of elastic bodies [17, 18], and also in this article when studying the deformation waves of nonlinear elastic shells containing a viscous fluid of constant density, taking into account the inertia of its motion. It is impossible to study the models of deformation waves using the methods of qualitative analysis in the case of filling the space between shells with a viscous incompressible fluid [2, 3, 19-23]. This leads to the necessity of applying numerical methods [4, 5].
This paper addresses the problem of obtaining mathematical models of the wave process in infinitely long physically nonlinear coaxial cylindrical elastic shells [16] with rigid nonlinearity by means of the perturbation method, the problem parameter being small. They differ from the known ones in that they take into account the presence of an incompressible viscous fluid between the shells. The research is carried out on the basis of the related hydroelasticity problems described by the shell dynamics and incompressible viscous fluid equations with the corresponding boundary conditions in the form of a system of generalized MKdV equations. The effects of the incompressible viscous fluid between the shells on the behavior of the deformation wave in the coaxial shells are revealed. The presence of a deformation wave in the outer shell leads to the appearance of a deformation wave in the inner shell, which was not present at the initial instant, and there is a "transfer of energy" (through the liquid layer) from the outer shell to the inner one, accompanied by a decrease in the wave amplitude in the outer shell. As a result, a decrease in the speed of its distribution takes place. In this case an increase in amplitude occurs in the inner shell. In the course of time the amplitudes become equal due to their changes. The fluid movement inertia changes the wave velocity.
A numerical study of the model constructed in the course of this work is carried out by using a difference scheme similar to the Crank-Nicolson scheme in the case of the heat equation [29-32].
2. Defining and resolving relations of the physically nonlinear theory of shells
A. A. Ilyushin's deformation plasticity theory [6, 24] in the case of rigid nonlinearity [7, 25] connects the components of the stress tensor ax, a@ with the components of the strain tensor ex, e© and the square of the strain intensity eu, in the form
+ + (2.D
(i)2 4
u 3
1 ■
— 1 + -(
3
M 4i)2+
- noe^e®
№ ~ 1 (1-A'o )2
1 -
2¿t0 - 1
where E is Young's modulus; m is the constant of the material, determined from the experiments on tension or compression; is Poisson's ratio of the shell material. Let us consider axisymmetric coaxial cylindrical shells.
We use the following notation: 5 is the width of the slit occupied by the fluid; R1 is the radius of the inner surface of the outer shell; R2 is the radius of the outer surface of the inner shell (Ri = R2 + 5); Rh are the radii of the median surfaces; h^ are the shell thicknesses;
Ri = R(1) -
ho
(i)
R2 = R(2) +
ho
(2)
; U (i) are the longitudinal elastic shells' displace-
22
ments; are the deflections directed to the center of curvature (i = 1 for the outer shell, i = 2 for the inner one).
Let us write down the relation between the components of the strains and elastic displacements in the form of [8]
=
-x
dU(i) 1 ( dW(i)
dx
+ 2
dx
— z-
g'2W(i)
dx2
-(i)
W(i)
£e R{t) ,
(2.2)
where x is the longitudinal coordinate along the median surface and z is the normal coordinate
in the shell I —— ^ z ^ J. We write down the square of the strain intensity in the form
,(i)2 -
+¿¿2
11
W(i)
W
dU{i) 1 dx + 2
OU& 1 dx + 2
'dW{i) dx
dx
— z
— z-
Q'2W{i)
dx2
d2WW
dx2
+
W (i)2
~W2
+
(2.3)
£
1
2
2
2
4
u
3
2
Fig. 1. Elastic infinitely long coaxial cylindrical shells. RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2020, 16(2), 275
We determine the forces in the middle surface of the shell and the moment using the following formulas
hd)
2
n0 2
n0 2
N® = j a« dz, N® = j a® dz, M« = f
a.
( z dz.
(2.4)
h(i)
di)
di)
The asymptotic analysis performed in [18] showed that the expressions in square brackets of formulas (2.3) can be considered on the median surface (z = 0) for longitudinal waves:
= \
dUW 1 / dWW
dx 2 I dx
+
W(i)2
~W2
+ H 2
W(i)
W
dUW 1 / dWW
dx 2 I dx
By substituting (2.1), (2.2), (2.5) into (2.4), we find
fifi'l /dU(l) | 1 fdw^
2 I dx
+
1 " lO \
1 (dW{i) dx
dx 2
2
¿¿0
W(i)
w
Il
W(i) 4 m + 3~E
dU{i) 1 / cWw dx 2 I dx
dU{i) dx
-+
N® =
Eh
+
(i)
W (i)
dU(i) 1 ( dW(i)
W(i)
+
1 - lO
4m
+ 3 ~E
dUW 1 / cW^
dx 2 \ <9.r
dU(i) 1 ( dW(i)'
1 / cW^ + 2 1 fib
dx 2
W(i) ---1-
W(i)
ii
' dU{€) dx
- +
+
' W(i)
+ I2
fi>[fW 1 / dWW \ * \
"SaT +2 \ fite J ) "W
. (2.5)
(2.6)
Mw =__^
w d2W{i) / 4 mi [" / 12 (1-jug)"^-+ 3£p +
1 / dW{i) + 2 1 dx
+
'w (i)
fi^W 1 /
^2 dx 2 \ <9.r
W(i) "rw
(2.7)
The asymptotic analysis [18] showed that in (2.7) the first term is much larger than the remaining terms and they can be discarded because Mx itself is much less than the force Nx. Therefore, we obtain from (2.7)
M(i) = -
Eh
(i)3
d 2 W(i)
12(1 - i2) dx2 '
(2.8)
0
0
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
The dynamics equations of the shells (Fig. 1) are written down as
dNf] dx
n h^&2U[l)
dr
dx
R
^Ml d_ f dWW^) J_ (i) =
dx2 dx dx ^ EMJVe
(2-9)
= po h
{l)d2W« , ^
dt2
— (-1)
qn-W^+U^ dr dx
R
(i) (i) where t is the time; p0 is the density of the shell material; qx , qn are the stresses from the
side of the liquid inside the annular section; r, x are the cylindrical coordinates; i = 1 for the
outer shell, i = 2 for the inner one. By substituting (2.6) into (2.9), we obtain the permutation
equations
Eh{j] d / dUW 1 /dW& 1 — dx \ dx 2 I dx
W(i) 4 m
dx
-+
+
1 / dWW
2 \ dx
2
At o
W(i)
w
IIl
dUW 1 / dW& dx 2 I dx
+
+
— po% -
W(i) ~W
dU(i) 1 ( dW(i)
W(i) ~W
dt2
dr dx
Eh0i)3 d2 id2W(i)'
+
Eh0) d I dW(i)
4 m
+3
1 — i2 dx \ dx dU(i) 1 ( dW(i) N 2
dUW 1 / dWW
dx 2 I dx
12(1 — i2) dx2 y dx2
o
W(i)
Ho-
dx + 2
dx
Ho-
W(i) w
Ii
dU(i) 1 dW(i)
dx
R(i) 2
dx " + 2
+
+
+
W(i) "RW
+I2
( dU(i)
l
1 / dWW
dx 2 I dx
W& 4 m
W(i)
' +
Eh
(i)
1 — I2 R(i)
Io
dU(i) 1 ( dW(i)
At0 1 ~dT + 2
dUW 1 / dWW
<9.r 2 I dx
W(i)
R(i)
(2-10)
+
Ii
1
"&T + 2
dx
+
W(i)
+ I2
dUW 1 "daT + 2
'dW(i)\ \ W(i)
dx
—Qp--
qn-W^+U^ dr dx
R
R(i)
2
2
2
2
2
2
2
2
2
1
2
x
2
2
x
3. The asymptotic method for studying the equations of shells with a liquid
We used the two-scale perturbation method for solving system (2.10). The estimates made in dimensionless variables characterize the problems under consideration. For wave problems the shell is considered infinite. For longitudinal waves, dimensionless variables and dimensionless parameters are introduced in the shell. We take the wavelength as the characteristic length l, and um, wm as the characteristic values of elastic displacements
W(i) = WmV^, U(i) =
(i)
Vm V1 j
X
Wm = ho, Vm =
X
T'
hpl
t* = c4t, r* =
r (i) ~R '
(3-1)
co
E
P(1 - Mo)
is the propagation velocity of longitudinal elastic waves in the shell. Set
h
(i)
R(i)
= £ < 1,
R(i)2 W
= 0(£), ^ = 1),
l2
h
(i)2
l /7M [ >' E [ >' P
(i) o
m2 i2
(3.2)
where e is the small parameter of the problem. We introduce independent variables in the form
£ = X* -J 1 - M2t*, T = £t*,
(3.3)
where t is the fast time. In these variables (3.3) we obtain the equations as in [9, 26, 33, 34] by keeping terms of order e and £2 in Eqs. (2.10) and discarding terms with higher degrees
d / um dui
(i)
d£ \ l d£
Mo
wm (i) , 4 m ( um dit^
(i)
R(i)
U* +TTTT
3 E \ l d£
Wm (i)
Mo^tt^u?' I x
R(i)
Mi
(i)
Vm
Um dll^ I
(1 - Ml)
+
Wm
2 (i)2
V
Vm dui Wm (i)
d2 ui
d£2
- 2£y 1 - Ml
cfiuf d(dr
l
Pohl) cl
Q^,
Mo-
um du
(i)
mi
Wm
4 m
u¿ + -— Mo-
urn du
(i)
mi
Wm
l d£ R(i) 3 3 E
l d£ R(i)
u
(i)
(3.4)
(i)
Mi
R{t)2 Wm l2 RW
Um dll^ I
(1 - Ml)
+
Wm
R(i)
22
u
(i)2
+M2
u
2 (i)
d 2u
d£2
- 2£ 1 - M2
2d2uf
Um QU,® Wm I R(i)"'
R(i)
d£dT
(-1)i-i Qn
Poho] c0
2
x
l
x
2
X
We represent the dependent variables as an asymptotic expansion
ui = + sui + ..., u« = ug + s4} + .... (3.5)
We obtain the system of equations by substituting (3.5) into (3.4) and keeping terms of order s
d / duS „ J£ml_(i)\ h „2X^42
ae \~W " / -(1"Ato)
du[Q wml (¿)
,, ----1U " „.\"l _ n
We get from this system
d{ umR(i)
Wml (i) du10 (Q 7\
umR(i> ^0 dt
(i)
Thus, uio is an arbitrary function, since the shell has infinite length, the wave velocity / E
is equal to \ the wave velocity in the rod. We obtain the system of equations in the approximation s2
d2u{w , ™ iu™\2o A ~2(............2A (^10 ^ 2 dVio
+ -
didr + +
:=12 2 ^ Zy/l^JZeUmPoWcZ
Ji) ., R i
4 -^oy(-l) ^
(3.8)
for
The resulting equations are generalized Modified Korteweg-de Vries (MKdV) equations
du(i]
no
dt '
In the absence of fluid the right-hand side of the equation is zero and the system of Eq. (3.8) disintegrates into two independent modified Korteweg-de Vries(MKdV) equations. It is necessary to determine the right-hand side by solving the equations of hydrodynamics.
4. The study of stresses acting on the shell from the side of the fluid inside
The equation of motion of an incompressible viscous fluid and the continuity equation in a cylindrical coordinate system (r, 0, x) in the case of an axisymmetric flow [3] are written as
dVr TrdVr dVr 1 dp (d2Vr 1 dVr d2Vr Vr
dt dr dx p dr \ dr2 r dr dx2 r2
dVx TrdVx dVx 1 dp (d2Vx 1 dVx c)2Vx \
t r r x x p x r2 r r x2
Vr Vr Vx
—- + — H--= 0.
t r x
The conditions for the fluid to stick to the boundary of the shells and the fluid are satisfied at r = Ri - W(i) (Fig. 1)
V, = V, = (4,2)
dt
dt
where t is the time; r, x are the cylindrical coordinates; Vr, Vx are the projections on the axis of the cylindrical coordinate system of the velocity vector; p is pressure in the fluid; p is the fluid density; v is the kinematic viscosity coefficient. The stresses from the side of the fluid layer are determined by the formulas
Qn
Qx =
Prrcos ( —n^\nr ) + Prxcos (
Prxcos ( —n^\nr ] + Pxxcos (
r=Ri-W «
r=Ri-W №
p ,o dyr p fovx OVA ovx
Prr = -p + Prx = pv "r— + -r— , Pxx = -p + 2pv—~
dr \ dr dx dx
(4.3)
According to Euler's approach, we have
cos I —, nr ] = —
Ri — W(i)
cos | —,
\N \ 1
COS I —, i I = —
i)
2\ 2
Ri - WW dWW |lV| dx ' dW(i)
—, cos I I =
dx
i
2\ 2
|iV| = I 1 +
dx
i
2\ 2
(4.4)
Here n is the normal to the middle surface of the ith shell, nr, n©, i are the unit vectors of the basis (r, 0, x) of the cylindrical coordinate system, the center of which is located on the geometric axis. If we carry down the stress on the unperturbed surface of the shell, we can
assume that — n = nr and cos ( —n^\nr ] = 1, cos ( ] = 0.
5. Ring cross section
We introduce dimensionless variables and parameters
Vr=wm%jr, Vx=wm^-vx, r = R-2 + 5r*, t* = ^-t, x* = j l 0 l l
pvcolRiWm 0
P =-p-P + Po, 0 = —= o(l),
wm wm R2 f e\ wm wm 0
X = — =R^T = 0UI
Wm _ Wm^_R4 _ S _ _
l 0 Ri l l Ri l
(5.1)
We obtain hydrodynamics equations in the introduced dimensionless variables
öc0 ö d2vr
+
ddvr ( dvr dvr
—--h A vr —--h vx ——
dt* \ dr* dx*
+
ö
vr
ö2vr
dP
dr * ö2 d2 vr
dr*2 R2 (1 + #*) dr* R2 (1 + ^r*Y
+
öc0 ö ~1
vr vr vr
■7;--1" A vr —--h vx ——
t* r* x*
+
vr dr *
+
dP _ 9V
dr * ö
+
ö
l2 dx*2 dvx
+
ö2 d2 vx
(5.2)
dr*2 R2 (1 + ^r*) dr* l2 dx*2'
vx
+
R2 (1 + ^r * ) dx
0
and boundary conditions
vx = -
ö um dul
(i)
du
(i)
l wm dt*
Vr = — ■
t*
(5.3)
at r* = 1 - Xu^ and r* = —Xu^.
Let us consider the asymptotic expansions of the velocity components and pressure in small
parameters of the problem, - <C 1 and — C 1. For the first terms of these expansions, that
l R2
is, T = 0 and = 0 (the hydrodynamic lubrication theory with the corresponding Reynolds l R
S
number y-jr < 1), we obtain hydrodynamics equations in the form
dP ~
—— = 0, Re
d)r *
vx vx vx
+ A vx —— + vr —— dt* \ dx* dr*
+
dP
x*
d2vx
r
2
vr vx r* x*
0, Re
and boundary conditions: vr = —
du
(i)
ö öc0
vx = 0 for r* = 1 — Xu^ and r* = —Xu^'. By expanding
(5.4)
(2)
dt*
pressure and velocity components in powers of a small parameter X
P = P0 + XP1 + ..., Vx = vX + Xvlx + .. we obtain equations for the first terms of the expansion
vx = v°r + Xv1 +
dP°
dr *
and boundary conditions
0, Re
dvX dP0
|2„,0
t*
+
x*
d2v, r
vr0 v
2
r*
+
_x_
x*
0 du^ v =--—
dt*
0 duf^ v =--—
dt*
vx = 0, where r* = 1, v°x = 0, where r* = 0.
(5.5)
(5.6)
(5.7)
Up to X we obtain
0
( i) wmco dvl
r2* = -Xuii) , r* = 1 - Xu^
(5.8)
pVCplWrn p0 ~ ¿3 ^ ■
ö
l
2
■I
0
■
We use the iteration method as in [18]. At the first step of the iteration we set Re = 0 (creeping fluid flows, hydrodynamic theory of lubrication) [23]. Taking the boundary conditions into account, we obtain from the equations of fluid motion
Pö = 12
du
(2)
du
(1)'
dt*
dt*
dx*
dx*
(d2 u{? d2u31)\
dt
*2
dt
*2
dx*. (5.9)
dv0
Substituting the found values into the equations of fluid dynamics (5.6), at the second step of the iteration we find
P 0
12
du
(2)
3
du
(1)
dv0
= "
dt* 6
dt*
5 I dt*2 a+*2
dt*2 j
dx dx ,
fd^ul
\dt*2 dt*2 J^-"™1
10
dt
-3-
dt*2
dx*
(5.10)
Taking into account the introduced variables £ = x* — t* \/l — Ho and r = et* at t <C 1, we
find
P0
«30 "M30 --Re(l-^)
du
(1)
30
du
(2)'
30
de de
dx*
dP°
12 V1 " Vo (4o " 4?) " (1 - ß2)
du
(1) 30
du
(2) 30
de de
dv0
= (2r* - 1) dr* v '
du
(1) 30
dv°x
dr *
1 = (4o} "4o ) - f^ (! -ßo)
dv g dr *
r* =0
dv g dr *
r*=1
du
(1) 30
du
(2) 30
de de
du
(2) 30
de de
(5.11)
The equations of system (3.8) include the expressions
(i) R(i dqn , Therefore, we obtain from (5.12) the following expressions:
R{1) dqn R{1)puc0lwm r ~
—äT = ——W1"^
du
(1) 30
du
(2) 30
de de
(2) ^ R{2) dqn
QÏ 1 + ßo t
Vo-
R(2 pvc0lwm
ö3
"12^1 ~ß2o
du
(2) 30
du
(1) 30
de de
u(1) u(2) u30 - u
30
1
1
ö
12 ß0R(1)
(2) _ (1) u30 - u
30
l-1--*-2 ß0RW
1ö
1
2 ß0R(1)
1
12 ß0R(1)
(5.12)
(5.13)
3
3
:
r* =
l
ö
1
Let us find the expression on the right-hand sides of the system of equations (3.8). Considering that wmlu30 = I0umR(i)uji0g and that R(1) = R(2) = R, h(1) = h(2) = h0 due to the smallness of A for the first equation of the system (3.8) (i = 1), we obtain
Pl
IV
p0h0 Rc0s V 5
2 (2)
2u
2„,(1)'
10
2u
10
dt
dt
1
15
2 ioR
2 (2)
2u
30
i2ji)'
2u
30
t2
t2
1
1__S_
12 i0R
(5.14)
For the second equation of this system (i = 2) we obtain
Pl
IV
p0h0 Rc0s V 5
.2„,(1)
2u
2 (2)
10
2u
10
t
t
2J1)
2u
30
2 (2)
2u
30
1
15
2 IoR
t2
t2
1
1__s_
12 J^R
(5.15)
3
3
6. The equations of coaxial shells dynamics
Taking into account the obtained right-hand sides (5.14), (5.15) of the system of equations (3.8), this system of equations takes the form
cMO m (um\2 d(dr Ee\ I
) 2\jl~nl (ii + I2V0 + /J>il4)
u
(1)' 10
t
d2(
+
l2
s l2
dt4
R pUColWm
-2 <TG I b> ^^
1\J\ — il eump0h04 I 83
u
(1)
30
u
(2) 30
t t
2 IoR, j-1 *
12 I0R
2 (2)
2 (2)
2u
10
+
l2
t4
2^/1 eumpohoc^l° I 63
2t
R pvc0lwm
+
u
(2) 30
u
(1)' 30
t t
1
_1__s_
12~ioR
(6.1)
2
1
2
2
l
1
x
2
s
It is possible to introduce the notation u^ = C3¿(1), u^ = c34>(2), n = C1 £, t = c2t, where
(2)
C2
P. 2 Pl
pohx£ \ ô
R
1
1ô
2 i0R J ôc0
Cl
C2£
R
V1 -Mo
li-
es
C2 E£ ( l
a m \UmJ 2-y/l - n20 (in + H2H0 + mul)
(6.2)
Taking
ai = 6^o" Pl
R\2 6 1 y/l - 1%
p0h0 \ ô J l £ 10
1
12i0 R
we obtain the system of equations
(6.3)
^ + 6^ ^ + + ¿(1) — ¿(2) — «71 — 0 ) = 0,
# + 6^(2) ¿(2) + + ^(2) — ^(1) — a1 (^2) — ¿M ) = 0.
(6.4)
The system of equations has the following exact solution:
¿(1) = ¿(2) = ±k ch-1 (kn + 4kst). (6.5)
Due to the presence of fluid between the shells, a numerical solution of the system of equations under the initial condition is required:
¿(1) = k ch"1 (kn), ¿(2) =0.
(6.6)
7. Computational experiment
We consider a difference scheme for the equations which is similar to the Crank-Nicolson scheme for the heat equation [27-32] for numerical modeling
, (1) 3n+1 (1) 3n+1\ u(i) ^—uw n j — uU ] _2-1— 2
(1)
3 n
(1)
3n
j-1 ) + ( u j+1 — uj-1
T
.(1) n+1 2u(1) n+1 + 2u(1) n+1 u(1) n+^ j+2 — 2uj+1 +2uj-1 — uj-2
+ ~-777-- +
4h
u(D n uj+2
+
—jn+jn—
4h3
4h3
+
(1)ni1 (1)n (2)ni1 (2)n uj - uj uj - uj
H------~--ci
((y^r1 — y^r1^ + (u(i1n — u(V n
2
2
(2) n+1
"j+1 uj-1 I + \uj+1 uj-1
4h
(2)
n+1
uj+1 uj-1
+ ( uj+1 uj-1
(2) n)
4h
= 0,
/
(7.1)
2
2
l
2
v
2
1
6
ô
u
(2)
n+1
— u
(2)n
T
+ 2
(2) Uj+1
3n+1
u
(2) j -1
33n+1
+i u^:n - u^:n
+
(2) n+1 0 (2) n+1 . 0 (2) n+1 (2) n+1" Uj+2 - 2Uj+1 + 2Uj-1 - Uj-2
4h
\ ( (2) n 0 (2) n . 0 (2) n (2) n\ J | (4+2 ~2n)+l +24-l ~Uj-2 )
+
4h3
U
(2)
n+l
+ -
(2)n (1)n+1 (l)n - Uj Uj - Uj
/ / (2) n+\- (2) n+\-
2
(l) n+l
(l)
n+l
Uj+1
- Uj-1
4h3
+ (U(2) n_ U(2) n + \ Uj+1 Uj-1
+
4h
Uj+1 Uj-1
+ ( Uj+1 Uj-1
4h
(1) n
)
0.
The step in the dimensionless coordinate is h = 0.2, and the step in dimensionless time t is chosen to be 0.5h. The obtained approximation is coordinated with the system of equations (6.4). As the differential scheme coordinated with it is not overt, it converges to the solution of the above-mentioned system of equations (6.4) with initial condition (6.6). This differential scheme possesses stability and the second order of accuracy in the coordinate and in time.
Fig. 2. Influence of the absence of the inertia of the fluid motion = 0).
2
Fig. 3. Influence of the inertia of the fluid motion (<1 = 1).
8. Results and conclusions
The presence of a deformation wave in the outer shell led to the appearance of a deformation wave in the inner shell, which was not present at the initial instant. The wave velocity decreases. According to Fig. 2, the effect of viscous fluid stress on the shells leads to a decrease in the deformation wave amplitude in the first shell, and in the second one the deformation wave amplitude grows until the average value of the amplitudes in both shells is achieved, the energy transfer taking place through the fluid layer. As a result, energy volumes in both shells become equal. The deformation wave velocity is subsonic. According to Fig. 3, the wave velocity decreases (the graph shifts to the left), due to the influence of the inertia of the fluid motion
(<71 = 1).
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