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Nonlinear rupture of thin free liquid sheets
174
Communications
Nonlinear Sheets 2
in Nonlinear
Science & Numerical
Rupture
of Thin
Simulation
Vo1.4, No.3 (Sep. 1999)
Free Liquid
Zhi LI’ and N. R. SIBGATULLIN** *(Department of Mechanics & Engineering Sciences, Peking University, Beijing 100871, China) **(Department of Mechanics & Mathematics, Moscow State University, Moscow 119899, Russia) Abstract: which has amplitude harmonic numerical given. Keywords:
We study nonlinear effects on wave propagation in a thin free liquid sheet finite midline deviation. A system of four equations for nonlinear waves of finite in liquid sheets is obtained in long-wave approximation. Waves generated by oscillations of the nozzle from which the sheet is issued are investigated by the simulation. The solutions demonstrating nonlinear rupture of the sheet are thin free liquid
sheet
Introduction Free liquid sheet, or plane liquid jet, is a continuous layer of fluid bounded by two interfaces between liquid and gas or liquid and two other liquids, and it moves almost in a constant direction of space on a distance of many cross-sectional sizes of its own. The theory of motion of a liquid jet under the action of surface tension was first studied by lord Rayleigh from his theoretical and experimental researches and now great progresses have been made in applications in connection with chemical and biomedical engineering (see, for example, reviews [l, 21). The instability and rupture of free liquid sheets in air was investigated for the first time by Squire131, Hagerty & Shea 141. The linear stability analysis with respect to infinitesimal perturbations shows the existence of two modes of waves in the sheet: the sinuous (bending, stretching, antisymmetric) mode and the varicose (squeezing, dilational, symmetric) mode, both of which can be detected experimentally 15t61. The effects of liquid viscosity on liquid sheet instability were considered by many authors[7-1’l. Up to now, much attention has been devoted to the numerical simulation of rupture of free liquid sheet with respect to varicose perturbations. In papers [12-161 (see also references in [IS]), the varicose waves in the sheet were described by nonlinear evolution equations where the influences of liquid viscosity, the van der Waals forces, surfactants and other factors on dynamics of the sheet were taken into account. The mentioned researches concerned the case, on which the medline of the sheet has zero or infinitesimal deviation from the undisturbed position. In paper [17], a method permitting to take into account finite medline deviation from the undisturbed position was developed, and a Hamilton nonlinear system was constructed to describe asymptotically the deformation of free liquid sheets without taking account of the dispersing terms. In this paper, we developed the method of paper [17] to study the sinuous waves of finite amplitude in liquid sheet and made a numerical simulation of waves generated by harmonic oscillations of the nozzle from which the sheet is issued. 2The paper was received on Sep.16, 1999
LI
No.3
1 Formulation
et
175
al.: Nonlinear Rupture of . . .
of the Problem
We consider a plane free liquid sheet issuing from a long plane nozzle in still air. The fluid is assumed to be inviscid, incompressible and has a density p, a coefficient of surface tension T. We study the sheet motion under the action of surface tension, neglecting the gravity and the influence of air on dynamics of the sheet. The undisturbed sheet with a thickness 2he (coincides with the thickness of the nozzle) flows with a constant velocity U. If the nozzle moves in the direction perpendicular to the undisturbed sheet, the plane sinuous disturbances are formed in the sheet. To describe the two-dimensional sheet motion, we introduce a mobile Cartesian coordinate system zz with the z axis parallel to the direction of the undisturbed sheet motion and moving with the basic velocity U of the sheet, with the L axis perpendicular to the undisturbed sheet and with z = 0 at the undisturbed midline of the sheet (see Fig. 1).
X
Fig.1
Thin
free
liquid
sheet
issuing
from
the moving
nozzle
With these assumptions, the sheet motion is described by the Laplace equation for the perturbed velocity potential cp(z, z, t) and the four boundary conditions on the free surfaces z = h*(x, t): cpsz+ cpzz= 0, h- < z < h+ (1)
T&z,
~(1 + h:,)3/2
’
z = h*
(2)
z = hi h&t = ~3, - v&z, (3) Eqs.(S) and (3) are the dynamic condition (the Cauchy-Lagrange integral) and the kinematic condition on the free surfaces, respectively. We consider the boundary conditions on the outlet of the nozzle. First of all the thickness of the sheet on the outlet coincides with the thickness of the nozzle. Let z = Z(t) at x = -Ut be the motion law of the nozzle in the mobile coordinate system and the characteristic velocity of the nozzle motion along the z axis be much smaller than the basic velocity of the sheet, that is, 2 - EU, where Z(t) = dZ/dt, E is a small parameter. In this case, we can simply think that the perturbed velocity of the fluid on the outlet of the nozzle coincides with the velocity of motion of the nozzle along the z axis. Thus, the idealized boundary conditions on the outlet of the nozzle are h* = &ho + Z(t),
cp = zi(t),
x = -Ut
(4
At small value E the waves in the sheet are long, therefore, we have the following evaluations: X-&/E, ZNZO, h+-20, p,-EU, cpze2U where 2s is the characteristic
scale along the .z axis.
176
Communications in Nonlinear Science& Numerical Simulation
2 The Evolution
System
In long-wave approximation power series p = z
(--;;;’
Vo1.4,No.3 (Sep. 1999)
of Equations a solution of the Laplace Eq.(l)
2
+ z
‘T:‘;;1 n
.
2,
will have the form of a
he < .z < h,
(5)
where functions f = f(z, t), w = w(z, t) have continuous all order derivatives. When the surface z = 0 is inside the sheet, the functions f and w may be interpreted as the perturbed potential and velocity of the fluid at z = 0 respectively, that is, f = cp(z, z = 0, t), w = &(Z,Z = o,t>. Denote the function cp at the free surfaces z = h& by @k respectively, that is, @& = cp(z, z = h*(z,t), t). The formula (5) allows to express the function cp through functions @k, hh and their derivatives. In fact, from formula (5) it follows that ‘p = f(z,t) From the definition
+ zw(z,t)
- iz2fxz
+ O(E5ZOU)
(6)
of 9& we have 9* = f + h&w - ;h:f,,
hence it is possible to asymptotically f = p,
1 - gz3w,,
+ $(h+ + h-)f,,
In view of the definition
+ O(E~ZOU)
get
+ ‘p-) - i(h+ + h-p:;:
w = ;;I;-
- ;h$w,,
+ O(e3Z~U)
+ ;(h:
+ h+h-
+ I$)( “,;I;r),,
+ O(E~U)
(7)
of the functions @h Eqs.(2) and (3) have the form
Substituting (6) and (7) into the last two equations, we will have an evolution system of four equations for the functions 9+ and hh. We introduce now four new required variables: 2h=h+-h-,
2Y=h++h-,
2x=9+-$-,
29=@++9-
(8)
Obviously, Y has the meaning of elevation of the midline and 2h is the approximate thickness of the sheet (the true thickness of the sheet may be defined by 2h/dm). Finally we will have the following evolution system hi + (h@,, - xYz)z = O(E~U) yt - $j(l+ Y," + h;) + @,Y, + xzh, + $h”(& Xt + (%x @t - j&l+
- qq
- g (
Y,” + h;) + $‘; -3m
= O(E~U) >
= 0(&W)
(9)
+ xl) + ghx(& - m>
= OwJ2)
In system (9) the force of surface tension (dispersing terms) is taken into account in its exact formulation.
No.3
LI et al.: Nonlinear
177
Rupture of..
It should be noted that system (9) may be obtained by the Hamilton formalism, in other words, the boundary conditions (2) and (3) in the new variables (8) may be written a&l71 yt+,
p-g,
ht+&
at=-g
4G = .I-( (a Irzh+ - cpzIzEh-) (W + Y,” + h:) + 2xYA) +((PzIzzh+ + (Pr(,,J (x(1 + y, + h2,) + 2wJ44~dXX, + -5) -2Y,(x9), + $ (Jl + (Yz + h,)2 + ,/l + (Yz - h,)2 - 2))dz Substituting
(6) and (7) in the Hamiltonian
(10)
G, after simple evaluations we see that
2G = J&(1 + Y, + h2,) + h(Q2, +x2 + xx,) - 2Y,+,x - +“x(f),, +$(,/l+ (Yz + h,)2 + dl+ (Yz - h,)2 - 2) + O(8ZeU2))dz Computing the variation of G, we again obtain Eqs.(S) from (10). The boundary conditions (4) in variables h, Y, x, + have the form h = ho,
3 Numerical
Y = Z(t),
x = hoi(t),
ip = Z(t)i(t),
z = -Ut
(11)
Simulation
We assume that up to the instant t = 0 the nozzle is in rest, and the sheet issuing from the nozzle has the velocity U and the constant thickness 2ho. From the instant t = 0, the nozzle oscillates along the z axis under the harmonic law with a frequency w and a amplitude z(): 2 = z(t) = Zs(1 - coswt) We study waves generated by the oscillation of the nozzle with the help of Eq.(9). Obviously, the small parameter E should be defined by E = Zcw/U. Let us consider now the fixed coordinate system z = x + Ut and the dimensionless variable: 5 = Ux’fw, t = t/w, z = hoz’, Z = h,,Z’ h = h,,h’, Y = h,,Y’, x = h;wx’, ip = h;w@’ System (9) has the dimensionless form (the primes at the dimensionless variables are omitted) ht + h, + ,u(hia, - xYz>z = 0 Yi + Y, + p(+,Yz + x&z) - fj(l + py,” + ph:) + $h2(&
xt -t
xz + P(%X - +),
- We
c&$&T -&l+K:+&)
= 0
+ &$?i?tF>
= O
(12)
cP,+~,+~(~~+x22)+~hx(%),
r qe
-we
- ~l+%z)2)3)
p = E2hi/Zi, We = T/2hopU2 is the Weber number. in the fixed system in the dimensionless form are
The boundary
where
h = 1,
Y = Z(t),
where Q = Zo/ho.
x = i(t),
Q = Z(t)+),
Z(t) = a(1 - cost),
=O conditions
x = 0
(11)
(13)
The initial conditions may be written as h=l,
Y=O,
x=9=0,
t=O
(14)
178
Communications
in Nonlinear
Science & Numerical
Simulation
Vo1.4, No.3 (Sep. 1999)
To carry on our numerical simulation we use the well-known MacCormack [IS]). The general idea of the MacCormack scheme for the equation
with the boundary
scheme (see
and the initial conditions
21(x,0) = u”(x)
40, t) = uo(t),
is as follows. Let the computational grid be rectilinear, At and Ax are the mesh widths used for t and x respectively. According to MacCormack method, as the first step (predictor) we find the approximate value Gy’l of the function u on the (n + 1)-th step of time: At ;il?+l = $? - -+y+l 3 Ax
_ .jn) _ At +y,
and as the second step (corrector) the time:
The boundary
u?+;,
uj”, ujn+l -zxt
+ “y-1)
we compute the final value u on the (n -t 1)-th step of
and the initial conditions in discrete form are --n+1 UO
= g+l 0
-
uo((71.
+
l)W,
u!j = u’(jAz)
At the step predictor the derivative u, is approximated by the differences forward, and at the step corrector u, is approximated by the differences back. The other version of the MacCormack scheme is also possible using the differences back on the step predictor and the differences forward at the step corrector. To eliminate any misalignment stipulated by the discretization by one-sided differences we use alternatively the differences forward and back at the steps predictor-corrector. Both versions of the MacCormack scheme have the second order approximation. We can not find the exact stability condition of the MacCormack method for Eq.(12) in connection with the large complexity. The particular computations showed that the numerical scheme is stable when At is small enough with respect to Ax (in our computations, At = 0.001-0.005, Ax = 0.0341). The numerical simulation was conducted with parameters: We = 0.1-10, E = 0.1-0.3 and Q = l-20. The following effects were found out. At first, the nozzle motion results in changes of the thickness of the sheet and the formation of complex waves in the sheet. The sheet segments are formed with a large thickness in the middle and a small thickness on the edges. With increase of time the thickness in some place may go to zero, that means the rupture of the sheet (see Fig. 2). The sheet does not rupture at a rather large Weber number, a small oscillation frequency of the nozzle or a rather small parameter E. In these cases, waves arising in the sheet can spread far away from the nozzle (see Fig. 3). Secondly, in the fixed coordinate system the wavefront set is spread with a velocity (1 + &%%)U, that coincides with the velocity of the sinuous wave in free liquid sheet. The waves arising in the sheet can be considered as the outcome of interaction of two modes:
LI et al.: Nonlinear
No.3
179
Rupture of . . .
the sinuous disturbances gradually detach from the varicose ones and extend forward, and the varicose disturbances extend much slower and disperse.
t
15
10
= 18.46
10
Z
Z 5
5 +
0
+ 0
0
5
10
15
20
X
25
30
35
0
5
10
15
20
25
30
X
Fig.2 Nonlinear rupture of liquid sheet. The free surfaces and the midline of the sheet are represented. The parameters of computation are: (I = 4, E = 0.2, We = 0.2
6
Fig.3 The sinuous wave in the liquid sheet. The free surfaces and the midline of the sheet at t = lO?r are represented. The parameters of computation are: Q = 2, E = 0.2, We = 0.2
35
180
Communications
in Nonlinear
Science & Numerical
Vo1.4, No.3 (Sep. 1999)
Simulation
The waveform near to the wavefront can be computed by the linear theory. In fact, it is easy to check up that the system of Eq.(12) at p = 0 has the exact solution h=l,
Y=g(c),
x=-mg’(t),
@=-;v%%/(g’(E))2d<
where < = z - (1 -t d%&)t and g is any continuously (~1(1 - cos c), (~1 = const, then h=l,
Y=or(l-cost),
X=or*sin<,
differentiable
function.
4 +=8VBG(sin2E-2E)
Let g(E) =
(15)
Thus, the wavefront described by the problem (12)-(14) has form (15) (< > 0, or > o) within constant phase. This fact was confirmed by the numerical simulation. References [l] Entov, V. M. and Yarin, A. L., Dynamics of free jets and films of viscous and rheologically complex liquids, - In: “Advances in Mechanics”, VINITI, Mekhanika Zidkosti i Gaza, 1984, 18: 112-197 (in Russian) [2] Oron, A., Davis, S. H. and Bankoff, S. G., Long-scale evolution of thin liquid films, Rev. Mod. Phys., 1997, 69(3): 931-980 [3] Squire, H. B., Investigation of the instability of a moving liquid film, Br. J. Appl. Phys., 1953, 4: 167-169 [4] Hagerty, W. W. and Shea, J. F., A study of the stability of plane fluid sheets, J. Appl. Mech., 1955, 22(3): 509-514 [5] Taylor, G. I., The dynamics of thin sheets of fluid. II, Waves on fluid sheets, Proc. Roy. Sot. A, 1959, 253(1274): 296-312 [6] Joosten, J. G. H., Spectral analysis of light scattered by liquid films. I, General considerations, J. Chem. Phys., 1984, 80(6): 2363-2381 [7] Dombrowski, N. and Johns, W. R., The aerodynamic instability and disintegration of viscous liquid sheets, Chem. Eng. Sci., 1963, 18(9): 203-214 [8] Lin, S. P., Lian, Z. W. and Creighton, B. J., Absolute and convective instability of a liquid sheet, J. Fluid Mech., 1990, 220: 673-689 [9] Li, X. and Tankin, R. S., On the temporal instability of a two-dimensional viscous liquid sheet, J. Fluid Mech., 1991, 226: 425-443 [lo] Li, X., Spatial instability of plane liquid sheets, Chem. Eng. Sci., 1993, 48: 2973-2981 [ll] Li, X., On the instability of plane liquid sheets in two gas streams of unequal velocities, Acta Mech., 1994, 106: 137-156 [12] Prevost, M. and GaIlez, D., Nonlinear rupture of thin free liquid films, J. Chem. Phys., 1986, 84(7): 4043-4048 [13] Sharma, A. and Ruckenstein, E., Finite-amplitude instability of thin free and wetting films: Prediction of lifetimes, Langmuir, 1986, 2(4): 480-494 [14] Erneux, T. and Davis, S. H., Nonlinear rupture of free films, Phys. Fluids. A., 1993, 5(5): 1117-1122 [15] De Wit, A., GaIlez, D. and Christov, C. I., Nonlinear evolution equations for thin liquid films with insoluble surfactants, Phys. Fluids, 1994, 6(10): 3256-3266 [16] Sharma, A., Kishore, C. S., SalanwaI, S. and Ruckenstein, E., Nonlinear stability and rupture of ultrathin free films, Phys. Fluids, 1995, 7(8): 1832-1840 [17] Sibgatullin, N. R. and Sibgatullina, A. N., Instability of capillary waves in free thin films, Moscow University Mechanics Bulletin, 1997, 6: 13-17 [18] Anderson, D. A., Tannehill, J. C. and Pletcher, R. H., Computational fluid mechanics and heat transfer, Hemisphere Publishing Corp., N. Y., 1984
Communications
Nonlinear Sheets 2
in Nonlinear
Science & Numerical
Rupture
of Thin
Simulation
Vo1.4, No.3 (Sep. 1999)
Free Liquid
Zhi LI’ and N. R. SIBGATULLIN** *(Department of Mechanics & Engineering Sciences, Peking University, Beijing 100871, China) **(Department of Mechanics & Mathematics, Moscow State University, Moscow 119899, Russia) Abstract: which has amplitude harmonic numerical given. Keywords:
We study nonlinear effects on wave propagation in a thin free liquid sheet finite midline deviation. A system of four equations for nonlinear waves of finite in liquid sheets is obtained in long-wave approximation. Waves generated by oscillations of the nozzle from which the sheet is issued are investigated by the simulation. The solutions demonstrating nonlinear rupture of the sheet are thin free liquid
sheet
Introduction Free liquid sheet, or plane liquid jet, is a continuous layer of fluid bounded by two interfaces between liquid and gas or liquid and two other liquids, and it moves almost in a constant direction of space on a distance of many cross-sectional sizes of its own. The theory of motion of a liquid jet under the action of surface tension was first studied by lord Rayleigh from his theoretical and experimental researches and now great progresses have been made in applications in connection with chemical and biomedical engineering (see, for example, reviews [l, 21). The instability and rupture of free liquid sheets in air was investigated for the first time by Squire131, Hagerty & Shea 141. The linear stability analysis with respect to infinitesimal perturbations shows the existence of two modes of waves in the sheet: the sinuous (bending, stretching, antisymmetric) mode and the varicose (squeezing, dilational, symmetric) mode, both of which can be detected experimentally 15t61. The effects of liquid viscosity on liquid sheet instability were considered by many authors[7-1’l. Up to now, much attention has been devoted to the numerical simulation of rupture of free liquid sheet with respect to varicose perturbations. In papers [12-161 (see also references in [IS]), the varicose waves in the sheet were described by nonlinear evolution equations where the influences of liquid viscosity, the van der Waals forces, surfactants and other factors on dynamics of the sheet were taken into account. The mentioned researches concerned the case, on which the medline of the sheet has zero or infinitesimal deviation from the undisturbed position. In paper [17], a method permitting to take into account finite medline deviation from the undisturbed position was developed, and a Hamilton nonlinear system was constructed to describe asymptotically the deformation of free liquid sheets without taking account of the dispersing terms. In this paper, we developed the method of paper [17] to study the sinuous waves of finite amplitude in liquid sheet and made a numerical simulation of waves generated by harmonic oscillations of the nozzle from which the sheet is issued. 2The paper was received on Sep.16, 1999
LI
No.3
1 Formulation
et
175
al.: Nonlinear Rupture of . . .
of the Problem
We consider a plane free liquid sheet issuing from a long plane nozzle in still air. The fluid is assumed to be inviscid, incompressible and has a density p, a coefficient of surface tension T. We study the sheet motion under the action of surface tension, neglecting the gravity and the influence of air on dynamics of the sheet. The undisturbed sheet with a thickness 2he (coincides with the thickness of the nozzle) flows with a constant velocity U. If the nozzle moves in the direction perpendicular to the undisturbed sheet, the plane sinuous disturbances are formed in the sheet. To describe the two-dimensional sheet motion, we introduce a mobile Cartesian coordinate system zz with the z axis parallel to the direction of the undisturbed sheet motion and moving with the basic velocity U of the sheet, with the L axis perpendicular to the undisturbed sheet and with z = 0 at the undisturbed midline of the sheet (see Fig. 1).
X
Fig.1
Thin
free
liquid
sheet
issuing
from
the moving
nozzle
With these assumptions, the sheet motion is described by the Laplace equation for the perturbed velocity potential cp(z, z, t) and the four boundary conditions on the free surfaces z = h*(x, t): cpsz+ cpzz= 0, h- < z < h+ (1)
T&z,
~(1 + h:,)3/2
’
z = h*
(2)
z = hi h&t = ~3, - v&z, (3) Eqs.(S) and (3) are the dynamic condition (the Cauchy-Lagrange integral) and the kinematic condition on the free surfaces, respectively. We consider the boundary conditions on the outlet of the nozzle. First of all the thickness of the sheet on the outlet coincides with the thickness of the nozzle. Let z = Z(t) at x = -Ut be the motion law of the nozzle in the mobile coordinate system and the characteristic velocity of the nozzle motion along the z axis be much smaller than the basic velocity of the sheet, that is, 2 - EU, where Z(t) = dZ/dt, E is a small parameter. In this case, we can simply think that the perturbed velocity of the fluid on the outlet of the nozzle coincides with the velocity of motion of the nozzle along the z axis. Thus, the idealized boundary conditions on the outlet of the nozzle are h* = &ho + Z(t),
cp = zi(t),
x = -Ut
(4
At small value E the waves in the sheet are long, therefore, we have the following evaluations: X-&/E, ZNZO, h+-20, p,-EU, cpze2U where 2s is the characteristic
scale along the .z axis.
176
Communications in Nonlinear Science& Numerical Simulation
2 The Evolution
System
In long-wave approximation power series p = z
(--;;;’
Vo1.4,No.3 (Sep. 1999)
of Equations a solution of the Laplace Eq.(l)
2
+ z
‘T:‘;;1 n
.
2,
will have the form of a
he < .z < h,
(5)
where functions f = f(z, t), w = w(z, t) have continuous all order derivatives. When the surface z = 0 is inside the sheet, the functions f and w may be interpreted as the perturbed potential and velocity of the fluid at z = 0 respectively, that is, f = cp(z, z = 0, t), w = &(Z,Z = o,t>. Denote the function cp at the free surfaces z = h& by @k respectively, that is, @& = cp(z, z = h*(z,t), t). The formula (5) allows to express the function cp through functions @k, hh and their derivatives. In fact, from formula (5) it follows that ‘p = f(z,t) From the definition
+ zw(z,t)
- iz2fxz
+ O(E5ZOU)
(6)
of 9& we have 9* = f + h&w - ;h:f,,
hence it is possible to asymptotically f = p,
1 - gz3w,,
+ $(h+ + h-)f,,
In view of the definition
+ O(E~ZOU)
get
+ ‘p-) - i(h+ + h-p:;:
w = ;;I;-
- ;h$w,,
+ O(e3Z~U)
+ ;(h:
+ h+h-
+ I$)( “,;I;r),,
+ O(E~U)
(7)
of the functions @h Eqs.(2) and (3) have the form
Substituting (6) and (7) into the last two equations, we will have an evolution system of four equations for the functions 9+ and hh. We introduce now four new required variables: 2h=h+-h-,
2Y=h++h-,
2x=9+-$-,
29=@++9-
(8)
Obviously, Y has the meaning of elevation of the midline and 2h is the approximate thickness of the sheet (the true thickness of the sheet may be defined by 2h/dm). Finally we will have the following evolution system hi + (h@,, - xYz)z = O(E~U) yt - $j(l+ Y," + h;) + @,Y, + xzh, + $h”(& Xt + (%x @t - j&l+
- g (
Y,” + h;) + $‘; -3m
= O(E~U) >
= 0(&W)
(9)
+ xl) + ghx(& - m>
= OwJ2)
In system (9) the force of surface tension (dispersing terms) is taken into account in its exact formulation.
No.3
LI et al.: Nonlinear
177
Rupture of..
It should be noted that system (9) may be obtained by the Hamilton formalism, in other words, the boundary conditions (2) and (3) in the new variables (8) may be written a&l71 yt+,
p-g,
ht+&
at=-g
4G = .I-( (a Irzh+ - cpzIzEh-) (W + Y,” + h:) + 2xYA) +((PzIzzh+ + (Pr(,,J (x(1 + y, + h2,) + 2wJ44~dXX, + -5) -2Y,(x9), + $ (Jl + (Yz + h,)2 + ,/l + (Yz - h,)2 - 2))dz Substituting
(6) and (7) in the Hamiltonian
(10)
G, after simple evaluations we see that
2G = J&(1 + Y, + h2,) + h(Q2, +x2 + xx,) - 2Y,+,x - +“x(f),, +$(,/l+ (Yz + h,)2 + dl+ (Yz - h,)2 - 2) + O(8ZeU2))dz Computing the variation of G, we again obtain Eqs.(S) from (10). The boundary conditions (4) in variables h, Y, x, + have the form h = ho,
3 Numerical
Y = Z(t),
x = hoi(t),
ip = Z(t)i(t),
z = -Ut
(11)
Simulation
We assume that up to the instant t = 0 the nozzle is in rest, and the sheet issuing from the nozzle has the velocity U and the constant thickness 2ho. From the instant t = 0, the nozzle oscillates along the z axis under the harmonic law with a frequency w and a amplitude z(): 2 = z(t) = Zs(1 - coswt) We study waves generated by the oscillation of the nozzle with the help of Eq.(9). Obviously, the small parameter E should be defined by E = Zcw/U. Let us consider now the fixed coordinate system z = x + Ut and the dimensionless variable: 5 = Ux’fw, t = t/w, z = hoz’, Z = h,,Z’ h = h,,h’, Y = h,,Y’, x = h;wx’, ip = h;w@’ System (9) has the dimensionless form (the primes at the dimensionless variables are omitted) ht + h, + ,u(hia, - xYz>z = 0 Yi + Y, + p(+,Yz + x&z) - fj(l + py,” + ph:) + $h2(&
xt -t
xz + P(%X - +),
- We
c&$&T -&l+K:+&)
= 0
+ &$?i?tF>
= O
(12)
cP,+~,+~(~~+x22)+~hx(%),
r qe
-we
- ~l+%z)2)3)
p = E2hi/Zi, We = T/2hopU2 is the Weber number. in the fixed system in the dimensionless form are
The boundary
where
h = 1,
Y = Z(t),
where Q = Zo/ho.
x = i(t),
Q = Z(t)+),
Z(t) = a(1 - cost),
=O conditions
x = 0
(11)
(13)
The initial conditions may be written as h=l,
Y=O,
x=9=0,
t=O
(14)
178
Communications
in Nonlinear
Science & Numerical
Simulation
Vo1.4, No.3 (Sep. 1999)
To carry on our numerical simulation we use the well-known MacCormack [IS]). The general idea of the MacCormack scheme for the equation
with the boundary
scheme (see
and the initial conditions
21(x,0) = u”(x)
40, t) = uo(t),
is as follows. Let the computational grid be rectilinear, At and Ax are the mesh widths used for t and x respectively. According to MacCormack method, as the first step (predictor) we find the approximate value Gy’l of the function u on the (n + 1)-th step of time: At ;il?+l = $? - -+y+l 3 Ax
_ .jn) _ At +y,
and as the second step (corrector) the time:
The boundary
u?+;,
uj”, ujn+l -zxt
+ “y-1)
we compute the final value u on the (n -t 1)-th step of
and the initial conditions in discrete form are --n+1 UO
= g+l 0
-
uo((71.
+
l)W,
u!j = u’(jAz)
At the step predictor the derivative u, is approximated by the differences forward, and at the step corrector u, is approximated by the differences back. The other version of the MacCormack scheme is also possible using the differences back on the step predictor and the differences forward at the step corrector. To eliminate any misalignment stipulated by the discretization by one-sided differences we use alternatively the differences forward and back at the steps predictor-corrector. Both versions of the MacCormack scheme have the second order approximation. We can not find the exact stability condition of the MacCormack method for Eq.(12) in connection with the large complexity. The particular computations showed that the numerical scheme is stable when At is small enough with respect to Ax (in our computations, At = 0.001-0.005, Ax = 0.0341). The numerical simulation was conducted with parameters: We = 0.1-10, E = 0.1-0.3 and Q = l-20. The following effects were found out. At first, the nozzle motion results in changes of the thickness of the sheet and the formation of complex waves in the sheet. The sheet segments are formed with a large thickness in the middle and a small thickness on the edges. With increase of time the thickness in some place may go to zero, that means the rupture of the sheet (see Fig. 2). The sheet does not rupture at a rather large Weber number, a small oscillation frequency of the nozzle or a rather small parameter E. In these cases, waves arising in the sheet can spread far away from the nozzle (see Fig. 3). Secondly, in the fixed coordinate system the wavefront set is spread with a velocity (1 + &%%)U, that coincides with the velocity of the sinuous wave in free liquid sheet. The waves arising in the sheet can be considered as the outcome of interaction of two modes:
LI et al.: Nonlinear
No.3
179
Rupture of . . .
the sinuous disturbances gradually detach from the varicose ones and extend forward, and the varicose disturbances extend much slower and disperse.
t
15
10
= 18.46
10
Z
Z 5
5 +
0
+ 0
0
5
10
15
20
X
25
30
35
0
5
10
15
20
25
30
X
Fig.2 Nonlinear rupture of liquid sheet. The free surfaces and the midline of the sheet are represented. The parameters of computation are: (I = 4, E = 0.2, We = 0.2
6
Fig.3 The sinuous wave in the liquid sheet. The free surfaces and the midline of the sheet at t = lO?r are represented. The parameters of computation are: Q = 2, E = 0.2, We = 0.2
35
180
Communications
in Nonlinear
Science & Numerical
Vo1.4, No.3 (Sep. 1999)
Simulation
The waveform near to the wavefront can be computed by the linear theory. In fact, it is easy to check up that the system of Eq.(12) at p = 0 has the exact solution h=l,
Y=g(c),
x=-mg’(t),
@=-;v%%/(g’(E))2d<
where < = z - (1 -t d%&)t and g is any continuously (~1(1 - cos c), (~1 = const, then h=l,
Y=or(l-cost),
X=or*sin<,
differentiable
function.
4 +=8VBG(sin2E-2E)
Let g(E) =
(15)
Thus, the wavefront described by the problem (12)-(14) has form (15) (< > 0, or > o) within constant phase. This fact was confirmed by the numerical simulation. References [l] Entov, V. M. and Yarin, A. L., Dynamics of free jets and films of viscous and rheologically complex liquids, - In: “Advances in Mechanics”, VINITI, Mekhanika Zidkosti i Gaza, 1984, 18: 112-197 (in Russian) [2] Oron, A., Davis, S. H. and Bankoff, S. G., Long-scale evolution of thin liquid films, Rev. Mod. Phys., 1997, 69(3): 931-980 [3] Squire, H. B., Investigation of the instability of a moving liquid film, Br. J. Appl. Phys., 1953, 4: 167-169 [4] Hagerty, W. W. and Shea, J. F., A study of the stability of plane fluid sheets, J. Appl. Mech., 1955, 22(3): 509-514 [5] Taylor, G. I., The dynamics of thin sheets of fluid. II, Waves on fluid sheets, Proc. Roy. Sot. A, 1959, 253(1274): 296-312 [6] Joosten, J. G. H., Spectral analysis of light scattered by liquid films. I, General considerations, J. Chem. Phys., 1984, 80(6): 2363-2381 [7] Dombrowski, N. and Johns, W. R., The aerodynamic instability and disintegration of viscous liquid sheets, Chem. Eng. Sci., 1963, 18(9): 203-214 [8] Lin, S. P., Lian, Z. W. and Creighton, B. J., Absolute and convective instability of a liquid sheet, J. Fluid Mech., 1990, 220: 673-689 [9] Li, X. and Tankin, R. S., On the temporal instability of a two-dimensional viscous liquid sheet, J. Fluid Mech., 1991, 226: 425-443 [lo] Li, X., Spatial instability of plane liquid sheets, Chem. Eng. Sci., 1993, 48: 2973-2981 [ll] Li, X., On the instability of plane liquid sheets in two gas streams of unequal velocities, Acta Mech., 1994, 106: 137-156 [12] Prevost, M. and GaIlez, D., Nonlinear rupture of thin free liquid films, J. Chem. Phys., 1986, 84(7): 4043-4048 [13] Sharma, A. and Ruckenstein, E., Finite-amplitude instability of thin free and wetting films: Prediction of lifetimes, Langmuir, 1986, 2(4): 480-494 [14] Erneux, T. and Davis, S. H., Nonlinear rupture of free films, Phys. Fluids. A., 1993, 5(5): 1117-1122 [15] De Wit, A., GaIlez, D. and Christov, C. I., Nonlinear evolution equations for thin liquid films with insoluble surfactants, Phys. Fluids, 1994, 6(10): 3256-3266 [16] Sharma, A., Kishore, C. S., SalanwaI, S. and Ruckenstein, E., Nonlinear stability and rupture of ultrathin free films, Phys. Fluids, 1995, 7(8): 1832-1840 [17] Sibgatullin, N. R. and Sibgatullina, A. N., Instability of capillary waves in free thin films, Moscow University Mechanics Bulletin, 1997, 6: 13-17 [18] Anderson, D. A., Tannehill, J. C. and Pletcher, R. H., Computational fluid mechanics and heat transfer, Hemisphere Publishing Corp., N. Y., 1984