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Nonlinear waves in two-fluid flows
PHYSICA ELSEVIER
Physica D 123 (1998) 112-122
Nonlinear w a v e s in two-fluid flows * M.E G6z 1 Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA
Abstract Typically, the interaction of two miscible fluids leads to the formation of segregation patterns in the form of plane concentration waves and two- or three-dimensional voids or clusters. The development of small-amplitude plane travelling waves and the subsequent onset of higher-dimensional instabilities can be elucidated analytically; in addition, numerical methods allow the computation of higher-amplitude solutions. As both direct approaches become costly, although at different amplitude levels, one is tempted to look for a simpler model problem. Therefore, a nonlinear wave equation has been derived for two-dimensional dispersed two-phase flow in fluidized beds near the stability boundary using the Froude number as small parameter. This equation is a two-dimensional perturbation of the Korteweg--de Vries equation containing all low-order terms. It allows to identify the transition point between voids and clusters, and to investigate these two regions for solitary and periodic waves. A Ginzburg-Landau equation is derived to describe transverse perturbations of one-dimensional wavetrains bifurcating at the stability boundary. Copyright © 1998 Elsevier Science B.V. Keywords: Multiphase flows; Fluidized beds; Extended Korteweg-de Vries equation; Pattern formation; Solitary waves
1. Introduction We consider equations describing the interaction of two miscible fluids as they are used for dispersed two-phase flows like g a s - s o l i d and liquid-solid mixtures in fluidized beds [1,2] or the rise of small gas bubbles in liquid columns [3]. Typical patterns observed in such systems are plane voidage or concentration waves, and two- or three-dimensional travelling wave patterns such as plumes of fluid ('bubbles', voids) rising in a dense fluidized bed [4-7], or clusters of particles moving through dilute fluidized beds [4]. The terms dense and dilute refer to the average volumetric packing, respectively, distribution of particles. As generic behaviour one can perceive a sequence of instabilities leading from a uniform state to plane wavetrains [8] to two-dimensional travelling waves [9-11] in the form of bubbles or clusters to more complex spatio-temporal patterns [12]. There exist some characteristic differences between gas- and liquid-fluidized beds concerning the formation and stability of bubbles [4-7,10,13], which have been known experimentally since long to be related to * Supported by the Deutsche Forschungsgemeinschaft. 1 Present address: Martin-Luther-Universit~itHalle-Wittenberg, FB Verfahrenstechnik, MVT/UST, D-06099 Halle, Germany. Tel.: +49 3461 462884; fax: + 49 3461 462878; e-mail: [email protected]. 0167-2789/98/$19.00 Copyright © 1998 Elsevier Science B.V. All rights reserved PII S01 67-27 89(98)001 16-X
M.E GOz/Physica D 123 (1998) 112-122
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the magnitude of the Froude number [14]. This relationship is still not fully understood, but recent numerical and analytical studies of standard modelling equations have brought forth the importance of the Froude number and how its value determines the growth rates of primary and secondary instabilities in fluidized beds [ 13,15]. As the full system of equations is rather unhandy, it is reasonable to use the Froude number as perturbation parameter and reduce the bulky system of coupled Navier-Stokes equations to a single wave equation for the voidage perturbation near the stability boundary [16] (see also [17], in which an inconsistent scaling is used, and [ 18], where the Froude number is not employed as parameter so that a similar but simpler equation is obtained which is less suited to describe the bifurcation behaviour; cf. the discussion in [16]). The result is a quite general nonlinear wave equation, namely a two-dimensionally perturbed Korteweg-de Vries equation that is complete in the low-order terms; its one-dimensional version comprises various well-known wave equations like the BurgersKdV and Kuramoto-Sivashinsky equations, the Benney or Kawahara equation, as well as the so-called modified Kawahara equation, all of which appear in a wealth of physical problems (cf. e.g. [19]). Here we shall focus on fluidized beds, but expect the analysis to be applicable to many other single- and multiphase systems like traffic flow or polymer-solvent mixtures due to the similarity in the governing equations and the completeness of the derived 1|onlinear wave equation. After presenting the basic equations and parameters we will briefly describe the stability and scaling analysis that leads to the reduced wave equation. We will take a look at its general properties and the more important coefficients, and then study the instability of the trivial state which gives rise to periodic and solitary waves. We identify the transition point between voids and clusters, which we associate with a switch from dense to dilute beds; this may also be viewed as a change in one phase dominating the other. Regarding plane solitary waves, it turns out that perturbations of KdV-solitons blow up in finite time in dense and very dilute fluidized beds. This is an artifact of the reduced equation, the original system admits homoclinic solutions. At the other end of the unstable spectrum, a branch of one-dimensional periodic travelling waves bifurcates subcritically from the uniform state of dense and possibly very dilute beds. In the model problem, this branch does not turn around to stabilize at moderately high amplitude, in contrast to numerical results for the original equations [ 11,13]. Two-dimensional solutions are also found to be unstable in this case. For dilute particle suspensions we have propagating particle clusters instead of voidage waves; finite-amplitude solitary waves do exist, one-dimensional periodic solutions bifurcate supercritically but may turn unstable at higher amplitude and are transversally unstable.
2. Basic equations and parameters The original two-fluid equations are [ 16] -Otck + V • [(1
- ~b)v] = 0
Ot4~ + V • (4m) = 0
(dispersed phase),
(continuous phase),
F(1 - ~b)(0tv + v . Vv) = - ( 1 - q~)k + B(~b)(u - v) - (1 - ( b ) V p - F G ( f b ) V ( b + # A v , F~ga(Oru
+ u . Vu) = - ~ b k - B(q~)(u - v) - ~bVp + v # A u ,
(1) (2) (3) (4)
with the voidage 4) ( = volume fraction of the continuous phase), the velocities v and u of the dispersed (particulate) and continuous (fluid) phase, respectively, an effective fluid pressure p, the coefficient G (4~) < 0 of the interparticle torce, and the drag coefficient B(~b) = (1
-
~ ) ~ - - ~ o n+l ,
(5)
M.F. GOz/Physica D 123 (1998) 112-122
114
with the Richardson-Zaki exponent n "- 2 - 4 [20] and the voidage value tP0 of the state of uniform fluidization, whose non-dimensional form is given by ~=~o,
uo=k,
vo=O,
Vpo=-[1
(6)
- ~o(1 - 3)]k.
Here, k represents the unit vector into the x-direction which points against gravity; the transverse coordinate will be denoted by y. The non-dimensional parameters are the Froude number F = u 2 / g r and the Reynolds number R = ruoPd/IXd, with gravity constant g, particle radius r, particle density Pd, particle phase viscosity /Zd, and fluidization velocity u0 = (2/9)(1 - 6)4)~+lgr2pd/lZc which corresponds in non-dimensional form to the relation (1 - 8)tp~ +l = (9/2)v/z characterizing uniform fluidization. For brevity, we have introduced/z = F / R ; in addition, the density and viscosity ratios of the two phases are denoted by 6 = Pc/Pd E [0, 1), and v = /Xc//Zd E [0, 1), respectively.
3. Stability and scaling analysis The stability of the state of uniform fluidization (6) is determined by an eigenvalue cr which expands for small longitudinal wave number ~ as well as small Froude number as (here we set 3 = v = 0 for simplicity) ~r = -i)~d[1 - O ( F ) ] + Fq~o[(d 2 - 092)~2 _ w2k2] _ O ( F 2 ) 4) _ O ( F 2 ) 2k 2, F2k4),
(7)
where k is the transverse wave number,
1 - 4'0
d -- - Bo
[Bo - ~boB0 + (1 - 8)4~0(1 - 2~bo)] -- (n -t- 2)(1 - tPo)
(8)
(Bo -- B(tPo), etc.) the kinematic wave velocity related to bouyancy, and 090 = ~ + O(6) the dynamic wave velocity at which plane periodic travelling waves bifurcate with wave number )2 ..~ (d 2 _ 092)/F. Obviously, a (long-wave) instability can occur only if d > wo (a complete analysis also shows that 090 > O(v) > 0 has to be satisfied, but this is irrelevant for the small Froude number asymptotics). In order to balance leading-order terms and include transverse contributions as well, we assume that the strength of the instability scales with F << 1 in the form of d - wo "~ F '~, hence the wave numbers must scale as )2 ~ F,~- l, k 2 ~ F2U- I. The appropriate choice of coordinates and scalings is given by d - wo ~ F such that )~0 ~ O(1), thus ot = 1 and 490(d 2 - 092) + 0 ( 3 ) = coF,
X = x - dr,
Y = FU2y,
T = Ft,
(9)
where co is an O(1) quantity, which is negative or positive depending on whether the base state is stable or unstable, respectively. Then we employ the perturbation 49 = dpo + Fdp,
p = poX + F~,
Vx = F v x ,
Ux = 1 + Ffix,
(1)y, Uy) = F3/2(1)y, fly).
(10)
4. The reduced equation and s o m e of its properties Using the leading-order relations between the perturbation variables allows us to express the pressure and velocity perturbations in terms o f q~ plus O ( F ) contributions, such that these other variables can be pushed to higher and higher orders in the Froude number. This gives the following equation:
q~T "q- al (~2)X -I- a2~PXXX + F[co(bxx -- bl(byy --1-a2~PYyx - 2b2dPXT -- b3(bxxz + b4(dPqbxx)x + (b5 - 2alb2)(~p2)xx + (b6 - a l b 3 ) ( ( ) 2 ) x x x
+ b7(q~3)x] = O ( F 2 ) ,
(ll)
M . E GOz/Physica D 123 (1998) 112-122
115
where the constant coefficients depend on 4'0 and the physical parameters in a complicated manner [ 16]. We note in particular that (1 -
24'o)d
- (1 4'0(1 - 4'0)
aj = -
n+2
24,0
B~ ~ (d + 4'0 - 1)
¢0) 2
4'o(1 - 4'0) (B~'/2 + 1 Bo
8)
In ÷ 1 - (n ÷ 3)4'0]
(12)
so that the nonlinear term in the KdV-part vanishes at 4'o = (n + 1)/(n + 3); this signals the transition from voids to clusters. It is also important that bl = - ( 1 - 3)4'0G0 > 0, providing stabilizing diffusion in the transverse direction (the Y-derivative can occur only in pairs because the original equations are invariant under reflections in this direction). Furthermore, b3,4 > 0 unconditionally, b 6 , 7 > 0 for n not too close to 0, b2 > 0 for 3 E [0, 1], and a2 > 0 for v E [0, 1]. As will be seen below, the sign and magnitude of the coefficient 1 / d-1 b5 = ] 4 ' o G 0 + ~ 0
[(n + 2)4'od + 0(8)]
(13)
is crucial for our considerations. It is reasonable to assume a monotonically decreasing (with increasing voidage) interparticle force such that G~ > 0, which has a destabilizing effect; we will, however, restrict our discussion to a constant value of G. Using the first-order (KdV) approximation to eliminate the time-derivatives in the O(F)-terms of (11), and rescaling the voidage perturbation, time, space, and the Froude number as qb = d l ~ u ,
t =fiT,
x = vX,
F=7//~,
y = pY,
14)
with (provided that al # 0)
a2y' dl=
signal,
(Icol ~,/2
a -- lall '
/~ = a2v3,
Y = \2a~/
,
(icoi],/2 P = F \ bl ,]
,
I 0 -- 2b2F'
15)
leads to the equation ,)
Idt ÷ (bl-)x ÷ Uxxx ÷ I7[EUxx -- Uyy ÷ ClUyy x ÷ Uxx,:x ÷ C2Uxxxx x ÷C3(UUxx)x ÷ C4(uZ)xx
÷ C5(U2)xxx ÷ C 6 ( t t 3 ) x ]
=
O(/~2).
16)
In the following we will neglect the terms of order ~2 and the tilde over the rescaled Froude number. The new coefficients are given by signco,
cl-
b5
c4-
2alb2'
a2F bl
,
¢2 --
b3F 2b2
b6v C5 --
2alb2'
,
C3 - -
b4F
- - , 2alb2
(17)
a2bTy C6 --
2a2b2
•
The associated energy equation (for periodic domains or an infinite domain with vanishing boundary conditions) 1 d 2 dt
[
u2dxdy = F
f
dxdy[Eu~ - u 2v - uxx . +
+ (c3 - 2cs)uUxUxx]
(18)
shows that not all terms of (16) are of equal importance for the global behaviour; moreover, the last contribution vanishes if u is an even function of x. We note that E = - 1 or +1 if the base state is stable, respectively, unstable, and that cl,2.6 > 0 , sign(c3,5) -----d i, and ~ ~ d l u, where d l = 1 for "dense" fluidized beds, i.e. 4'0 < (n + 1)/(n + 3)[= 2/3 for n = 31, and d l = --1
M.E GOz/Physica D 123 (1998) 112-122
116
!
S
i
''''12'''0.
014
i
i
/ i
i
I
0.6
i
I I I I I
' ' S ~ .
-i
-2
-3
Fig. 1. The coefficient c4 in dependence of g00 e (0, 1) for n = 3. Solid c u r v e : c4(~ = 0), dashed c u r v e : c 4 ( 3 = 1). The singularity occurs at 4,0 = (n + 1)/(n + 3), the zero and crossover of both curves at (n + 1)/(n + 2), and the further intersection of the two curves at 3(n + 1)/(3n + 4). for "dilute" fluidized beds. Notice that particulate flows have a close-packing limit of ~cp "~ 0.35, so that al changes sign approximately in the middle o f the accessible voidage region for n = 3; in general, however, the transition point depends strongly on the assumptions on the drag force. Therefore, Eq. (16) describes either propagating voidage perturbations in dense, or particle clusters in dilute weakly unstable fluidized beds. We now proceed to the investigation of the coefficient c4, which for G~ = 0 reads
c4=
n+l-(n+2)~0 n+l
(n+2)~0d+6{½+(1-~0)[n(n+~)-(n+2)2O0]}
-(n+3)O0
(n+2)[~0d+3(1
-~0)(d-
1)]
(19)
The value o f C4(~) lies between those for ~ = 0 and 3 = 1; the c u r v e s c4(t~ -~- 0) and c4(3 = 1) vs. ~b0 are shown in Fig. 1 for n = 3. The limits o f c4 as 4~0 approaches the endpoints of the admissible voidage values are given by 1 l i m ¢4 =
~0~0
1 + n(2n + 3) 2(n + 1)(n + 2)
if3=0, if ~ ~ 0,
J 1/2
if3=0
C4 ----> |
as q~0 ---> 1. cx~
(20)
if 6 ~ 0
For later use we note that c4 is greater than 1/2 or 2/3 in the dense regime for n > 1 or n > 5/2, respectively, and in the very dilute regime for all n but only 8 > 0. In the sequel we will assume at least n > 1. Two-dimensional perturbations ,-~ e i(;~x+ky) of the trivial solution to (16) possess growth rates Re ~r = F [Z 2 (e Z 2) - k2], so that u = 0 is linearly unstable for ~ = 1 and Z2(1 - Z 2) > k 2, leading to the bifurcation of periodic travelling waves u(x + cot, y) with onset velocity cok = 12(1 -- Fc2 Z2) Jr- Fclk 2.
(21)
The stability boundaries are given by Z = k = O, where a solitary wave of infinite period occurs, and k = O, Z0 = 1, where a family of one-dimensional wavetrains bifurcates with velocity coo = 1 - Fc2. These relations correspond,
M.E GOz/Physica D 123 (1998) 112-122
117
and provide a small Froude number approximation, to the stability result (7) of the original system, taking into account the transformations (9) and (14), which in particular map the stability boundary Z0 to 1.
5. Plane
solitary
waves
Let us seek solitary wave solutions of (16) as perturbations of KdV-solitons. As has been shown in [16], only plane solitary waves can be obtained in this way. Thus we introduce T=
Ft,
O = A ( T ) [ x - f 2 - xo(T, y)] w i t h 0 r f 2 = 4 A 2 / F
f
(and OyA = 0 in order to avoid terms ~ u = uo + FUl + O(F2),
(22)
O y A 2 d T / F ) and expand
u0 = 6A2(T) sech20.
(23)
The solvability and boundary conditions for u l give the equations for the amplitude and the phase shift, which indicates also the propagation direction of the wave (for more details in the case of dense fluidized beds see [16]): A=-~[E-/:2
4- 4 ( 1 2 c 4 - 5 ) A 2 ] A 3,
x o = x o l ( A ) 4-/:y.
(24)
From this result and the properties of c4 described above we conclude that - Finite-amplitude solitary waves exist and are: - unstable for all ~: if the base state is stable and c4 > 5/12, this holds in the dense and very dilute regimes; - unstable for/:2 > 1 and c4 > 5/12 if the base state is unstable, again this holds in the dense and very dilute regimes; - stable for/:2 < 1 and c4 < 5/12 if the base state is unstable, this applies to the dilute regime except a region near ~b0 ~ 1. - Blow-up occurs for/:2 < 1, c4 > 5/12, e = 1, i.e. for unstable dense or very dilute states of uniform fluidization. In the original system, this is the regime in which plane wavetrains u(x, y, t) = fi(x + wt 5:/:y) with/: = k / Z bifurcate from the base state and develop into homoclinic connections [8,16].
6. One-dimensional
periodic
solutions
and their stability
We focus on the family of one-dimensional periodic travelling waves bifurcating at (Xo, w0). This solution branch can be approximated by the amplitude expansion u(l)(z) : ~ae i= 4-~2a201e2iZ 4- E3a3fle 3i: 4 - . - - 4- c.c., , , Z = L0 + E'X2 + . . . , co = wo + e~oo2 + . . . ,
(25)
with the travelling wave coordinate z = X(x 4- wt), the small amplitude 4, an arbitrary phase factor a, the wave number and velocity expansion terms Zo = 1,
(.o0 = 1 - Fc2,
Z2 = - F r / ( 2 F ) ;
0)2 = -[Yi + (1 - 2Fc2)I/r/F],
(26)
and the coefficients 2Fc4 - i[1 - F(c3 + 4c5)] 13,'=
312F - i(1 - 5Fc2)]
=
1
g [1 4- F(5c2
--
c3
--
2i 4c5)] + ~-(c4 3
-
-
1)F + O(F2),
(27)
M.E GOz/Physica D 123 (1998) 112-122
118
ot[6Fc4-i(1-F(18cs+5c3))]-iFC6z[3F_i(l_lOFc2)] =--121 [ 1 + F ( 3 0 c 2 - 7 c 3 - 2 6 c 5 + 3 c 6 )
1
5i
"~ "]'~ (C4 -- 1)F + O(F2),
y = 3iFc6 - ot[2Fc4 - i(2 - F(5c3 + 2c5))] ~ Yr + iyi, Yi = 2 + I F ( 7 c 3 _ 10c2 + 10c5 - 9c6) + O(F2).
(28) Yr = 2(~ - c4)F + O(F2), (29)
O b v i o u s l y , )~2 > 0 if Yr < 0; hence, the bifurcation is subcritical into the region )~ > Z0 in the case of c4 >
2/3 + O(F). This happens in the dense regime for n > 5/2 or not too small q~0 (certainly for 4,o > ~bcp),and at the very dilute end if 8 > 0. In contrast, the bifurcation is supercritical in most of the dilute regime. For a two-dimensional stability analysis of the primary wave (25), we linearize (16) at u 0) and expand the twodimensional perturbation, which is assumed to have transverse wave number k and the same longitudinal wavelength as the primary wave, with respect to the amplitude e. This yields the eigenvalue expansion 0- = 0-o+~2a2 + ' " ', with 0-0~ = - F (1 q: icl )k 2, and 0-2 to be determined from a quadratic equation. For pure one-dimensional perturbations, i.e. k = 0, we get 0-~: = 0, 0-2O) = 0 due to the translational symmetry of u (j) expressed by the phase factor a, and the stability-determining perturbed eigenvalue 0-2(2) = - 2 y r . For k # 0, Re 0-~ = - F k 2, so the real part of the leading-order eigenvalues is negative, and since the perturbation terms are small the stability boundary cannot be reached. However, the above expansion for 0- becomes non-uniform for small transverse wave numbers, where the unperturbed eigenvalues 0-~ are close to zero. It can be checked that 0-2 "" 0"2(2) -4- O(k2), so that 0"~:(k2) and 0"2 become the same order when k = O(e). Hence, assuming that Yr = O(1) with respect to E, the rescalings 0" = e2&, k = Ek, and a redesigned expansion yield the dispersion relation ~.2 + 2(yr + Fk2)o'o + 2F(yr - Cl yi)~2 + F2(1 + c~)]~4 = 0.
(30)
The solutions of (30) start at/~ = 0 with the values 0"2(!'2)as two branches of real eigenvalues; one stays negative, the other rises to a maximum and then drops below zero after the critical wave number
~2 = 2(c1~/i - yr)/[F(1 + c12)].
(31)
Then the two real eigenvalues merge into a complex-conjugate pair and fall off like 0-f (/~2) as ~2 ...+ CX~,cf. Fig. 2. Therefore the one-dimensional wavetrains are most unstable to a two-dimensional disturbance; the stabilization at ke indicates the secondary bifurcation to a two-dimensional quasi-steady wavetrain. Notice that the existence of kc follows from the assumptions F << 1 and Cl > 0. Such a secondary long-wave instability has already been proven to occur in the original fluidized bed equations [9], and it was one of the main reasons for the derivation of (16) to facilitate the further investigation of the secondary and other solutions. As a major difference to the above result we mention that in the original problem the perturbed eigenvalue has a 1/k 2 singularity which requires a scaling of the transverse wave number with v~- instead of E. This singularity can be ascribed to the presence of two additional perturbation modes, which are pure transverse modes at the bifurcation point but grow into 2D modes along the 1D primary wave; simultaneously the above pair of 2D modes (with unperturbed eigenvalues 0-~:) gains additional transverse structure. Originally, each of these modes contributes to the generation of the secondary instability and may become unstable at the secondary bifurcation point. Often, though not always, the transverse modes are damped and their impact has obviously been eliminated in the process of deriving (16). Indeed, one of the transverse modes represents a pure velocity disturbance of the uniform state and thus cannot show up in Eq. (16) for the perturbation of the volume fraction; the other pure transverse mode is still present in (16) but does not enter the dispersion relation (30). Moreover, the secondary instability can be either stationary or oscillatory in the original system, whereas (30) allows a stationary instability only.
M.E GOz/Physica D 123 (1998) 112-122
119
0.5
-(]. 5
1
(a)
•
.
.5
-0.5
-1
-1.5
-2
(b) Fig. 2. The solutions Re6- 0 versus/~ o f the dispersion relation (30) for (a) c4 = 1, (b) c4 = 1 / 3 . H e r e w e h a v e used the leading-order expressions (29) for Y,../ and have set F = 1 / 3 , c t = 1.
7. A m p l i t u d e equations In order to derive a modulation equation describing transverse perturbations along a one-dimensional wavetrain, w e define a new small parameter ~ which measures the distance from the bifurcation point via )~ - ~o = ~2~ 2 with 1~-21 = 1, write the wave velocity in the form o9 = k2(1 - Fc2)~ 2) Jr- ~ , introduce the new time and space scales T = ~2t, Y = ~y according to the rescalings in Section 6, and expand the solution as u =
~ V l Jr- ~ 2 v 2 q - • • - ,
with
vl =
A(T, Y)
• e i= + c . c .
(32)
120
M.F. GOz/Physica D 123 (1998) 112-122
This leads to d~o,1 = 0 and the complex Ginzburg-Landau equation
OvA = F(1 - icl)02A -- (i~b2 + 2FZ2)A - yAIAI 2 + O(~2).
(33)
The simplest solutions are uniform travelling waves
A = Ake i(kr+sT),
IAk 12 = --(2)v2 + k2)F/2/r, (34)
(02
+ S = F c l k 2 - YilAkl 2 = [22/iZ2 + (ClYr + yi)k2]F/}.'r.
The wave velocity s can be absorbed into (D2. In case o f ~2 -~- 1, such solutions exist for arbitrary values of k if Yr < 0, while no solution exists for Yr > 0. I f ~2 = - 1, solutions exist only for Yr < 0 and k 2 > 2, or Yr > 0 and k 2 < 2 (this is also the domain in which the trivial solution is unstable). For k = 0 we obtain the non-trivial steady state of (33), which corresponds to the one-dimensional wavetrain analysed in the previous section. Obviously, 6 2 = r-o21A012 - 2(1 - 2Fc2)~2 in this case. The linear stability of A = A0 is described by (30), with the scale factor • being replaced by IA0l. In accordance with the above results, a subcritical branch to the right exists for Yr < 0, or a supercritical branch to the left of Z0 = 1 for Yr > 0; in any case, the bifurcating branch is unstable to perturbations with wave numbers k 2 ~ (0, IA012k2), since Cl Yi - Y~ > 0. The stability of the solutions (34) with k ~ 0 is determined by a generalization of (30); because the coefficients of the generalized dispersion relation are complex, instabilities will be of oscillatory type. It is found that the plane waves (34) are all unstable if y~ < 0, while a window
k 2 < 2Tr(Yr - ClYi)/[2(D 2 + 1,'2) + Yr(Yr - ClYi)] of stable solutions exists for Fr > 0 provided that Yr - cl Yi > 0 (cf. e.g. [21,22]). This last condition, however, is not satisfied in our case, because Fr = O ( F ) << 1 and 0 < clyi = O(1). The secondary quasi-steady wave solution bifurcating at kc = IA01kc can be constructed by another amplitude expansion with respect to V ' ~ - kc I. The bifurcation is supercritical into the region k < kc if F~ > 0, i.e. in the (not too) dilute regime, and subcritical to k > kc otherwise. As can be seen from the expressions for cr~2) and Ak, a degeneracy occurs when y r / F is small, and this happens if c4 is close to c4° = 2/3 4- O(F). In this case we must include the next higher-order terms in (32) and (33), and observe in particular that Z - )~0 becomes of higher order, too; in addition it is necessary to rescale the transverse variable once more. It is reasonable to use ~2 = (c4 - c ° ) / h with h 2 = 1 as yet another expansion parameter. Thus, replacing ~2 by ~2~ 4 and Yr by - 2 h F ~ 2, Eq. (33) becomes (1 - 2i~2~ IAI2)OtA = -i(ff~2 + Yi IAI2)A + ?2F(1 - iq)OZA - ~ 2 [ k b 4 4- 2 F ~ 4 -+- 2 ( ~ b 2 - hF)IAI 2 + olAI4]A,
(35)
with 7~ = ~2t, I? = ~2y and = ~[i + F ( 4 - 3c°)][1 + O(F)],
rl = 1~[3i + F(17 - 20c°)][1 + O(F)].
(36)
The travelling wave solutions A~ = IA~ lexp(i/d~) are now determined by if)2 = --Yi IA~[ 2,
~4 --
1 + -5-4 O ( F ) Ia~[ 4 + F c l k 2,
23-4 + ~¢2 _ 2hla~12 _ ~la/,14 = O ( F ) .
(37)
M.E Grz/Physica D 123 (1998) 112-122
121
The temporal stability of the steady states
IZ~, 2 = - - f 5 4 ( - h - k - ~ / h 2 + 5 ~ 4 / 2 7 ) + O ( F )
(38)
is given by -/21
+ 4 F ~ 2 [ v / h 2 + 5~4/27 + O(F)]lA0~l 2 +
O(e'4).
(39)
It is obvious that for h = - 1 the supercritically bifurcating solution exists to the left of ~-4 = 0 until - 2 7 / 5 but then turns around into an unstable branch extending into the positive ~-4 region. For h = 1 we have only the subcritical branch that exists and is unstable for all 3. > 3.o (until the approximation breaks down). The instability with respect to spatial disturbances remains unaffected as long as Cl Yi is positive and of order 1. For the same reason all plane waves A~ are unstable (cf. [21]).
8. Discussion Starting from a two-fluid model for particle suspensions in fluidized beds we have derived a very general nonlinear wave equation that should find applications in other areas as well. We have then studied this equation for solitary and periodic wave solutions, and have derived amplitude and Ginzburg-Landau equations from it. It has also allowed us to identify a point in parameter space at which the domination of one phase by the other is reversed, in the sense of one phase falling out from the mixture and building agglomerates under the influence of gravity and the mutual interaction forces. The crucial parameter for most of our considerations has been the coefficient of the nonlinear diffusion term, c4; in addition, linear transverse dispersion as measured by the coefficient cl plays a significant role with regard to the two-dimensional instability of both one- and two-dimensional periodic solutions. In the case of dense and very dilute fluidized beds it turned out that usually c4 > 2/3 and therefore the bifurcating periodic solutions are all unstable. That the primary bifurcation is subcritical within the small Froude number/weakly unstable approximation is in accordance with a numerical bifurcation analysis for the full equations [23]; there it was found, however, that the branch subsequently turns and gains stability, and that farer away from the stability limit the bifurcation is supercritical with the branch staying 1D stable to higher amplitudes. In any case, these one-dimensional branches undergo a secondary bifurcation (which can be sub- or supercritical) to a two-dimensional travelling wave. Moreover, no finite-amplitude solitary wave-perturbations of KdV-solitons exist in dense and very dilute fluidized beds for the reduced wave equation, whereas homo- and heteroclinic orbits have been shown to exist in the original system [8]. Therefore, although the qualitative bifurcation behaviour of the full system is mimicked by Eq. (16), it is unsuited to describe higher-amplitude solutions in such fluidized beds and one has to return to the full equations, cf. also the discussion of various Froude number scalings in [ 17,24]. The situation appears more regular in dilute fluidized beds, which are populated by propagating particle clusters instead of voidage waves. Here, finite-amplitude solitary waves and stable wavetrains do exist, although the supercritical 1D branch turns into an unstable high-amplitude solution and is of course 2D unstable; however, the bifurcating 2D travelling wave is stable to perturbations of the same wave number in this case. In the frame of the modified Kawahara equation, which is obtained from (16) by dropping the transverse terms and those with coefficients ci except c4, it was shown recently that the 1D wavetrain bifurcating supercritically at 3. = I is unstable to perturbations of different longitudinal wave number; but there exists a range of (c4, 3.) values for which the periodic solutions, obtained as perturbations of cnoidal waves, are 1D stable against perturbations
122
M.E G6z/Physica D 123 (1998) 112-122
with arbitrary w a v e n u m b e r [19]. It will be interesting to generalize these results to Eq. (16) and investigate in particular the stability o f these solutions to transverse perturbations.
References [ 1] T.B. Anderson, R. Jackson, Fluid mechanical description of fluidized beds, Ind. Eng. Chem. Fund. 7 (1968) 12-21. [2] M.E G6z, On the origin of wave patterns in fluidized beds, J. Fluid Mech. 240 (1992) 379-404. [3] A. Sokolichin, G. Eigenberger, Gas-liquid flow in bubble columns and loop reactors: Part I. Detailed modelling and numerical simulation, Chem. Eng. Sci. 49 (1994) 5735-5746. [4] G. Hetsroni (ed.), Handbook of Multiphase Systems, McGraw-Hill, New York, 1982. [5] M.M. E1-Kalssy, G.M. Homsy, Instability waves and the origin of bubbles in fluidized beds. Part I: experiments, Int. J. Multiphase Flow 2 (1976) 379-395. [6] A.K. Didwania, G.M. Homsy, Flow regimes and flow transitions in liquid fluidized beds, Int. J. Multiphase Flow 7 (1981) 563-580. [7] A.K. Didwania, G.M. Homsy, Resonant side-band instabilities in wave propagation in fluidized beds, J. Fluid Mech. 122 (1982) 433-438. [8] M.E G6z, Bifurcation of plane voidage waves in fluidized beds, Physica D 65 (1993) 319-351. [9] M.E G6z, Transverse instability of plane wavetrains in gas-fluidized beds, J. Fluid Mech. 303 (1995) 55-81. [10] K. Anderson, S. Sundaresan, R. Jackson, Instabilities and the formation of bubbles in fluidized beds, J. Fluid Mech. 303 (1995) 327-366. [11] B.J. Glasser, I.G. Kevrekidis, S. Sundaresan, One- and two-dimensional travelling wave solutions in gas-fluidized beds, J. Fluid Mech. 306 (1996) 183-221. [12] M.E Grz, Bifurcation with symmetry in multi-phase fows, in: D.A. Drew, D.D. Joseph, S.L. Passman (Eds.), Particulate Flows: Processing and Rheology, IMA Proceedings, Springer, Berlin, 1998, pp. 55-66. [13] B.J. Glasser, I.G. Kevrekidis, S. Sundaresan, Fully-developed travelling wave solutions and bubble formation in fluidized beds, J. Fluid Mech. 334 (1997) 157-188. [ 14] R.H. Wilhelm, M. Kwauk, Fluidization of solid particles, Chem. Engng. Prog. 44 (1948) 201-218. [15] M.E Grz, S. Sundaresan, The growth, saturation, and scaling behaviour of one- and two-dimensional disturbances in fluidized beds, J. Fluid Mech. 362 (1998) 83-119. [16] M.E G6z, Small Froude number asymptotics in two-dimensional two-phase flows, Phys. Rev. E 52 (1995) 3697-3710. [17] S.E. Harris, D.G. Crighton, Solitons, solitary waves and voidage disturbances in gas-fluidized beds, J. Fluid Mech. 266 (1994) 243-276. [18] T.S. Komatsu, H. Hayakawa, Nonlinear waves in fluidized beds, Phys. Lett. A 183 (1993) 56-62. [19] D.E. Bar, A.A. Nepomnyaschy, Stability of periodic waves governed by the modified Kawahara equation, Physica D 86 (1995) 586-602. [20] J.E Richardson, Incipient fluidization and particulate systems, in: J.E Davidson, D. Harrison (Eds.), Fluidization, Academic Press, London, 1971, pp. 26-64. [2 l] W. Eckhaus, G. Iooss, Strong selection or rejection of spatially periodic patterns in degenerate bifurcations, Physica D 39 (1989) 124-146. [22] M.C. Cross, P.C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65 (1993) 851-1112. [23] B. J. Glasser, One- and two-dimensional travelling wave solutions in fluidized beds, Ph.D. Dissertation, Princeton University, 1996. [24] S.E. Harris, The growth of periodic waves in gas-fluidized beds, J. Fluid Mech. 325 (1996) 261-282.
Physica D 123 (1998) 112-122
Nonlinear w a v e s in two-fluid flows * M.E G6z 1 Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA
Abstract Typically, the interaction of two miscible fluids leads to the formation of segregation patterns in the form of plane concentration waves and two- or three-dimensional voids or clusters. The development of small-amplitude plane travelling waves and the subsequent onset of higher-dimensional instabilities can be elucidated analytically; in addition, numerical methods allow the computation of higher-amplitude solutions. As both direct approaches become costly, although at different amplitude levels, one is tempted to look for a simpler model problem. Therefore, a nonlinear wave equation has been derived for two-dimensional dispersed two-phase flow in fluidized beds near the stability boundary using the Froude number as small parameter. This equation is a two-dimensional perturbation of the Korteweg--de Vries equation containing all low-order terms. It allows to identify the transition point between voids and clusters, and to investigate these two regions for solitary and periodic waves. A Ginzburg-Landau equation is derived to describe transverse perturbations of one-dimensional wavetrains bifurcating at the stability boundary. Copyright © 1998 Elsevier Science B.V. Keywords: Multiphase flows; Fluidized beds; Extended Korteweg-de Vries equation; Pattern formation; Solitary waves
1. Introduction We consider equations describing the interaction of two miscible fluids as they are used for dispersed two-phase flows like g a s - s o l i d and liquid-solid mixtures in fluidized beds [1,2] or the rise of small gas bubbles in liquid columns [3]. Typical patterns observed in such systems are plane voidage or concentration waves, and two- or three-dimensional travelling wave patterns such as plumes of fluid ('bubbles', voids) rising in a dense fluidized bed [4-7], or clusters of particles moving through dilute fluidized beds [4]. The terms dense and dilute refer to the average volumetric packing, respectively, distribution of particles. As generic behaviour one can perceive a sequence of instabilities leading from a uniform state to plane wavetrains [8] to two-dimensional travelling waves [9-11] in the form of bubbles or clusters to more complex spatio-temporal patterns [12]. There exist some characteristic differences between gas- and liquid-fluidized beds concerning the formation and stability of bubbles [4-7,10,13], which have been known experimentally since long to be related to * Supported by the Deutsche Forschungsgemeinschaft. 1 Present address: Martin-Luther-Universit~itHalle-Wittenberg, FB Verfahrenstechnik, MVT/UST, D-06099 Halle, Germany. Tel.: +49 3461 462884; fax: + 49 3461 462878; e-mail: [email protected]. 0167-2789/98/$19.00 Copyright © 1998 Elsevier Science B.V. All rights reserved PII S01 67-27 89(98)001 16-X
M.E GOz/Physica D 123 (1998) 112-122
113
the magnitude of the Froude number [14]. This relationship is still not fully understood, but recent numerical and analytical studies of standard modelling equations have brought forth the importance of the Froude number and how its value determines the growth rates of primary and secondary instabilities in fluidized beds [ 13,15]. As the full system of equations is rather unhandy, it is reasonable to use the Froude number as perturbation parameter and reduce the bulky system of coupled Navier-Stokes equations to a single wave equation for the voidage perturbation near the stability boundary [16] (see also [17], in which an inconsistent scaling is used, and [ 18], where the Froude number is not employed as parameter so that a similar but simpler equation is obtained which is less suited to describe the bifurcation behaviour; cf. the discussion in [16]). The result is a quite general nonlinear wave equation, namely a two-dimensionally perturbed Korteweg-de Vries equation that is complete in the low-order terms; its one-dimensional version comprises various well-known wave equations like the BurgersKdV and Kuramoto-Sivashinsky equations, the Benney or Kawahara equation, as well as the so-called modified Kawahara equation, all of which appear in a wealth of physical problems (cf. e.g. [19]). Here we shall focus on fluidized beds, but expect the analysis to be applicable to many other single- and multiphase systems like traffic flow or polymer-solvent mixtures due to the similarity in the governing equations and the completeness of the derived 1|onlinear wave equation. After presenting the basic equations and parameters we will briefly describe the stability and scaling analysis that leads to the reduced wave equation. We will take a look at its general properties and the more important coefficients, and then study the instability of the trivial state which gives rise to periodic and solitary waves. We identify the transition point between voids and clusters, which we associate with a switch from dense to dilute beds; this may also be viewed as a change in one phase dominating the other. Regarding plane solitary waves, it turns out that perturbations of KdV-solitons blow up in finite time in dense and very dilute fluidized beds. This is an artifact of the reduced equation, the original system admits homoclinic solutions. At the other end of the unstable spectrum, a branch of one-dimensional periodic travelling waves bifurcates subcritically from the uniform state of dense and possibly very dilute beds. In the model problem, this branch does not turn around to stabilize at moderately high amplitude, in contrast to numerical results for the original equations [ 11,13]. Two-dimensional solutions are also found to be unstable in this case. For dilute particle suspensions we have propagating particle clusters instead of voidage waves; finite-amplitude solitary waves do exist, one-dimensional periodic solutions bifurcate supercritically but may turn unstable at higher amplitude and are transversally unstable.
2. Basic equations and parameters The original two-fluid equations are [ 16] -Otck + V • [(1
- ~b)v] = 0
Ot4~ + V • (4m) = 0
(dispersed phase),
(continuous phase),
F(1 - ~b)(0tv + v . Vv) = - ( 1 - q~)k + B(~b)(u - v) - (1 - ( b ) V p - F G ( f b ) V ( b + # A v , F~ga(Oru
+ u . Vu) = - ~ b k - B(q~)(u - v) - ~bVp + v # A u ,
(1) (2) (3) (4)
with the voidage 4) ( = volume fraction of the continuous phase), the velocities v and u of the dispersed (particulate) and continuous (fluid) phase, respectively, an effective fluid pressure p, the coefficient G (4~) < 0 of the interparticle torce, and the drag coefficient B(~b) = (1
-
~ ) ~ - - ~ o n+l ,
(5)
M.F. GOz/Physica D 123 (1998) 112-122
114
with the Richardson-Zaki exponent n "- 2 - 4 [20] and the voidage value tP0 of the state of uniform fluidization, whose non-dimensional form is given by ~=~o,
uo=k,
vo=O,
Vpo=-[1
(6)
- ~o(1 - 3)]k.
Here, k represents the unit vector into the x-direction which points against gravity; the transverse coordinate will be denoted by y. The non-dimensional parameters are the Froude number F = u 2 / g r and the Reynolds number R = ruoPd/IXd, with gravity constant g, particle radius r, particle density Pd, particle phase viscosity /Zd, and fluidization velocity u0 = (2/9)(1 - 6)4)~+lgr2pd/lZc which corresponds in non-dimensional form to the relation (1 - 8)tp~ +l = (9/2)v/z characterizing uniform fluidization. For brevity, we have introduced/z = F / R ; in addition, the density and viscosity ratios of the two phases are denoted by 6 = Pc/Pd E [0, 1), and v = /Xc//Zd E [0, 1), respectively.
3. Stability and scaling analysis The stability of the state of uniform fluidization (6) is determined by an eigenvalue cr which expands for small longitudinal wave number ~ as well as small Froude number as (here we set 3 = v = 0 for simplicity) ~r = -i)~d[1 - O ( F ) ] + Fq~o[(d 2 - 092)~2 _ w2k2] _ O ( F 2 ) 4) _ O ( F 2 ) 2k 2, F2k4),
(7)
where k is the transverse wave number,
1 - 4'0
d -- - Bo
[Bo - ~boB0 + (1 - 8)4~0(1 - 2~bo)] -- (n -t- 2)(1 - tPo)
(8)
(Bo -- B(tPo), etc.) the kinematic wave velocity related to bouyancy, and 090 = ~ + O(6) the dynamic wave velocity at which plane periodic travelling waves bifurcate with wave number )2 ..~ (d 2 _ 092)/F. Obviously, a (long-wave) instability can occur only if d > wo (a complete analysis also shows that 090 > O(v) > 0 has to be satisfied, but this is irrelevant for the small Froude number asymptotics). In order to balance leading-order terms and include transverse contributions as well, we assume that the strength of the instability scales with F << 1 in the form of d - wo "~ F '~, hence the wave numbers must scale as )2 ~ F,~- l, k 2 ~ F2U- I. The appropriate choice of coordinates and scalings is given by d - wo ~ F such that )~0 ~ O(1), thus ot = 1 and 490(d 2 - 092) + 0 ( 3 ) = coF,
X = x - dr,
Y = FU2y,
T = Ft,
(9)
where co is an O(1) quantity, which is negative or positive depending on whether the base state is stable or unstable, respectively. Then we employ the perturbation 49 = dpo + Fdp,
p = poX + F~,
Vx = F v x ,
Ux = 1 + Ffix,
(1)y, Uy) = F3/2(1)y, fly).
(10)
4. The reduced equation and s o m e of its properties Using the leading-order relations between the perturbation variables allows us to express the pressure and velocity perturbations in terms o f q~ plus O ( F ) contributions, such that these other variables can be pushed to higher and higher orders in the Froude number. This gives the following equation:
q~T "q- al (~2)X -I- a2~PXXX + F[co(bxx -- bl(byy --1-a2~PYyx - 2b2dPXT -- b3(bxxz + b4(dPqbxx)x + (b5 - 2alb2)(~p2)xx + (b6 - a l b 3 ) ( ( ) 2 ) x x x
+ b7(q~3)x] = O ( F 2 ) ,
(ll)
M . E GOz/Physica D 123 (1998) 112-122
115
where the constant coefficients depend on 4'0 and the physical parameters in a complicated manner [ 16]. We note in particular that (1 -
24'o)d
- (1 4'0(1 - 4'0)
aj = -
n+2
24,0
B~ ~ (d + 4'0 - 1)
¢0) 2
4'o(1 - 4'0) (B~'/2 + 1 Bo
8)
In ÷ 1 - (n ÷ 3)4'0]
(12)
so that the nonlinear term in the KdV-part vanishes at 4'o = (n + 1)/(n + 3); this signals the transition from voids to clusters. It is also important that bl = - ( 1 - 3)4'0G0 > 0, providing stabilizing diffusion in the transverse direction (the Y-derivative can occur only in pairs because the original equations are invariant under reflections in this direction). Furthermore, b3,4 > 0 unconditionally, b 6 , 7 > 0 for n not too close to 0, b2 > 0 for 3 E [0, 1], and a2 > 0 for v E [0, 1]. As will be seen below, the sign and magnitude of the coefficient 1 / d-1 b5 = ] 4 ' o G 0 + ~ 0
[(n + 2)4'od + 0(8)]
(13)
is crucial for our considerations. It is reasonable to assume a monotonically decreasing (with increasing voidage) interparticle force such that G~ > 0, which has a destabilizing effect; we will, however, restrict our discussion to a constant value of G. Using the first-order (KdV) approximation to eliminate the time-derivatives in the O(F)-terms of (11), and rescaling the voidage perturbation, time, space, and the Froude number as qb = d l ~ u ,
t =fiT,
x = vX,
F=7//~,
y = pY,
14)
with (provided that al # 0)
a2y' dl=
signal,
(Icol ~,/2
a -- lall '
/~ = a2v3,
Y = \2a~/
,
(icoi],/2 P = F \ bl ,]
,
I 0 -- 2b2F'
15)
leads to the equation ,)
Idt ÷ (bl-)x ÷ Uxxx ÷ I7[EUxx -- Uyy ÷ ClUyy x ÷ Uxx,:x ÷ C2Uxxxx x ÷C3(UUxx)x ÷ C4(uZ)xx
÷ C5(U2)xxx ÷ C 6 ( t t 3 ) x ]
=
O(/~2).
16)
In the following we will neglect the terms of order ~2 and the tilde over the rescaled Froude number. The new coefficients are given by signco,
cl-
b5
c4-
2alb2'
a2F bl
,
¢2 --
b3F 2b2
b6v C5 --
2alb2'
,
C3 - -
b4F
- - , 2alb2
(17)
a2bTy C6 --
2a2b2
•
The associated energy equation (for periodic domains or an infinite domain with vanishing boundary conditions) 1 d 2 dt
[
u2dxdy = F
f
dxdy[Eu~ - u 2v - uxx . +
+ (c3 - 2cs)uUxUxx]
(18)
shows that not all terms of (16) are of equal importance for the global behaviour; moreover, the last contribution vanishes if u is an even function of x. We note that E = - 1 or +1 if the base state is stable, respectively, unstable, and that cl,2.6 > 0 , sign(c3,5) -----d i, and ~ ~ d l u, where d l = 1 for "dense" fluidized beds, i.e. 4'0 < (n + 1)/(n + 3)[= 2/3 for n = 31, and d l = --1
M.E GOz/Physica D 123 (1998) 112-122
116
!
S
i
''''12'''0.
014
i
i
/ i
i
I
0.6
i
I I I I I
' ' S ~ .
-i
-2
-3
Fig. 1. The coefficient c4 in dependence of g00 e (0, 1) for n = 3. Solid c u r v e : c4(~ = 0), dashed c u r v e : c 4 ( 3 = 1). The singularity occurs at 4,0 = (n + 1)/(n + 3), the zero and crossover of both curves at (n + 1)/(n + 2), and the further intersection of the two curves at 3(n + 1)/(3n + 4). for "dilute" fluidized beds. Notice that particulate flows have a close-packing limit of ~cp "~ 0.35, so that al changes sign approximately in the middle o f the accessible voidage region for n = 3; in general, however, the transition point depends strongly on the assumptions on the drag force. Therefore, Eq. (16) describes either propagating voidage perturbations in dense, or particle clusters in dilute weakly unstable fluidized beds. We now proceed to the investigation of the coefficient c4, which for G~ = 0 reads
c4=
n+l-(n+2)~0 n+l
(n+2)~0d+6{½+(1-~0)[n(n+~)-(n+2)2O0]}
-(n+3)O0
(n+2)[~0d+3(1
-~0)(d-
1)]
(19)
The value o f C4(~) lies between those for ~ = 0 and 3 = 1; the c u r v e s c4(t~ -~- 0) and c4(3 = 1) vs. ~b0 are shown in Fig. 1 for n = 3. The limits o f c4 as 4~0 approaches the endpoints of the admissible voidage values are given by 1 l i m ¢4 =
~0~0
1 + n(2n + 3) 2(n + 1)(n + 2)
if3=0, if ~ ~ 0,
J 1/2
if3=0
C4 ----> |
as q~0 ---> 1. cx~
(20)
if 6 ~ 0
For later use we note that c4 is greater than 1/2 or 2/3 in the dense regime for n > 1 or n > 5/2, respectively, and in the very dilute regime for all n but only 8 > 0. In the sequel we will assume at least n > 1. Two-dimensional perturbations ,-~ e i(;~x+ky) of the trivial solution to (16) possess growth rates Re ~r = F [Z 2 (e Z 2) - k2], so that u = 0 is linearly unstable for ~ = 1 and Z2(1 - Z 2) > k 2, leading to the bifurcation of periodic travelling waves u(x + cot, y) with onset velocity cok = 12(1 -- Fc2 Z2) Jr- Fclk 2.
(21)
The stability boundaries are given by Z = k = O, where a solitary wave of infinite period occurs, and k = O, Z0 = 1, where a family of one-dimensional wavetrains bifurcates with velocity coo = 1 - Fc2. These relations correspond,
M.E GOz/Physica D 123 (1998) 112-122
117
and provide a small Froude number approximation, to the stability result (7) of the original system, taking into account the transformations (9) and (14), which in particular map the stability boundary Z0 to 1.
5. Plane
solitary
waves
Let us seek solitary wave solutions of (16) as perturbations of KdV-solitons. As has been shown in [16], only plane solitary waves can be obtained in this way. Thus we introduce T=
Ft,
O = A ( T ) [ x - f 2 - xo(T, y)] w i t h 0 r f 2 = 4 A 2 / F
f
(and OyA = 0 in order to avoid terms ~ u = uo + FUl + O(F2),
(22)
O y A 2 d T / F ) and expand
u0 = 6A2(T) sech20.
(23)
The solvability and boundary conditions for u l give the equations for the amplitude and the phase shift, which indicates also the propagation direction of the wave (for more details in the case of dense fluidized beds see [16]): A=-~[E-/:2
4- 4 ( 1 2 c 4 - 5 ) A 2 ] A 3,
x o = x o l ( A ) 4-/:y.
(24)
From this result and the properties of c4 described above we conclude that - Finite-amplitude solitary waves exist and are: - unstable for all ~: if the base state is stable and c4 > 5/12, this holds in the dense and very dilute regimes; - unstable for/:2 > 1 and c4 > 5/12 if the base state is unstable, again this holds in the dense and very dilute regimes; - stable for/:2 < 1 and c4 < 5/12 if the base state is unstable, this applies to the dilute regime except a region near ~b0 ~ 1. - Blow-up occurs for/:2 < 1, c4 > 5/12, e = 1, i.e. for unstable dense or very dilute states of uniform fluidization. In the original system, this is the regime in which plane wavetrains u(x, y, t) = fi(x + wt 5:/:y) with/: = k / Z bifurcate from the base state and develop into homoclinic connections [8,16].
6. One-dimensional
periodic
solutions
and their stability
We focus on the family of one-dimensional periodic travelling waves bifurcating at (Xo, w0). This solution branch can be approximated by the amplitude expansion u(l)(z) : ~ae i= 4-~2a201e2iZ 4- E3a3fle 3i: 4 - . - - 4- c.c., , , Z = L0 + E'X2 + . . . , co = wo + e~oo2 + . . . ,
(25)
with the travelling wave coordinate z = X(x 4- wt), the small amplitude 4, an arbitrary phase factor a, the wave number and velocity expansion terms Zo = 1,
(.o0 = 1 - Fc2,
Z2 = - F r / ( 2 F ) ;
0)2 = -[Yi + (1 - 2Fc2)I/r/F],
(26)
and the coefficients 2Fc4 - i[1 - F(c3 + 4c5)] 13,'=
312F - i(1 - 5Fc2)]
=
1
g [1 4- F(5c2
--
c3
--
2i 4c5)] + ~-(c4 3
-
-
1)F + O(F2),
(27)
M.E GOz/Physica D 123 (1998) 112-122
118
ot[6Fc4-i(1-F(18cs+5c3))]-iFC6z[3F_i(l_lOFc2)] =--121 [ 1 + F ( 3 0 c 2 - 7 c 3 - 2 6 c 5 + 3 c 6 )
1
5i
"~ "]'~ (C4 -- 1)F + O(F2),
y = 3iFc6 - ot[2Fc4 - i(2 - F(5c3 + 2c5))] ~ Yr + iyi, Yi = 2 + I F ( 7 c 3 _ 10c2 + 10c5 - 9c6) + O(F2).
(28) Yr = 2(~ - c4)F + O(F2), (29)
O b v i o u s l y , )~2 > 0 if Yr < 0; hence, the bifurcation is subcritical into the region )~ > Z0 in the case of c4 >
2/3 + O(F). This happens in the dense regime for n > 5/2 or not too small q~0 (certainly for 4,o > ~bcp),and at the very dilute end if 8 > 0. In contrast, the bifurcation is supercritical in most of the dilute regime. For a two-dimensional stability analysis of the primary wave (25), we linearize (16) at u 0) and expand the twodimensional perturbation, which is assumed to have transverse wave number k and the same longitudinal wavelength as the primary wave, with respect to the amplitude e. This yields the eigenvalue expansion 0- = 0-o+~2a2 + ' " ', with 0-0~ = - F (1 q: icl )k 2, and 0-2 to be determined from a quadratic equation. For pure one-dimensional perturbations, i.e. k = 0, we get 0-~: = 0, 0-2O) = 0 due to the translational symmetry of u (j) expressed by the phase factor a, and the stability-determining perturbed eigenvalue 0-2(2) = - 2 y r . For k # 0, Re 0-~ = - F k 2, so the real part of the leading-order eigenvalues is negative, and since the perturbation terms are small the stability boundary cannot be reached. However, the above expansion for 0- becomes non-uniform for small transverse wave numbers, where the unperturbed eigenvalues 0-~ are close to zero. It can be checked that 0-2 "" 0"2(2) -4- O(k2), so that 0"~:(k2) and 0"2 become the same order when k = O(e). Hence, assuming that Yr = O(1) with respect to E, the rescalings 0" = e2&, k = Ek, and a redesigned expansion yield the dispersion relation ~.2 + 2(yr + Fk2)o'o + 2F(yr - Cl yi)~2 + F2(1 + c~)]~4 = 0.
(30)
The solutions of (30) start at/~ = 0 with the values 0"2(!'2)as two branches of real eigenvalues; one stays negative, the other rises to a maximum and then drops below zero after the critical wave number
~2 = 2(c1~/i - yr)/[F(1 + c12)].
(31)
Then the two real eigenvalues merge into a complex-conjugate pair and fall off like 0-f (/~2) as ~2 ...+ CX~,cf. Fig. 2. Therefore the one-dimensional wavetrains are most unstable to a two-dimensional disturbance; the stabilization at ke indicates the secondary bifurcation to a two-dimensional quasi-steady wavetrain. Notice that the existence of kc follows from the assumptions F << 1 and Cl > 0. Such a secondary long-wave instability has already been proven to occur in the original fluidized bed equations [9], and it was one of the main reasons for the derivation of (16) to facilitate the further investigation of the secondary and other solutions. As a major difference to the above result we mention that in the original problem the perturbed eigenvalue has a 1/k 2 singularity which requires a scaling of the transverse wave number with v~- instead of E. This singularity can be ascribed to the presence of two additional perturbation modes, which are pure transverse modes at the bifurcation point but grow into 2D modes along the 1D primary wave; simultaneously the above pair of 2D modes (with unperturbed eigenvalues 0-~:) gains additional transverse structure. Originally, each of these modes contributes to the generation of the secondary instability and may become unstable at the secondary bifurcation point. Often, though not always, the transverse modes are damped and their impact has obviously been eliminated in the process of deriving (16). Indeed, one of the transverse modes represents a pure velocity disturbance of the uniform state and thus cannot show up in Eq. (16) for the perturbation of the volume fraction; the other pure transverse mode is still present in (16) but does not enter the dispersion relation (30). Moreover, the secondary instability can be either stationary or oscillatory in the original system, whereas (30) allows a stationary instability only.
M.E GOz/Physica D 123 (1998) 112-122
119
0.5
-(]. 5
1
(a)
•
.
.5
-0.5
-1
-1.5
-2
(b) Fig. 2. The solutions Re6- 0 versus/~ o f the dispersion relation (30) for (a) c4 = 1, (b) c4 = 1 / 3 . H e r e w e h a v e used the leading-order expressions (29) for Y,../ and have set F = 1 / 3 , c t = 1.
7. A m p l i t u d e equations In order to derive a modulation equation describing transverse perturbations along a one-dimensional wavetrain, w e define a new small parameter ~ which measures the distance from the bifurcation point via )~ - ~o = ~2~ 2 with 1~-21 = 1, write the wave velocity in the form o9 = k2(1 - Fc2)~ 2) Jr- ~ , introduce the new time and space scales T = ~2t, Y = ~y according to the rescalings in Section 6, and expand the solution as u =
~ V l Jr- ~ 2 v 2 q - • • - ,
with
vl =
A(T, Y)
• e i= + c . c .
(32)
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M.F. GOz/Physica D 123 (1998) 112-122
This leads to d~o,1 = 0 and the complex Ginzburg-Landau equation
OvA = F(1 - icl)02A -- (i~b2 + 2FZ2)A - yAIAI 2 + O(~2).
(33)
The simplest solutions are uniform travelling waves
A = Ake i(kr+sT),
IAk 12 = --(2)v2 + k2)F/2/r, (34)
(02
+ S = F c l k 2 - YilAkl 2 = [22/iZ2 + (ClYr + yi)k2]F/}.'r.
The wave velocity s can be absorbed into (D2. In case o f ~2 -~- 1, such solutions exist for arbitrary values of k if Yr < 0, while no solution exists for Yr > 0. I f ~2 = - 1, solutions exist only for Yr < 0 and k 2 > 2, or Yr > 0 and k 2 < 2 (this is also the domain in which the trivial solution is unstable). For k = 0 we obtain the non-trivial steady state of (33), which corresponds to the one-dimensional wavetrain analysed in the previous section. Obviously, 6 2 = r-o21A012 - 2(1 - 2Fc2)~2 in this case. The linear stability of A = A0 is described by (30), with the scale factor • being replaced by IA0l. In accordance with the above results, a subcritical branch to the right exists for Yr < 0, or a supercritical branch to the left of Z0 = 1 for Yr > 0; in any case, the bifurcating branch is unstable to perturbations with wave numbers k 2 ~ (0, IA012k2), since Cl Yi - Y~ > 0. The stability of the solutions (34) with k ~ 0 is determined by a generalization of (30); because the coefficients of the generalized dispersion relation are complex, instabilities will be of oscillatory type. It is found that the plane waves (34) are all unstable if y~ < 0, while a window
k 2 < 2Tr(Yr - ClYi)/[2(D 2 + 1,'2) + Yr(Yr - ClYi)] of stable solutions exists for Fr > 0 provided that Yr - cl Yi > 0 (cf. e.g. [21,22]). This last condition, however, is not satisfied in our case, because Fr = O ( F ) << 1 and 0 < clyi = O(1). The secondary quasi-steady wave solution bifurcating at kc = IA01kc can be constructed by another amplitude expansion with respect to V ' ~ - kc I. The bifurcation is supercritical into the region k < kc if F~ > 0, i.e. in the (not too) dilute regime, and subcritical to k > kc otherwise. As can be seen from the expressions for cr~2) and Ak, a degeneracy occurs when y r / F is small, and this happens if c4 is close to c4° = 2/3 4- O(F). In this case we must include the next higher-order terms in (32) and (33), and observe in particular that Z - )~0 becomes of higher order, too; in addition it is necessary to rescale the transverse variable once more. It is reasonable to use ~2 = (c4 - c ° ) / h with h 2 = 1 as yet another expansion parameter. Thus, replacing ~2 by ~2~ 4 and Yr by - 2 h F ~ 2, Eq. (33) becomes (1 - 2i~2~ IAI2)OtA = -i(ff~2 + Yi IAI2)A + ?2F(1 - iq)OZA - ~ 2 [ k b 4 4- 2 F ~ 4 -+- 2 ( ~ b 2 - hF)IAI 2 + olAI4]A,
(35)
with 7~ = ~2t, I? = ~2y and = ~[i + F ( 4 - 3c°)][1 + O(F)],
rl = 1~[3i + F(17 - 20c°)][1 + O(F)].
(36)
The travelling wave solutions A~ = IA~ lexp(i/d~) are now determined by if)2 = --Yi IA~[ 2,
~4 --
1 + -5-4 O ( F ) Ia~[ 4 + F c l k 2,
23-4 + ~¢2 _ 2hla~12 _ ~la/,14 = O ( F ) .
(37)
M.E Grz/Physica D 123 (1998) 112-122
121
The temporal stability of the steady states
IZ~, 2 = - - f 5 4 ( - h - k - ~ / h 2 + 5 ~ 4 / 2 7 ) + O ( F )
(38)
is given by -/21
+ 4 F ~ 2 [ v / h 2 + 5~4/27 + O(F)]lA0~l 2 +
O(e'4).
(39)
It is obvious that for h = - 1 the supercritically bifurcating solution exists to the left of ~-4 = 0 until - 2 7 / 5 but then turns around into an unstable branch extending into the positive ~-4 region. For h = 1 we have only the subcritical branch that exists and is unstable for all 3. > 3.o (until the approximation breaks down). The instability with respect to spatial disturbances remains unaffected as long as Cl Yi is positive and of order 1. For the same reason all plane waves A~ are unstable (cf. [21]).
8. Discussion Starting from a two-fluid model for particle suspensions in fluidized beds we have derived a very general nonlinear wave equation that should find applications in other areas as well. We have then studied this equation for solitary and periodic wave solutions, and have derived amplitude and Ginzburg-Landau equations from it. It has also allowed us to identify a point in parameter space at which the domination of one phase by the other is reversed, in the sense of one phase falling out from the mixture and building agglomerates under the influence of gravity and the mutual interaction forces. The crucial parameter for most of our considerations has been the coefficient of the nonlinear diffusion term, c4; in addition, linear transverse dispersion as measured by the coefficient cl plays a significant role with regard to the two-dimensional instability of both one- and two-dimensional periodic solutions. In the case of dense and very dilute fluidized beds it turned out that usually c4 > 2/3 and therefore the bifurcating periodic solutions are all unstable. That the primary bifurcation is subcritical within the small Froude number/weakly unstable approximation is in accordance with a numerical bifurcation analysis for the full equations [23]; there it was found, however, that the branch subsequently turns and gains stability, and that farer away from the stability limit the bifurcation is supercritical with the branch staying 1D stable to higher amplitudes. In any case, these one-dimensional branches undergo a secondary bifurcation (which can be sub- or supercritical) to a two-dimensional travelling wave. Moreover, no finite-amplitude solitary wave-perturbations of KdV-solitons exist in dense and very dilute fluidized beds for the reduced wave equation, whereas homo- and heteroclinic orbits have been shown to exist in the original system [8]. Therefore, although the qualitative bifurcation behaviour of the full system is mimicked by Eq. (16), it is unsuited to describe higher-amplitude solutions in such fluidized beds and one has to return to the full equations, cf. also the discussion of various Froude number scalings in [ 17,24]. The situation appears more regular in dilute fluidized beds, which are populated by propagating particle clusters instead of voidage waves. Here, finite-amplitude solitary waves and stable wavetrains do exist, although the supercritical 1D branch turns into an unstable high-amplitude solution and is of course 2D unstable; however, the bifurcating 2D travelling wave is stable to perturbations of the same wave number in this case. In the frame of the modified Kawahara equation, which is obtained from (16) by dropping the transverse terms and those with coefficients ci except c4, it was shown recently that the 1D wavetrain bifurcating supercritically at 3. = I is unstable to perturbations of different longitudinal wave number; but there exists a range of (c4, 3.) values for which the periodic solutions, obtained as perturbations of cnoidal waves, are 1D stable against perturbations
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M.E G6z/Physica D 123 (1998) 112-122
with arbitrary w a v e n u m b e r [19]. It will be interesting to generalize these results to Eq. (16) and investigate in particular the stability o f these solutions to transverse perturbations.
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