Pattern formation for oscillatory bulk-mode competition in a two-layer B~nard problem

Transcription

1 Z angew Math Phys 47" (1996) /96/ / 1996 BirkNiuser Verlag, Basel Pattern formation for oscillatory bulk-mode competition in a two-layer B~nard problem Yuriko Y. Renardy Abstract. Two immiscible liquids lie between parallel plates and are heated from below. The focus is on the case where the interfacial mode is strongly stabilized by surface tension and a suitable density stratification. A mechanism for a Hopf bifurcation is the competition between the least stable of the bulk modes in each fluid. The well known criterion for balancing the effective Rayleigh numbers in both fluids is augmented with a criterion for non-self-adjointness of the system, yielding a heuristic method for picking suitable fluids when Hopf modes are desired. The pattern formation problem in three dimensions is addressed for the case of doubly periodic solutions on a hexagonal lattice. Of the solutions with maximal symmetry, the travelling rolls are found to be stable. Mathematics Subject Classification (1991). 76E15, 76E3. Keywords. Fluid dynamics, convection, bifurcation, pattern formation. 1. Introduction The BSnard problem is concerned with thermal instabilities that result when a layer of fluid is heated from below. In this paper, we examine the situation with two immiscible liquids in layers between two plates, heated from below. The equations governing this system, together with the interface conditions for a deformable interface, are written in full in Chap. III of Joseph and Renardy (1993), so only a brief summary is given in this paper. There has been recent interest among experimentalists (Rasenat et al 1989, Andereck et al 1995, Cardin and Nataf 1991) in observing an oscillatory onset that has been theoretically studied in the twolayer B~nard problem. The weakly nonlinear analysis in the literature applies to conditions close to criticality and this is pursued here. In order to set up an oscillatory onset, there needs to be a competition between two modes. Two Research supported by the Newton Institute (Cambridge), NSF Grant CTS and ONR Grant N14-92-J The author is grateful to D. C. Andereck and M. Renardy for discussions.

2 568 Y.Y. Renardy ZAMP such mechanisms are as follows. First, a bulk mode which is destabilized by the temperature difference between the plates may compete with a stabilizing interfacial mode due to a favorable stratification in the fluid properties (Renardy and Joseph 1985, Y. and M. Renardy 1985, M. and Y. Renardy 1988, Y. Renardy 1986, Wahal and Bose 1988, Fujimura and Renardy 1995). Contour plots in 2d for the vertical component of the velocity and perturbation temperature for the critical Hopf mode in this mechanism are shown in Fujimura and Renardy (1995), and show cells that extend the whole way between the plates; these rock back and forth or travel in a time-periodic manner. Secondly, when the interfacial mode is strongly stabilized, the competition between the least stable of the bulk modes associated with each fluid can lead to an oscillatory onset (Rasenat et al 1989, Gershuni and Zhukovitskii 1982, Colinet and Legros 1994). Contour plots for the critical Hopf modes for this mechanism are quite different from the other, in that there are cells in each fluid (see See. IV). This case appears at the moment to be more accessible to experiments (Rasenat et al 1989, Andereck et al 1995) and is the subject of this paper. The experiments of Andereck et al (1995) concern a layer of Fluorinert lying below a layer of silicone oil 47v1. The properties of these fluids are discussed in See. II C. Following the criterion for the effective Rayleigh numbers of each fluid to be equal (Rasenat et al 1989, Gershuni and Zhukovitskii 1982, Colinet and Legros 1994), it is found (Andereck et al 1995) that this system could have an oscillatory onset when the liquid depths are approximately.436. In choosing fluids for the purpose of observing such an onset, it is useful to have a rule of thumb so that a larger window of parameters would be involved, and with desirable short periods for ease of measurements. This is developed in Sec. III. In Sec. IV, a linear stability analysis of the Fluorinert - 47v1 system and the fictitious Anderinert - 47v1 system is described and the critical conditions are given. In Sec. V, the weakly nonlinear analysis for doubly periodic solutions on a hexagonal lattice (Roberts, Swift and Wagner 1986) is investigated. This methodology has been applied in Renardy and Renardy (1988) to the interfacial instability case. The results for the bulk-mode competition case are found to be qualitatively different from those of the interfacial instability case. For solutions with two degrees of freedom, eleven classes of solutions are analysed. It is found that the travelling rolls are stable while the other solutions are supercritical but unstable. This 3D analysis extends that of Colinet and Legros (1994) which concerns infinite Prandtl number liquids, equal thermal conductivities, and the density difference is large so that the interface is nondeformable. Since the thermal conductivities are equal, the perturbation temperature and temperature gradient are continuous across the interface. They derive amplitude equations for left and right travelling waves. They show that one of these generally appears rather than standing waves in sufficiently large cells. They show numerically that these waves have a limited range of existence because of a transition to stationary convection as Rayleigh number is increased. This reflects the property that the normal form analysis

3 Vol. 47 (1996) Pattern formation for oscillatory bulk-mode competition 569 has a limited range of application. In a weakly nonlinear analysis, the amplitude equations are derived on the assumption that certain terms are small compared with others. For example, the growth rate is assumed to be small. It can be seen from the data of Colinet and Legros (1994) and of Sec IV B that the growth rate may become the same order of magnitude as the frequency when the Rayleigh number is of order i above critical. This gives a rough estimate for the limit of applicability of the normal form equation, and is discussed in Renardy (1996). 2. Governing equations Two immiscible fluids lie in layers between two parallel plates in the (x*, y*, z*) plane. Asterisks denote dimensional variables. The upper plate at z* = l* is kept at a constant temperature ~), and the lower boundary at z* = is kept at a higher constant temperature ~ = ~ + A*. The location of the interface is given by z* = h*(x*, y*, t*). The average value of h* is denoted by l~, the average height of the lower fluid (fluid 1). The average height of the upper fluid (fluid 2) is l* - l~ = l~. The velocity is denoted by v* = (u*, v*, w*), the pressure by p* and the temperature by ". At temperature ~, fluid i has coefficient of cubical expansion &~, thermal diffusivity a~, thermal conductivity ki, viscosity #i, density Pi, and kinematic viscosity ui = ]ti/fli. S* is the dimensional interfacial tension coefficient. Dimensionless variables (without asterisks) are as follows: (x,y,z) = (x*,f,z*)/l*, t = lt*/1 v = o = (o* - p = h(x, y, t) = h*(x*, y*, t*)/l* - ll. There are six independent dimensionless ratios arising from the stratification in the fluid properties: Pl m:--, Pl r=--, fs1 7=--, kl ~=-~2, &l /~= =-' ll=l;/l*=l-12, P2 f12 l';2 ~2 where 12 = l~/l*. (2.1) There are four more independent parameters: a Rayleigh number R, a Prandtl number P, a dimensionless measure of the temperature difference between the plates &l A* which should be sufficiently small to be consistent with the Oberbeck- Boussinesq approximation, and a surface tension parameter. These are, respectively, R = g&lao*l*a/(~l~q), P = /21/t~1, &IAO* : RP/G, where G- g(l*)3 a~ ' (2.2) S = S*I*/(~;l#l). In each fluid, the governing equations are Fourier's law of heat conduction, the Navier-Stokes equations and incompressibility. The boundary conditions are zero velocity and constant temperature: = 1 at z =, = at z = 1. The interface is at z = 11 +h(x, y, t). The conditions to be satisfied at the interface are continuity of velocity, temperature and heat flux, and balance of tractions.

4 57 Y.Y. Renardy ZAMP A. Base solution A base solution to the problem is given by 1 - Alz for < Z < 11, (2.3) h =, v =, = A2(1 - z) for 11 < Z < 1, where 1 A1 - ll + ~12' A2 = ~A1. Note the corresponding pressure field is found in Joseph and Renardy (1993) and enters into the interface continuity conditions. B. Perturbation equations We denote by the difference between and the base solution (2.19), and by/5 the difference between p and the base solution. The perturbation stress tensor is denoted P[Vv + (Vv) r] in fluid 1, = P [Vv + (Vv) T] -/51 in fluid 2, where (Vv) T denotes the transpose of Vv. O<z</i, The governing equations are, for -- Alw -/k = -(v. V), (2.4) - PAv + V/5- RPOez = -(v. V)v, where A denotes the Laplacian operator and for 11 < z < 1, - A2w - = -(v. 7 - LpAv + rv/5 - R- Pez = -(v. V)v, m P together with div v =. The boundary conditions at the walls z = and z = 1 arev =, =. Let tl, t2 denote two unit tangent vectors and let n be a unit normal to the interface. By [.~ we denote the jump of a quantity across the interface, i.e. its value in fluid 1 minus its value in fluid 2. The interface conditions at z = ll + h are the continuity of velocity, shear stress, the jump in the normal stress is balanced by surface tension and curvature, continuity of temperature, continuity of heat flux, and the kinematic condition. These are, respectively, ~v~ =, [ti.t. n~ =, i = 1, 2, (2.5) [n. q2. n~ = -Mlh + M2h 2

5 Vol. 47 (1996) Pattern formation for oscillatory bulk-mode competition 571 psr, o h(l + lo m + + ( c9h'12~ h OhOh } [ Ox 2 ~'~ J ] Oy 2 \ ~ J ] OxOy Ox Oy (1 + (gh)2)3/2 EO~ = h(a1 - A2), [kn. V~ =, Oh h + + Vyy = w, where and M, = G(1-1)r + RPA212(-~ - 1), A2 _ A1). M2 = - (77 For the three-dimensional bifurcation analysis, terms up to third order are required in the computations. The equations (2.4) contribute quadratic nonlinearities. The interface conditions (2.5), expanded in Taylor series about the known position z = 11 and truncated, yield quadratic and cubic nonlinearities. These are lengthy expressions and can be found in Chap. III of Joseph and Renardy (1993) and Renardy and Renardy (1988). We denote by 9 the solution vector (u,v, w,~,o, h) and write the system of equations, boundary and interface conditions in the schematic form L(~) = N2(~, ~) + N3(~,~,~), (2.6) where L, N2, Na represent the linear, quadratic and cubic operators, respectively. C. Bulk-mode competition in the silicone oil/ Fluorinert system For the numerical results, we focus on the fluids used in Andereck et al (1995). Fluid 1 (bottom layer) is Fluorinert FC-7. Fluid 2 (top layer) is silicone oil Rhodorsil 47%. Their properties are listed in Table 1. We also consider a second two-layer system, consisting of the same silicone oil which overlies Anderinert, a fictitious fluid with 3.24 (or (9/5) 2) times the thermal diffusivity of Fluorinert; i.e. s = x 1 -a cm2/s. Anderinert was created by a software package used by Andereck et al (1995) to convert units for the thermal diffusivity to the cgs units. Since Anderinert creates conditions more favorable to Hopf bifurcation than true Fluorinert, we shall analyze both systems here. The total depth is l* = 1.26 cm. For the silicone oil and Fluorinert system, this yields P = 46.3, G = 1.65 x 1 l~ /3 = , 7 =.4651, r = , ~ = , m = The interfacial tension value is not known, but based on an estimate of S* ~ 2, we have S = x 15. For the silicone oil and Anderinert system, the parameters remain the same except for P = , G = x 19, 7 = , S = The effect of changing the value of interfacial tension from 1

6 572 Y. Y. Renardy ZAMP Table 1 Properties of the fluids. density g/cm a kinematic viscosity cm2/s thermal diffusivity cm2/s thermal expansion/k thermal conductivity g cm/k s a surface tension with air g/s 2 Fluorinert 47v1 pl = 1.94 p2 =.9273 Ul =.14 z~2 =.1 ~1 = ~2 = 8.61 x 1-4 &l =.1 &2 = 1.8x1-3 kl = 7x13 k2 = 1.3x to 4 makes only small changes to the least stable mode. At this magnitude, the interfacial tension strongly stabilizes the interfacial mode and its effect on the bulk modes, through the normal stress condition, is small. The effective Rayleigh numbers in each fluid are ^ **3 /g /j R1 = = 9 zxo q /( 2 2). (2.7) Since [kdo*/dz~ =, the ratio A;/A~ is 11/(~12). Using this, the ratio R1/R2 is (/~r)/(~mta4), where a = 12/11. When this is approximately 1, an oscillatory onset is expected, on the basis of the competition between the least stable of bulk modes in each layer. The depth ratio is = (2.s) This is the condition discussed in Sec. III of Colinet and Legros (1994). Their {#, u, a, ~;, p, A, a} correspond to our {m, re~r, fi, 7, r, C, 12/11 } respectively. For the silicone oil and Fluorinert system, ll = For the silicone oil and Anderinert system, a =.967 or ll =.58, so that for approximately equal depths, we expect a Hopf bifurcation. Oscillatory states occur only in small windows and may be difficult to find (Colinet and Legros 1994). 3. The oscillatory bulk-mode interaction In order to make observations of the Hopf modes in experiments, it is helpful if we can find a parameter that governs whether their period will be short or long, since long periods are difficult to measure. In this section, we develop a criterion for the "non-self-adjointness" of the system. This helps to compare one fluid system against another when choosing the component fluids with the aim of setting up oscillations. If a system is far from self-adjoint, then there is chance for a larger window of oscillatory modes and shorter periods of oscillation, whereas the closer the system is to self-adjoint, the more difficult it is to set up oscillations. Eq. (2.8) gives a rule of thumb for balancing the effective Rayleigh numbers in each fluid, for which on physical grounds, one may expect an oscillatory onset, but this is not enough information to guarantee that. For instance, we show below for the case

7 Vol. 47 (1996) Pattern formation for oscillatory bulk-mode competition 573 of infinite Prandtl numbers that it will not be oscillatory if fltr = 1 (the onset will be real). In order to look for oscillations that are easy to detect, one would require both (2.8) and that the system be far from self-adjoint. Since the silicone oil and Fluorinert possess large Prandtl numbers and the interracial tension stabilizes the interface, we turn to the case of P = oc, h =. The linear stability equations for modes proportional to exp(iax+c~t) are, dropping the tildes, (subscripts denote fluid i): a1 = Alwl + A1, (3.1) 1 ~~ = A2we + --A2, (3.2) 7 A2Wl -- R~I21 =, (3.3) Ra2m A2w =. (3.4) r9 This is the same as the system considered in Colinet and Legros (1994) except that they set the thermal conductivity ratio r = 1. We multiply (3.1) by A2/A11, where A2/A1 = 4, and (3.2) by 72 and add them. We multiply (3.3) by (A2~I)/(A1R) and (3.4)by (A2r/~2)/(A1Rm), and use these to eliminate the cross products such as Wl~. The result is integrated over x and z. The integration over x yields simply the factor 1. The terms at z = ll that arise from integration by I! It parts with respect to z are handled with wl = w2 =, (~ = O~,w 1 = w2, rnw 1 = W 2 ". The normal stress condition is passive in this case where the interface is not deformed. This yields + '1 + Z? + ( [Awll 2 + ]Aw2[ 2) + - (7/~r- -- (3.5) m ~ I)L Oz Oz 2 jz=ll. The boundary terms vanish when 7~r = 1, where 7 is the ratio of thermal diffusivities,/3 is the ratio of thermal expansions and r is the ratio of densities measured at the temperature of the top plate; the system is then self-adjoint and the eigenvalues are real. As the parameter 7r begins to differ from 1, one would expect that the imaginary parts of the critical eigenvalues in the situation with balanced effective Rayleigh numbers would increase, and the oscillations would have shorter periods and become easier to detect experimentally. This conjecture was put to the test for the silicone oil/anderinert system (Sec. II C), in which P is not infinity, and h is not exactly. Table 2 shows several criticalities for three cases, in each of which one of the parameters r,/3, "y is varied while the others are fixed. In

8 574 Y.Y. Renardy ZAMP Table 2 Critical conditions are shown related to the Anderinert/47v1 system of Sec. II C, for oscillatory onsets as the parameter ~Tr is varied from 1 to 5, while the effective Rayleigh numbers are balanced by Eq. (2.8). In case A, r and ~ are fixed as in the Anderinert/47v1 system and V is varied. In case B, 7 and r are fixed as in Anderinert/47v1 and/3 is varied. In case C, ~ and 7 are fixed as in Anderinert/47v1 and r is varied. This illustrates that as ~vr increases from i, the frequency increases. Case A Case B Case C , i , i , i , i , , i , i , i , real eigenvalues , i , i , i , real eigenvalues case A, r and/~ are fixed as in Sec. II C and "y is varied such that the parameter r/37 take on the values 1 to 5 as listed. For each entry, 11 is chosen so that the effective Rayleigh numbers are balanced as in (2.8). Thereafter, a search is done for the critical Rayleigh number Rc, the critical wavenumber ac and the critical eigenvalues at. For the self-adjoint case flvr = 1, the criticalities in the range.42 _< It _<.45 yield simple zero eigenvalues. In case B, 3' and r are fixed as in Sec. II C and/3 is varied. In case C,/3 and 7 are fixed and r is varied. For both cases B and C, 7 is the same, so that the values of I1 for balancing the Rayleigh numbers according to Eq. (2.8) give the same values even though the /3 and r are different: we choose (12/11) 4 = 1/(Cm7) times fir, where fir=l/7 times the integers 1 to 5 for the 5 cases. The combination/3r, rather than the/3 or r separately, is important in determining the stability in the situation where interface deformation is small. The effective densities at the unperturbed interface z = It are given by the Oberbeck-Boussinesq term pi(1 - (&ia*)) where A* is the temperature difference between the plates, and the constants pi and &i are measured at the temperature of the top plate. From this, the ratio of the density gradients at the interface 8p/, made dimensionless, is/3r. This difference in the density gradients appears on Eq. (3.4) in combination with the Rayleigh number, and is the term that affects the stability for BSnard convection. The difference in the densities, on the other hand, enters through the normal stress

9 Vol. 47 (1996) Pattern formation for oscillatory bulk-mode competition 575 Table 3 Critical conditions for the Fluorinert/47v1 system. ll Rc ac cr.39 21, E , E , E , E , E , E , E , E-3 jump condition in the case of interracial deformation, but is unimportant when the interface is almost nondeformed as here. The small differences in the eigenvalues for cases B and C are due to the small amount of deformation in the interface; the critical conditions are almost identical. At fitr = 2, there are only real eigenvalues at criticality. Table 1 provides some numerical evidence that the conjecture may be useful as a guideline for choosing fluids that would yield oscillations with desirable periods; this is not a proof. The situation presented in Fig. 2 of Colinet and Legros (1994) has m = 1, m/(r3) =.5, 7 = 2., so that ~Tr = 4, as compared to our system which is 2.5. Thus, their system is further from the self-adjoint situation than ours, and therefore we expect that they have larger Im a and a somewhat larger window for the Hopf modes, as borne out in the next section. 4. Linear stability results For the Fluorinert/silicone oil system, no Hopf onsets are found in the vicinity of 11 =.43. Modes do cross, for instance, Figure 1 shows the least stable eigenvalues for R = 246, a = 5.3 plotted against the depth 11. The vertical axis is the real part of the eigenvalues. The eigenvalues are complex conjugate pairs for 11 between.415 and.417 and these are stable. However, for a = 5.4 and higher, the modes do not cross though they become close, and they remain real. Table 3 shows the critical situations where the Rayleigh numbers in both fluids are almost balanced. It is possible to have a neutral Hopf situation by raising the Rayleigh number above criticality. For example, Figure 2 shows the real part of the eigenvalues versus wavenumber at R = 26,6,11 =.423. There are Hopf modes where the branches merge into one, for wavenumbers between 4 and The case a = , ~r = -.11E is a neutral situation. The mode of maximum growth rate is wavenumber 6.1, with a = Note that there is linear stability for wavenumbers greater than 8. Thus, the neutral Hopf mode may be important if the size of the apparatus were constrained so that multiples of that wavelength were excited. The parameter r which should be as far away from 1 as possible in order to optimize the window for Hopf modes, is.77, not good

10 576 Y. Y. Renardy ZAMP 4 "d MOO.1 5 I I L1.9-5 i i i L1 Figure 1. Plot of eigenvalues a versus depth 11, for R = 24,6, a = 5.3 for the Fluorinert/silicone oil system. The eigenvalues are complex conjugates for 11 =.415 to.417, where Recr is plotted. 2-2 ~ "~ R = 266, LI=.423, i i i i, ~ 1 i i i i wavenumber ] Figure 2. Graph of real parts of the eigenvalues versus wavenumber for R = 26,6,11 =.423 for the silicone oil/fluorinert system. news for Hopf onsets. The situation is more optimistic for the Anderinert/silicone oil system for which vflr is 2.5. The Anderinert system does have a window for Hopf onsets, and the application of the weakly nonlinear theory to that is perhaps more meaningful than at the neutral situation of the Fluorinert system.

11 Vol. 47 (1996) Pattern formation for oscillatory bulk-mode competition 577 Table 4 This is the list of critical conditions for the Anderinert/47v1 system of Sec. II C for.3 11 <.52. The first set of data concern real onsetsl the fifth column shows whether the cells are thermally coupled (TC), in transition, or mechanically coupled (MC). The second set of data concern the Hopf onsets which occur for.486 < II <.54. ~1 /r~c O~c O'c MC MC MC transition TC TC TC TC TC see Hopf onsets MC MC MC The following are Hopf onsets i i t i i E i E i Table 4 shows the critical conditions for the Anderinert/silicone oil system of Sec. II C in the range.3 _< ll _<.52. The interfacial mode is stable and the least stable modes are those associated with the bulk modes. Real onsets are found for 11 _<.486 and 11 _>.55. Hopf onsets are found for.486 < ll _<.54, so that the criterion for balancing the effective Rayleigh numbers (2.8) is close. Compared with this, the system in Colinet and Legros (1994) yields Hopf modes for.483 _< 11 <.52, a somewhat larger window because their system is more non-self-adjoint than ours. At 11 ~.486,.55, both steady and Hopf modes are simultaneously at onset, but with wavenumbers that lead to non-resonant interactions. This type of interaction has been documented in various flow fields (Fujimura and Kelley 1993, Chossat and Iooss 1994, Renardy et al 1996). The center manifold is 6-dimensional and the equilibrium solutions include the travelling wave, the steady mode, the standing wave, a symmetric mixed mode and an asymmetric mixed mode. Depending on the fluids, it may be possible to set up the wavenumbers for the steady and Hopf modes in a ratio 2:1, in which case the interaction would be resonant. In the interracial instability case, this type of resonant interaction can be set up by adjusting the thermal conductivity stratification to attain the Hopf mode at a

12 578 Y.Y. Renardy ZAMP Figure 3a. Contour plot for the perturbation temperature for the critical eigenfunction for the Anderinert/silicone oil system at ll =.42, R = 7984, ~c = 4.6, ~c =,29. The vertical coordinate is < z < 1, the horizontal coordinate is < x < 2~r/ac. desired wavenumber and then balancing the surface tension and effective density difference to attain the steady mode at twice that wavenumber (Fujimura and Renardy 1995, Hill and Stewart 1991). The two modes generate a two-parameter bifurcation. The resonance occurs through quadratic terms in the amplitude equations. The simple equilibrium solutions include the steady solution, the travelling waves, and the mixed standing waves. In the particular situation of Fujimura and Renardy (t995), there is a region of stability for the standing waves, and another equilibrium solution, the asymmetric mixed mode. Parts A and B below concern the Anderinert system. A. Steady onsets For.3 _< 11 <.42, the onsets show mechanical coupling; that is, a cell extends through the upper fluid and a cell extends through the lower fluid, and at the interface, they rotate in the same direction. An example is the case tl =.42, where the ratio of the effective Rayleigh numbers RI/R2 of (2.8) is/%14/(~m714) =.26, so that the upper fluid has a much higher Rayleigh number. Figure 3 (a) shows a contour plot for the perturbation temperature for the critical eigenfunction. The critical mode consists of rolls in the upper fluid, and the lower fluid is passively coupled. This type of mode is mentioned in Sec. 3 of Rasenat et al (1989). Figure 3 (b) shows the vertical component of the velocity w(, z) at x = versus the fluid depth _< z < 1, showing that w changes sign across approximately the interface position z -- ll and is of opposite sign in the fluids. The horizontal component of the velocity u(x, z) at x =.25 2~/ac versus z, shows no change in sign across the interface. These are symptoms of mechanical coupling. The picture of the temperature field in figure 3 (a) shows the dominant cells in the upper fluid, but in this situation where so little is going on in the lower fluid, either a velocity vector plot or the plot of w or u versus the depth of the

13 Vol. 47 (1996).2 Pattern formation for oscillatory bulk-mode competition L1 =.42, R = 7984, waveno. 4.6, eva , , z Figure 3b. The vertical component of the velocity versus fluid depth z at x =. fluid is needed to make the diagnosis of mechanical coupling. At R = 7, the same mode remains the mode of maximum growth rate. An examination of the velocity vector plot at 1l =.46 for the critical mode shows small cells in the lower fluid just below the interface position, and larger cells below them. This case may be considered to be a transition case, between mechanical coupling and thermal coupling. In thermal coupling, the cells in each fluid rotate in the opposite directions and u changes sign across the interface position. The contour plot for the perturbation temperature at 11 =.48 is similar to the case of figure 3 (a) even though it is thermally coupled, whereas figure 3(a) depicts mechanical coupling. In order to distinguish the thermal coupling, it is helpful to look at the vector velocity plot. In this case, it shows that cells in each fluid rotate in opposite directions; the small cells at the interface that adjust for this are ignored in this diagnosis. Figure 4 shows the perturbation temperature at x = versus the depth z for several values of I1; for the range ll <.486, the variation is similar whether the coupling is thermal or mechanical, and for 11 >.55, the variation shows the symptom of mechanical coupling with strong cells in each fluid. When the Rayleigh number is increased at 11 =.48, e.g., R = 15,, the mode of maximum growth rate becomes mechanically coupled. This trend is

14 .5 58 Y. Y. R.enardy ZAMP.15 Variation of perturbation temperature with z at x=o I I I I I I I I.1.5 II ~ c N I I I L I I I I z Figure 4. The perturbation temperature versus depth z at x = for the Anderinert/silicone oil system is compared for several values of ll. mentioned in Colinet and Legros (1994). The interracial mode is shown with the two least stable bulk modes in Figure 5, where the growth rate Re (~) is shown versus the wavenumber of the perturbation a for 11 =.48, Rc = 1312; this figure shows long waves since the interfacial mode is effectively stabilized for larger a by surface tension and the density stratification. As the 11 approaches.486, the two least stable bulk modes begin to branch together to form complex conjugates. Figure 6 shows a contour plot for the perturbation temperature (9 for the critical mode at 11 =.55, showing the type of mechanical coupling that is described in Fig. 1 of Colinet and Legros (1994). The upper cell is weaker at 11 =.52, which is still mechanically coupled.

15 Vol. 47 (1996) O( Pattern formation for oscillatory bulk-mode competition 581 R , L1 =.48 I I I l I bulk mode bulk mode cat).~ interfacial mode O - 12 p i, ~ I wavenumbers Figure 5. The growth rate Re cr versus wavenumber of perturbation a at the critical Rayleigh number R = 1312 for 11 =.48 for the Anderinert system. Figure 6. Contour plot for the perturbation temperature for the critical eigenfunction for the Anderinert system at ll =.55. The verticai coordinate is <_ z < 1, the horizontal coordinate is < x < 27r/ac. B. Oscillatory onsets Figures 7 (a-c) show the critical wavenumbers ac, critical Rayleigh numbers Rc and

16 582 Y. Y. Renardy ZAMP.... I I I ' I..... I - O Hopf onset real onset. ',., I..... i depth of lower fluid L1 Figure 7, Critical conditions for the Anderinert system are shown for the range where oscillatory onsets occur. (a) the critical wavenumber versus the depth of the lower liquid. imaginary parts of the eigenvalues IIm(cr)[ versus the depth of the lower fluid. Over the range of the Hopf modes, these show the linear variation of ac and Rc with l 1. The wavenumbers do not vary much and is the reason that a is fixed throughout in Fig. 2 of Colinet and Legros (1994), Our minimum period is achieved at 11 ~.495 and in dimensional units corresponds to 2~rl*2/([Im a[~l) ~ 4 minutes. The growth rates versus the wavenumber of the perturbation for 11 =.5 is shown in Figure 8 (a) for the critical situation, showing the two branches of real-valued bulk modes coalescing over the range 3 < a < 9 to form a complex conjugate pair. The frequencies are shown in Figure 8(b). A contour plot for the vertical component of the velocity w(x, z) reveals that there is similar level of action in both fluids. When the Rayleigh number is increased for an oscillatory onset case, the mode of maximum growth rate becomes real and mechanically coupled. For instance, at ll =.5, the Hopf modes are the only unstable ones for Rayleigh numbers up to 14,5 but by R = 14,8, real modes have larger growth rates than the Hopf modes. The situation at R = 15, is shown in Figure 9 (a-b) where the maximum growth rate mode is at a = 5,7, a = 5 and this is mechanically coupled with action in each fluid.

17 Vol. 47 (199s) Pattern formation for oscillatory bulk-mode competition i O Hopf onset real onset,9, I I I I , depth of lower fluid L1 Figure 7b. Tile critical Rayleigh number versus the depth of the lower liquid. 5. Weakly nonlinear analysis with double periodicity on a hexagonal lattice A. Spatial periodicity on the hexagonal lattice In the x-and y-directions~ the solution is assumed to be doubly periodic, e.g., (x + nix1 + nsx2, t) = (x, t) for every pair of integers (nl, n2), where the vectors xl and x2 span a hexagonal lattice of period W: Xl = W. (v~/2, 1/2, ), x2 = W. (, 1, ). The lattice obtained from this double periodicity is invariant under the symmetries of the hexagon; that is, rotation by multiples of 6 degrees, reflection across the vectors a~ defined by 47r 4~ 1 v~ O" al =- Wx/~(1,,), a2 - WV~( 2' 2 ' )' a3 = -al-a2, (5.1) and reflection across axes perpendicular to the ai. The same periodicity condition holds for, v, h and/3. These variables are expanded in Fourier series; e.g., oo O(Z, y, Z, t) : E ~kl[[z' ~e "xq-i/a2'x. (5.2)

18 584 Y. Y. Renardy ZAMP 3.5 J e. ~' I I I depth of lower fluid L1 Figure 7c. The frequency IIm(a)l versus the depth of the lower liquid. For the linearized problem (Sec. IV), the method of separation of variables yields kz = e~t(z), and similarly for v,/5, h. This leads to an eigenvalue problem for ~, in which the results do not depend on the direction of the vector kal + la2, but on its magnitude a: = Ikal +/a2[ - v~w47r (k kl) ~/2. (5.3) where the wavenumber a denotes the critical value determined in Sec. IV. Equation (5.3) then determines the period W of the lattice. The factor k 2 + l 2 - kl can be, 1,3,4,7,... The mean flow mode k = l = is not of interest in the linear problem, but will enter into the nonlinear interactions. The smallest nonzero value of k kl is 1, for which W is 4~/(v~a). This occurs for six possible pairs (k,/): (4-1, ), (, (1, 1), and (-1, -1), yielding a sixfold degeneracy of the corresponding eigenvalue. We pursue this case where the lattice size fits exactly into the critical period, and look at the nonlinear interaction generated by the 6 Fourier modes for Eq. (5.2). (If k kl is chosen larger than one, then we would seek solutions on a larger lattice with period a multiple of the critical one.) At criticality, there is a pair of complex conjugate eigenvalues +iw which has six eigenfunctions denoted (k, corresponding to the six values of (k, l). The eigenfunction computed in Sec. IV corresponds to the case of 1 =, k = 1 and is

19 Vol. 47 (1996) Pattern formation for oscillatory bulk-mode competition R = 13962, Hopf modes - i' real -15.~ real real modes -5 I I I I i ~ I 1 I wavenumbers Figure 8a. Growth rate versus wavenumber of the perturbation for the critical situation Re = 13962,/1 =.5, for the Anderinert system showing the two branches of real bulk modes coalescing to form the Hopf modes for 3 < a < 9. denoted r --- ~(z) exp(ial, x), where al -- (Ctc,,). Tile vectors 4-a2 and -t-a3 emanate from the center of a hexagon and terminate at its six vertices. The critical eigenfunctions are waves propagating in the directions of the vertices. B. Amplitude equations in normal form The parameter A denotes a bifurcation parameter, e.g., the difference between the Rayleigh number and its critical value /~ - Re. Close to criticality, the eigenfunctions Ck(A) depend on exp[-p(a)t]. At criticality, A =, #() = -icj and Re #~() <. Instability occurs for A >. We denote the complex time-dependent amplitude function of the wave propagating in the direction of ai by zi and the amplitude of the wave propagating in the direction of -ai by zi+3. In order to obtain an amplitude evolution equation for the weakly nonlinear analysis, Eq. (2.6) is reduced to a system of ordinary differential equations in the 12-dimensional space IR 12 by invoking the Center Manifold Theorem which says roughly that in the neighborhood of criticality q) =, the dynamics is governed by interactions

20 586 Y. Y. Renardy ZAMP R = 13962, L1 = o o I I... I, I I I I I wavenumbers Figure 8b. The frequencies are shown for the Hopf modes. among the six critical modes. This allows us to write the solution in the form = ~1 + ~2 + higher order terms, 6 6 ~71 "~- E Zi~i "~ E Zi~i, (5.4) i=1 i=1 6 ~2 z 2 Re ( E zizj~ij -~ zi2j~ij)~ i,j=l where the interaction terms r Xij are found by certain projections in the center manifold reduction scheme. The following normal form equation is obtained: dzi d--t -]-Fi(Zl'Z2'Z3'Z4'Z5'Z6'/\) :, i : 1,...,6, (5.5) where F~, i = 2,.., 6 can be retrieved from F1 by use of symmetry properties and +~3(A)(Iz2] 2 + ]z312)zl + c~4(a)(iz512 + Izal2)zl + c~5(a)(z2z5 +z3z6) , (5.6)

21 Vol. 47 (1996) 5 Pattern formation for oscillatory bulk-mode competition 587 R = 15, L1 =.5 [ I -5-1' "~ - 15 " I I I I I I wavcnumber alpha Figure 9a. Growth rate versus wavenumber of the perturbation for R = 15,, ll =.5, for the Anderinert system showing the two branches of real bulk modes coalescing to form the Hopf modes for 9 3 < c~ < 5 and the mode of maximum growth rate is real. where the dots denote terms of higher than third degree. The coefficients c~i are the Landau constants which need to be computed to determine the stability of bifurcating solutions. In Roberts, Swift and Wagner (1986), group theoretic methods are used to analyze solutions to general equations of the form (5.5). A classification is given for those time-periodic solutions with two degrees of freedom out of the full twelve degrees for the {zi}, i.e., one of the zi is free. Eleven qualitatively different classes of such bifurcated solutions are found: standing rolls, standing hexagons, standing regular triangles, standing patchwork quilt, travelling rolls, travelling patchwork quilt (1) which is never stable, travelling patchwork quilt (2), oscillating triangles, wavy rolls (1), twisted patchwork quilt, and wavy rolls (2). Illustrations of these solutions are found in Joseph and Renardy (1993). A complete discussion of the stability of the eleven solutions is given in Roberts, Swift and Wagner (1986). Note that a supercritical branch can be unstable in this situation where the critical eigenvalue is not simple. The theory has been applied to the case of interfacial instability in the two-layer Bfinard problem in Renardy and Renardy (1988) and Joseph and Renardy (1993).

22 588 Y. Y. Renardy ZAMP i R = 15, L1 =.5 i I I I wavenumber I I Figure 9b. The frequencies are shown for the Hopf modes. C. Numerical results For the numerical evaluation of the Landau coefficients, the code of Renardy and Renardy (1988) was used. They treat the two-layer problem with a deformable interface and focus on three mechanisms for an oscillatory onset due to the competition between the least stable of the bulk modes and the interfacial mode. In their situation, the temperature difference destabilizes a bulk mode, leading to a cellular motion that extends between the plates. This mode is counterbalanced by a stabilizing interracial mode under three separate mechanisms: (i) surface tension, (ii) a stabilizing local density difference across the interface and (iii) a favorable stratificaiton in thermal conductivity which optimizes heat transfer. Cases 2 and 3 of their Table I were reproduced. Their case I is probably in error; we have al = -.7E E-5i, a2 = -.9E E-6i, a3 =.15E E-6i, a4 = -.35E E-6i, a5 =.79E E-6i. This does not change their conclusion that there are no stable branches. In the cases they treated, there are a number of subcritical branches, including the travelling rolls. The results for the neutral situation for the Fluorinert system at R = 26,6, ll =.423 discussed in Sec. IV is that all eleven solutions are unstable. They are supercritical unstable except for the travelling patchwork quilt (1) and oscillating

23 Vol. 47 (1996) Pattern formation for oscillatory bulk-mode competition 589 triangles which are subcritical. Computations were done for the oscillatory onset conditions shown in Table 4, all of which yielded stable travelling rolls. The other ten solutions are supercritical and unstable. Unlike the interracial mechanism case, there are no subcritical branches. In the case of 11 =.495, we have ~ =.2E i, al = i, a2 = i, c~3 = i, O~4 : i, a5 = i. In the case of the two dimensional analysis, there is a twofold degeneracy of the critical eigenvalue (eigenvalues for a critical wavenumber a, and also for -a). The possible bifurcating solutions are standing rolls and travelling rolls (Ruelle 1973). If either is subcritical, then both are unstable. If both are supercritical, then one of them is stable. This analysis is performed in Colinet and Legros (1994) for their system at 7 = 2, m/(r~) =.5, m = 1, ~ = 1, a = 2.7, 12/11 =.95; they find that travelling waves are stable. Our analysis concerns the same mechanism of Hopf bifurcation due to the competition between bulk modes, but for different fluids, and extended to 3d where the stability of a variety of solutions is investigated. The qualitative agreement in our results may imply that travelling rolls are preferred by this mechanism. References [1] D. C. Andereck, P. W. Colovas and M. Degen, Multi-layer convection. Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Multi-Fluid Flows and Interracial Instabilities, Seattle (1995). [2] P. Cardin and H.-C. Natal, Nonlinear dynamical coupling observed near the threshold of convection in a two-layer system. Europhys. Lett. 14 (1991), 655. [3] P. Chossat and G. Iooss, The Couette-Taylor Problem. Springer, New York [4] P. Colinet and J. C. Legros, On the Hopf bifurcation occurring in the two-layer Rayleigh- B~nard convective instability. Phys. Fluids 6 (1994), [5] K. Fujimura and 1~. E. Kelly, Mixed mode convection in an inclined slot. J. Fluid Mech. 246 (1993), 545. [6] K. Pajimura, and Y. Y. Renardy, The 2:1 steady-hopf mode interaction in the two-layer B~nard problem. Physica D 85 (1995), 25. [7] G. Z. Gershuni and E. M. Zhukovitskii, Monotonic and oscillatory instabilities of a twolayer system of immiscible liquids heated from below. Soy. Phys. Dokl. 27 (1982), 531. [8] A. Hill and I. Stewart, Hopf-steady-state mode interactions with (2) symmetry. Dynamics and Stability of Systems 6 (1991), 149. [9] D. D. Joseph and Y. Y. Renardy, Fundamentals of Two-Fluid Dynamics, Springer Verlag New York (1993). [1] S. Rasenat, F. It. Busse and I. Rehberg, A theoretical and experimental study of doublelayer convection. J. Fluid Mech. 199 (1989), 519. [11] M. Kenardy, Hopf bifurcation on the hexagonal lattice with small frequency. Adv. Diff. Eq. I (1996), [12] M. P~enardy, Y. Renardy, R. Sureshkumar and A. Beris, Hopf-Hopf and steady-hopf mode interactions in a viscoelastic Taylor-Couette problem. J. Non-Newt. Fluid Mech. 63 (1996), 1. [13] Y. Renardy, Interfacial stability in a two-layer B~nard problem. Phys. Fluids 29 (1986), 356.

24 59 Y.Y. Renardy ZAMP [14] Y. Renardy and D. D. Joseph, Oscillatory instability in a two-fluid Bfinard problem. Phys. Fluids 28 (1985), 788. [15] M. Renardy and Y. Renardy, Bifurcating solutions at the onset of convection in the Bfinard problem of two fluids. Physica D 32 (1988), 227. [16] Y. Renardy and M. Renardy, Perturbation analysis of steady and oscillatory onset in a B~nard problem with two similar liquids. Phys. Fluids 28 (1985), [17] M. Roberts, J. W. Swift, and D. H. Wagner, The Hopf bifurcation on a hexagonal lattice. in: Multiparameter Bifurcation Theory, M. Golubitsky and J. M. Guckenheimer, Eds., AMS Series: Contemporary Mathematics, 56 (1986), 283. [18] D. Ruelle, Bifurcations in the presence of a symmetry group. Arch. Rat. Mech. Anal. 51 (1973), 136. [19] S. Wahal and A. Bose, Rayleigh-B~nard and interfacial instabilities in two immiscible liquid layers. Phys. Fluids 31 (1988), 352. Yuriko Y. Renardy Dept of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA U.S.A. (e-maih renardyy@math.vt.edu) (Received: July 24, 1995; revised: January 2, 1996)