a r X i v :p a t t -s o l /9507001v 1 7 J u l 1995
Long-Wavelength Instability in Surface-Tension-Driven B´e nard Convection
Stephen J.VanHook ∗,Michael F.Schatz †,William D.McCormick,
niiJ.B.Swift,and Harry L.Swinney ‡
Center for Nonlinear Dynamics and Department of Physics
University of Texas at Austin,Austin,Texas 78712
(February 9,2008)
Laboratory studies reveal a deformational instability that leads to a drained region (dry spot)in an initially flat liquid layer (with a free upper surface)heated uniformly from below.This long-wavelength ins
tability supplants hexagonal convection cells as the primary instability in viscous liquid layers that are sufficiently thin or are in microgravity.The instability occurs at a temperature gradient 34%smaller than predicted by linear stability theory.Numerical simulations show a drained region qualitatively similar to that en in the experiment.
PACS numbers:47.20.Dr,47.20.Ky,47.54.+r,68.15.+e
B´e nard’s obrvation in 1900[1]of hexagonal convec-tion patterns launched the modern study of convection,pattern formation,and instabilities;yet understanding of the surface-tension-driven regime in which B´e nard per-formed his experiments is still far from complete.Block [2]and Pearson [3]first showed how temperature-induced surface tension gradients (thermocapillarity)caud the instability obrved in B´e nard’s experiments.Pearson’s linear stability analysis with a nondeformable liquid-gas interface yielded an instability at wavenumber q =1.99(scaled by the mean liquid depth d )and Marangoni num-ber M c =80,where M ≡σT △T d/ρνκ(e Fig.1)express the competition between the destabilization by thermocapillarity and the stabilization by diffusion.
A deformable free surface allows a cond type of primary instability in which a perturbation creates a nonuniform liquid depth and temperature profile [4,5].Thermocapillarity caus cool,elevated regio
ns to pull liquid from warm,depresd regions (e Fig.1).The instability appears with a long wavelength since sur-face tension stabilizes short wavelengths.Linear stability analys that allow for deformation reveal this instability (e Fig.2)at zero wavenumber (q =0)and M c =2驴羚
3G
<80,so the long-wavelength mode should
巧克口语become the primary instability (e Fig.2).This long-wavelength instability has not been experimentally inves-tigated.
In this Letter we describe experimental obrvations of the ont of the long-wavelength instability and com-pare the obrvations to linear stability theory.The instability leads to a large-scale drained region with di-ameter ∼100d .The qualitative features of the instabil-ity are compared to nonlinear theory bad on a long-wavelength evolution equation.We also explore the com-petition between the long-wavelength and hexagonal in-stabilities and study the physical mechanism that lects which pattern will appear.For a range of liquid depths,both patterns coexist.
d
不可能的英文d g L
FIG.1.A surface-tension-driven B´e nard (Marangoni)con-vection cell (not to scale)contains both a liquid (silicone oil)layer of mean depth d and local depth h (x,y,t )and a gas (air)layer of mean thickness d g .The mean temperature drop across the liquid layer is ∆T .The liquid has density ρ(0.94g/cm 3),kinematic viscosity ν,thermal diffusivity κ(0.0010cm 2/s),and temperature coefficient of surface ten-sion σT ≡|dσ/dT |(0.069dynes/cm ◦C).
01234
200
400
M
wavenumber (q)
FIG.2.Marginal stability curves for layers of 10cS silicone oil at two thickness:d =0.1cm (---),hexagonal convection cells form at M c =80with q =1.99(G =104);d =0.01cm (—–),the long-wavelength (q =0)instability forms at M c =6.7(G =10).
We study a thin layer of silicone oil that lies on a heated,gold-plated aluminum mirror and is bounded
above by an air layer(e Fig.1).A single-crystal sap-phire window(0.3-cm-thick)above the air is cooled by a temperature-controlled chloroform bath.The temper-ature drop across the liquid layer is calculated assuming conductive heat transport[7]and is typically0.5–5◦C. We u a polydimethylsiloxane silicone oil[8]with a vis-cosity of10.2cS at50◦C.The circular cell(3.81cm inner diameter)has aluminum sidewalls who upper surface is made non-wetting with a coating of Scotchgard.The ex-periments are performed with0.005cm<d<0.025cm and0.023<d g<0.080cm(typically d g=0.035cm);the corresponding fundamental wavevectors are in the range 0.008<q=2πd/L<0.040(≪1).The liquid layers are sufficiently thin that buoyancy is negligible[9].The gap between the sapphire window and the mirror is uniform to1%,as verified interferometrically.The liquid surface is initiallyflat and parallel to the mirror to1%in the central90%of the cell,with a boundary region near the sidewalls due to contact line pinning.This initial depth variation is accentuated by thermocapillarity as∆T is incread;measurements of depth variation show a10% surface deformation(in the central90%of the cell)at 3%below ont.For visualization,we u interferome-try,shadowgraph and infrared imaging(256×256InSb staring array,nsitive in the range3-5µ
m).
FIG.3.Dry spot in both experiment and numerical simu-lation.The experimental picture shows the measured bright-ness temperature as a function of position along the interface just above(∼10%)ont;d=0.011cm.Numerical simula-tion of a long-wavelength evolution equation shows the depth as a function of position for1%above ont of linear insta-bility.The depth of the liquid goes to zero in the drained region.
Above a critical∆T,the liquid layer becomes unsta-ble to a long-wavelength draining mode that eventually forms a dry spot[e Fig.3(a)].The drained region takes veral hours(of order a horizontal diffusion time, L2/κ)to form.The dry spot is not completely dry since an adsorbed layer∼1µm remains[10].The size of the drained region is typically one quarter to one third the diameter of the entire cell.
2019年12月四级答案Our measurements for the ont of instability are com-pared to the prediction of linear stability theory[6,11,12] in Fig.4.The results are given in terms of the dynamic Bond number,B≡G/M=ρgd2/σT∆T,which is the relevant control parameter for the long-wavelength insta-bility;B is a measure of th
e balance between gravity’s stabilizing influence and thermocapillarity’s destabilizing effect.Figure4shows that the measured critical values of B−1are independent of liquid depth and viscosity,as predicted by theory.However,B−1c in the experiment is34%smaller than predicted.We do not believe this discrepancy is due to systematic errors in the character-ization of our ,geometry,fluid proper-ties)since experiments in the same convection cell using thicker liquid layersfind ont of hexagons in agreement with another experiment[13]and linear stability theory [3].
B
-1
d(cm)
FIG.4.Measurements of ont compared to the theoretical prediction,B−1c=2
hexagons do not form at the ont of the long-wavelength mode,but can form in the elevated region for ∆T suffi-ciently above ont.Similar mode competition phenom-ena have
been studied theoretically for solutocapillary
convection [14].When d ≪d c ,increasing ∆T above ∆T c increas the area of the dry spot;no qualitatively new structures are obrved.At fixed ∆T ,the dry spot is stable.
FIG.5.Infrared images from experiments with increasing fluid depth (increasing G ).(a)10%below ont of instability (d =0.033cm).(b)For d =0.011cm,the long-wavelength mode is the primary instability.The dark region is the dry spot.(c)At d =0.022cm,the long-wavelength and hexago-nal modes coexist.A droplet (white circle)is trapped within the dry spot (dark oval);the liquid layer is strongly deformed (white annulus)between the dry spot and the hexagonal pat-tern.(d)For d >0.025cm,hexagons are the primary insta-bility (here,d =0.033cm).
无忧雅思网To understand the long-wavelength instability more fully,we study a long-wavelength evolution equation de-rived by Davis [15,16]for an insulating upper boundary and later considered by other authors [17,18]for more general upper boundary conditions.The evolution equa-tion for the local liquid depth h (x,y,t )(scaled by d )with an insulating upper boundary is 3
2B
h 2
魔兽英文名字
▽h −h 3
▽h +
(2π)2
3,
公务员考试复习资料at which point it disappears since the
unstable solution past that point is unphysical (h <0);nowhere does the backwards unstable curve tur
4级分数线n over in a saddle-node bifurcation.Simulations using more gen-eral upper thermal boundary conditions [11]also find no stable deformed solutions.
In conclusion,we have obrved the formation of a large-scale dry spot due to a long-wavelength deforma-tional instability.The formation of such dry spots could be a rious problem in the planned u of liquid layers (even as thick as one centimeter)in microgravity environ-ments,where fluid motion is driven primarily by surface tension gradients (buoyancy effects are negligible).
The structure of the dry spot agrees qualitatively with numerical simulations.Linear stability theory correctly predicts the functional dependence of the ont on ex-perimental parameters,but the ont in the experiment
occurs at B −1
c (or ∆T c )34%smaller than predicte
d by linear theory.This discrepancy in th
e ont may be due to the difference in lateral boundaries between theory and experiment.For example,all theory to date has as-sumed periodic boundary conditions,which become sus-pect when
the wavelength o
失落的战场f the mode under consider-ation is of order the system size.The long-wavelength evolution equation is not valid near a boundary and it is not clear what boundary conditions other than periodic might be ud for the evolution equation.Future nu-
merical simulations employing realistic,finite horizontal boundaries may answer this question.
We thank S.H.Davis,R. E.Kelly,and E.L. Koschmieder for uful discussions.This rearch is sup-ported by the NASA Microgravity Science and Appli-cations Division(Grant No.NAG3-1382).S.J.V.H.is supported by the NASA Graduate Student Rearchers Program.
3
(1+H),where the Biot number
H characterizes the thermal properties of the liquid-gas
interface.In the long-wavelength limit,H=k g d/k f d g, where k and k g are the thermal conductivities of the liq-uid and gas,respectively[e L.Hadji,J.Safar and M.
Schell,J.Non-Equilib.Thermodyn.16,343(1991)].H ranges from0.01to0.08in our experiments.Values in Fig.4include the(1+H)correction factor[that is,they are B−1c/(1+H)].The pattern and the topology of the bifurcation diagram are independent of H.
[12]The results are for an infinite system.Afinite system
has a lowest order mode of q=2πd/L,which becomes unstable at a slightly higher(6%)value of the control parameter.We do not include this correction in Fig.4.
[13]M.F.Schatz,S.J.VanHook,W.D.McCormick,J.B.
Swift and H.L.Swinney,submitted to Phys.Rev.Lett.
[14]A.A.Golovin,A.A.Nepomnyashchy and L.M.Pismen,
Phys.Fluids6,34(1994).
[15]S.H.Davis,in Waves on Fluid Interfaces,edited by R.
E.Meyer(Academic Press,New York,NY,1983),p.291.
[16]S.H.Davis,Annu.Rev.Fluid Mech.19,403(1987).
[17]B.K.Kopbosynov and V.V.Pukhnachev,Fluid Mech.
Sov.Res.15,95(1986).
[18]A.Oron and P.Ronau,J.Phys.II(France)2,131
(1992).
