Subject Index

Aa remarquable, (2.27), relation; 39

dimensionless form, (2.30); 40, 88adjustment-transient behavior; 50–52adverse conduction temperature gradient; 10,

11, 137adverse conduction gradient with TA; 112alternative; 2, 198Alfvén (A) number; 333amplitude equations; 150, 152, 154, 160,162, 248–250, 351–352, 359anelastic equations; 359–363

hydro-static case; 363Zeytounian dissipative system; 362, 363

approximate law state, 39asymptotic modelling; 104–108atmospheric thermal convevtion; 277averaged evolution equations; 209, 238–240

linear system; 240averaged IBL approach; 345–349

Bbasic adverse temperature gradient; 11, 79Bénard cells; 57Bénard problem in unbounded atmosphere;

319Benney equation; 223, 224

discussion; 223–225Biconv = B(H); 13, 204Biconv = B(T ); 13BH-C equations; 284, 285Biot numbers; 11BM equations: 200–204, 272–274BM instability; 197, 240, 349

BM long-wave equations; 207model problem; 211–214

BM problem with θ ; 252–254BM upper free surface conditions; 202–204body, electric, force f; 336Bois and Kubicki approach; 309Boltzmann’s constant; 35Bond (Bd) number; 64, 65, 197boundary conditions; 6Boussinesq approximation; 6Boussinesq hydrostatic convection

equations; 284–286Boussinesq number; 8breeze pronlem; 279–281breeze simple mechanism; 286, 287

two simple problem; 286–292Brunt-Väisälä frequency; 318, 329

CCharle’s law; 35characteristic equation; 217characteristic velocity Uc; 210chemical convection; 338, 339coefficient of thermal expansion; 7, 37

of isothermalcompressibility, β; 34, 37,56

coefficient χ ; 34coefficient of thermal expansion for water;

38, 56combined thermocapillary-buoyancy

convection; 168, 169comments concerning some recent

references; 377–384competition between hexagonal and roll

391

392 Subject Index

patterns; 258–260complementary references; 378–383conduction basic state Biot number; 11, 89conduction state; 10, 56, 93, 94conduction temperature �(z′); 199conduction temperature θ(z′); 200constant temperature Td ; 11constitutive relations; 5constitutive theory; 52, 53continuum regime; 3, 9–14convection Biot number; 11convection down a free-falling vertical liquid

film; 219, 220convection equations in atmosphere; 315

a simple case; 316, 317convection in the Earth’s outer core, 327–331convection in rotating cylinder; 342, 343convection over a curved surface; 298–300

Noe approach; 300–304Coriolis parameter; 24Cp; 32, 36, 37crispation (capillary) number; 16, 17, 95critical Rayleigh number; 193curvilinear coordinates; 288, 289cutoff wave number; 217, 227, 235Cv ; 33, 36

DDavis (1987) approach; 12–14Davis (1987) upper condition; 14, 46deep convection; 174

equations; 175, 176formula for Ra; 181linear theory; 177–181

matrix of Dff

; 178rigorous results; 189–192route to chaos; 182–188

deep thermal convection with viscousdissipation; 271, 272

dimensionless conditions; 103dimensionless reduced pressure; 15, 77dimensionless reduced pressure πs(z

′); 66dimensionless temperature θ ; 8, 14, 46, 90dimensionless temperature θs(z

′); 65dimensionless free surface; 93dimensionless temperature �; 10, 47, 89,198dimensionalization; 62, 133–135dispersion relation; 227, 245, 247

dispersion relation for Pr �= 0; 247dispersive similarity parameter; 233dissipation number Di = ε Bo; 8, 65, 271

of the specific kinetic energy; 311divergence of u; 5dominant equation for u′; 100dominant equation for θ ; 101dominant equation for �; 113double limiting process for anelastic

convection; 361dynamical system; 228, 235, 244

EE(t); 166Earth’s outer core convection 327–331

values; 330, 331EHD; 336

body force f; 336energy equation; 31enthalpy; 37entropy production inequality; 52epilogue: 383, 384ε2/ �≡ K0 = O(1); 88equation for the deformation of the free

surface; 16, 140equation for the temperature; 36

of a liquid; 39equation of state; 4, 30–32, 37, 43, 44, 80equation of state relative to TA and pA; 113estimation for E(t); 167estimation for the temperatures difference,

�T ; 21equation for the specific energy; 5, 31equation for T (dS/dt); 32, 34equation for ush with a term θ2

sh; 81estimation for the thickness, d , of the liquid

layer; 18, 20, 21evolution equations; 37, 40expansible liquid; 4, 38expansibility parameter, ε; 7, 20expansibility parameter, ε′; 198

Ffactor which affect breeze; 305, 306Feigenbaum period duobling scenario; 184–186FHD; 338film falling down (geometry); 126film falling down an inclined plane; 126–128

Convection in Fluids 393

Fourier law; 5four significant convections; 15, 102, 103free falling vertical film; 219–223free surface equation; 9, 41, 43free surface upper dimensionless conditions;

106–108Froude (Frd ) number; 2, 7, 87Froude (FrAd ) number; 198function �∗(χ); 213function �∗(H); 221function (χ); 214function (H); 222function q(t, x); 221

GGalileo number, Ga; 87gas constant; 36geometry of the Bénard problem; 93geostrophy; 292, 293Gibbs energy; 52Golovin et al. interaction approach; 349–351gradient; 4Grashof (Gr) number; 7, 18, 20, 63, 285

Hheat capacity; 32, 33, 36, 40heat flux (Fl); 312heat flux Rayleigh (Rl) number; 312Hills and Roberts’ equations; 53, 54

limit process; 75second-order model equations; 76

Howard & Krishnamurti DS; 160hydrostatic limit process; 282

equations; 282, 283hydrostatic parameter; 24, 282

IIBL isothermal model; 238IBL non-isothermal model; 239, 345–349initial conditions; 49–52isothermal coefficient of compressibility; 34

Jjump condition for T ; 12jump condition for θ ; 14

KKapitza (T ) number; 128Kazhdan computations; 352–354

Ki = (RF l)2/3Ta−5/6 Pe−1 � 1; 312

Kibel (Ki) number; 24, 310kinematic condition, 43Knudsen number, 163K0; 99Kronecker delta tensor; 4KS equation; 225, 256KS energy equation; 228KS–KdV equation; 233

DS system, 244generalized; 245

LLandau equation; 230

constant; 231leading-order system; 123limiting process; 16, 18, 21 137, 207

à la Boussinesq; 8, 66, 71DC; 18incompressible; 21N-ADH; 341quasi-hydrostatic; 24

linearization; 97, 98, 114linearized upper condition for �′ at z′ = 1;

114linear deep thermal convection theory;

176–181liquid hanging below a solid ceiling; 130liquid Mach number; 17local coordinates; 341local in time model; 208, 366local Prandtl convection model; 278local steady thermal convection problem;

294, 295long-wave (λ) parameter; 118, 282Lorenz dynamical system; 152, 154

strange attractor; 155–157lower bound for d; 7low Re and Ma theory; 231–233LS equation; 248–250

transition to chaos, 250, 251lubrication equations; 215, 216lubrication, one-dimensional equation; 217,

256lubrication theory; 196

MMach (atmospheric) number; 25Mach (liquid) number; 17

394 Subject Index

magneto-hydrodynamics; 331–336first integrals; 334quasi-steady limit equations; 334static equilibrium approximation; 334

magnetic induction, B; 333main effects in Bénard problem; 85Marangoni (Ma) number; 10, 128, 247Marangoni problem for film falling

down; 127, 128Ma with TA; 111matching; 50, 208material motion; 3Maxwell equations; 337Maxwell relations; 4, 32,33

for Cp − Cv ; 4, 33mean curvature; 41, 45mechanical pressure; 5, 31meso-scale prediction, (M-SP); 341, 342MHD convective equations; 333MHD equations; 333middle deck; 296model convection problem; 315, 316

simple problem; 316, 317modified Ma and We; 95mountain slop wind (Zeytounian); 319–322multi-scale approach; 145

NN , N1, N2; 202N-ADH limit process; 341Navier–Stokes equations; 30new coordinates and functions; 205, 206Noe approach; 300–304Newton’s cooling law; 10, 12, 14, 42, 220nonlinear stability for the deep convection;

190–192NS equation for ω = rot u; 311NS-F equations,

expansible liquid; 40for the Bénard problem, 96, 97thermally perfect gas; 37

NS-F 2D equations; 120Nusselt (Nu) number; 310

OOberbeck–Boussinesq equations; 357O–B simplified equations; 357, 358ocean circulation; 340–342Ohm’s law; 337

oscillatory convection; 307

Pparameter Bo′; 113parameter δ; 205parameters, W , W∗ and M ; 210parameter R∗; 207parameter F 2; 210parameters M∗ and W∗∗; 215parameters for atmospheric convection; 24Pearson approach; 115–117Pearson parameter L; 91, 116Pearson upper condition for �; 48penetrative convection; 343, 344Pellew and Southwell results; 69, 70phenomenological features; 304–306Pomeau–Manneville scenario; 187, 188Prandtl (Pr) number; 7, 64 201

mountain slope problem; 278pressure, p = (1/3)Tij

Qquasi-hydrostatic dissipative (Q-HD)

equations; 282, 283quasi-hydrostatic limiting process; 24, 282

RRa = ε Ga; 104Ra < Ta5/4; 313rate of change of surface tension; 9rate of loss Q(T ); 115rate of strain (deformation) tensor; 5, 30rate of viscous dissipation �; 5ratio, Cp/Cv = γ ; 33rational analysis; 104–110rational analysis and asymptotic modelling;104Rayleigh dimensionless problem; 62–66

conditions, 61Rayleigh equation for θ ; 64Rayleigh linear problem; 68–70Rayleigh number 7, 59, 63Rayleigh problem quations; 60RB convection patterns; 141, 142RB equations; 138, 140RB problem; 134, 135

in rarefied gases; 163–165RB rigid-rigid problem, 67, 68

second-order problem; 72

Convection in Fluids 395

RB standard model problem; 165, 166RB thermal shallow convection; 266–270reduced regularized model; 348, 349relation between, Td − TA and Tw − Td ; 13,

46relation between thermocapillay and

buoyancy effects; 259relationship between M∗∗, We∗ and R∗; 246Reynolds (Re) number; 50, 117Reynolds (Red ) number; 205Reynolds (Reλ) number; 207rigorous mathematical results; 189Rossby (Ro) number; 24, 283rotating RB convection; 308, 342Ruelle–Takens scenario; 183

Ssecond-order boundary layer solution for

temperature; 317second-order model equations; 72, 100, 101second-order model equations for RB; 143,

144short-scale – long-scale interaction; 349

amplitudes equations; 350attracteurs; 352, 353DS system; 351–354

short time for adjustment problem; 50similarity relations; 8, 16, 25, 77, 136, 210similarity rule between χ and α2; 40simple conduction problem; 93simple equation of state ρ = ρ(T ); 6simple-hump solitary waves; 375slope wind local problem; 288–292solar convection; 339solitary waves phenomena; 369–377

from IBL model; 376, 377head-on collision of the dissipativesolitons; 375homoclinic trajectories; 374localized structures, 374phase space representation;profile of a dissipative solitary wave;two-hump; 375

specific energy; 5specific entropy equation; 32specific entropy for a perfect gas; 37specific volume; 31sphericity parameter; 24squared sound speed; 34

stability results for KS equation; 226–231starting equations and conditions; 95, 96Stokes relation; 5strange attractors; 155–157, 159, 160,

183–188and intermittency; 186, 187by period doubling; 185from torii; 184infuence of the deep parameter; 188

stratification parameter; 285streamlines over a mountain slope; 302stress tensor; 3, 5Stuart–Landau (LS) equation; 244sub-critical instability; 166, 167surface gradient; 41surface tension; 9, 42, 48

TTakashima results; 241–243Taylor (Ta) number; 307, 307temperature �S(z′); 113temperature parameter τ ; 25thermal conductivity; 10thermal wind equation; 311thermally perfect gas; 4, 35, 36thermocapillary effect; 42, 196

convection with temperature-dependentsurface tension; 272–274

thermodynamic pressure; 5thermodynamic relations; 31, 34, 35, 37thermosolutal convection; 354–359thickness (lower bound); 7thickness dBM; 20thin liquid film over a rotating disk; 364–369

analyze via a lubrication equation; 368,369inner (in time) problem; 366outer equations; 364

three main facets of Bénard convection; 265three significant cases; 15, 102triple deck view point; 292–297typical values of Pr; 88

Uultra-thin film; 109unit outward normal vector; 41, 45unit tangent vectors; 44, 45upper bound for d; 136upper free surface conditions, 41–49, 76, 78,

396 Subject Index

202–204, 273, 274along the free surface; 78for the temperature; 12–14, 22, 42, 46, 48,199for the temperature θ ; 108, 200

linearized; 241for the temperature �; 112, 199for the motionless conduction state; 22for the pressure difference; 41, 44, 76upper deck; 296, 297

Vvector of rotation of the Earth; 24viscous dissipative function; 5, 32, 61viscous lower deck equations; 295, 296

WWe at TA; 111Weber (We) number; 10Weber (We∗) relations; 246

ZZeytounian anelastic dissipative equations;

362, 363Zeytounian approach for averaged IBL non-isothermal equations; 345, 346Zeytounian thermal atmospheric convection

approach (mountain slope wind); 320–322Zeytounian thermal deep convection; 172

equations; 174, 175