Rigorous bounds on scaling laws in fluid dynamics Felix Otto Notes written by Steffen Pottel Contents Main Result on Thermal Convection Derivation of the Model. A Stability Criterion........................ Dynamics and Boussinesq Aroximation........... 6.3 Conservation Laws and Dissiation................4 Boussinesq & Dissiation.....................5 Nondimensional Parameters and Nondimensionaliation... 3 Sketch of the Proof 3 3. Identities and Inequalities Involving Nu and Ra........ 4 3. The Constantin-Doering 96 Bound............... 6 3.3 The Constantin-Doering 99 Bound Pr =......... 6 3.3. Ignoring Logarithms.................. 6 3.3. Rigorous Argument [à la Choffrut-Nobili-O 4].... 7 3.4 Our New Bound......................... 3.4. Ignoring Logarithms.................. 3.4. Rigorous Argument.................... Main Result on Thermal Convection Consider a fluid between two arallel lates with unit distance, heated from below and cooled from above with unit temerature difference. =, u =, T = t T + u T T = Pr t u + u u u + = Ra T e u = =, u =, T = {{ eriod L
Here we denote the temerature by T, the Prandtl number by Pr, the velocity by u, the Rayleigh number by Ra and the vector of unit length in direction by e. Furthermore we note that the term Ra T e relates to the exansion of hotter fluid articles, thus lighter articles, while in contrast we assume incomressibility of the fluid by u =. We write t T + T u {{ T = heat flux T u T e uwards heat flux and the average uwards heat flux, the Nusselt number Nu, is given by where Nu.= T u T e d f = lim su t t t L d [,L] d fdx dt Theorem A. Choffrut, C. Nobili, FO 4 + []. Indeendently of initial data and for Ra { Ra ln Ra 3 for Pr Ra ln Ra 3 Nu Ra ln Ra for Pr Ra ln Ra 3 Pr Remark. Here and in the following the symbol means u to a constant only deending on the dimension d. Derivation of the Model. A Stability Criterion Consider a liquid in local thermodynamic equilibrium described by ε = εv, s ρ = v with internal energy er mass ε, secific volume v, secific entroy s and density ρ. Here we assume that ε is strictly convex and smooth. In the gravity field g = g the energy is given by E = ρε + gdx
It is our goal to comute and interret first and second variation under infinitesimal erturbations described by velocity field u, i.e. Clearly, this will involve the ressure and the temerature t ρ + ρu = t s +u s =.= T.= ε v s ε s v where A means that the quantity A is held fixed. With the following claims, we establish a condition on the temerature such that the system is stable under infinitesimal erturbations.. The time-derivative of the energy is given by de dt = ρg u dx. At initial time, we have de dt = t= 3. If de dt t= = then d E dt = t= v ε v s {, ρ only deend on & d d = gρ u + v ε v s + g 4. For the convexity of the energy E, we obtain d E dt t= s u g v ε v s v s v s u g dx s 5. Reexressing the revious condition by erforming v, s, T s v s v T v T 3 g s v + T {{ > T
6. If additionally α > ln T αg c with α thermal exansion coefficient and c secific heat caacity. Now tells us that the temerature cannot decrease too fast as a function of height. Argument for : Enough to show Indeed t ρ + ρu = v= ρ t ρ + ρu = t ε + u ε = v u t ρε + g + ρε + gu = u ρg u t v + u v = v u t ε + u ε = ε v u t s + u s = v = t ρε + ρεu = u t ρ + ρu = t ρg + ρgu = g u = ρg u Argument for : Clearly { { d E = = dt = = ρg = gρ = gρ { ρ = ρ t= Argument for 3: By comuting d E dt = ρg t u dx + t u t ρ u gdx d E dt = t u + t ρ u gdx t= it is enough to show t u + t ρ u g = t u v t v u g ε = v u + u g v s v ε v s + g v ε v s This follows from t v + u v = v u ρ=ρ t v = dv u g + v u v=v g d t s + u s = t + u = v v u = t = s g 4 d u g + v d s v s u g u v s
Hence t u v t v u g = quadratic exression in u u g Use further = v s Argument for 4: Clear from Argument for 5: Because of ε v s >, ε v s >. s d = s T d d = g v d d = d d = g V v s dv d + s T + T ds s v d. it is enough to show v >, s v T s T v = T This follows from Legendre transform calculus. Consider the Gibbs free energy Because of φ.= ε + v T s ε + φ = v + T s = ε v + v s ε s s v we see that φ is the Legendre transform of ε in variables, T. The benefit of changing from extensive variables s and v to intensive variables and T can be seen by the hysical meaning Hence = ρg mechanical equation & T {{ = φ + ρ = thermal equation on level of first derivative φ v = T = φ, s = T φ T = φ T in articular s T = φ v T = T 5
on level of second derivative in articular = ε v ε s v ε s v ε s ε v s v ε s s v ε v v T > φ φ T φ T φ T φ v = T v φ T T φ s v T > Argument for 6: The thermal exansion coefficient ln v α.= = v > tyically T v T = = describes the relative volume change if temerature increases by K, while the secific heat caacity at constant ressure w c = w = ε + v = φ + T s enthaly form of heat T s =T T describes the heat er mass taken u when temerature is increased by K. Hence v v g s T T v + T = αv αv g T v + c T T ln T = αv αg + c. Dynamics and Boussinesq Aroximation We derive the dynamics from Newton s axiom mass acceleration mass acceleration volume ρ t u + u u = force = force volume t u + u u = ρ + g = first variation of E = ρg Since we noticed in the revious section that it is reasonable to work with the variables and T instead of s and v, we are interested in t ρ + ρu = t s + u s = t u + u u = + g ρ 6 we want to get to u = t T + u T = t u + u u + ρ = αt T g 3
with T reference temerature and ρ density at temerature T. On the right hand side of 3, the Boussinesq aroximation is already imlemented. In the aroximation for our variables s and v, the density variations are negligibly small excet when they are multilied by the gravitational acceleration g. How this is incororated in the descrition by the variables and T is derived in the following five stes.. By a change of variables v =, s, T and the hysical interretation of the ρ Jacobian we write ρ ρ T s s T T = v v T = = φ v > T v v T = c c=seed of sound at constant temerature = αρ = v = α T T ρ = c T dρ = d + αρdt c ds = α ρ d + c T dt. Provided ρc osc αosc T then we have to leading order dρ = αρdt, ds = c T dt 3. dρ = αρdt and ds = c dt imly T t ρ + ρu = t s + u s = { u = t T + u T = 4. Provided ρc osc αosc T then we obtain ρ + g = ρ αt T g 5. Let h denote the height of the system. Then free fall seed = gh seed of sound c α osc T = relative variation in secific volume is consistent with osc αosc T. 4 ρc 7
The requirement given by 4 is the condition for the Boussinesq aroximation exressed by and T. It states that the ressure variation relative to the density is small in comarison to the total thermal exansion, where the latter is small itself. This means hysically that we may only consider terms roortional to g describing relative forces namely buoyancy among fluid cells of unit mass with different temeratures, thus different exansions. Argument for : That the seed of sound c at constant temerature is given by c = = ρ ρ T follows from T t ρ + ρu = t u + u u + ρ = g tρu + ρu u + = neglect neglect t ρ + ρu = t ρu + = = t ρ ρ = ρ ρ T ρ T if thermal equilibrium i.e. T = =c Argument for : Clearly osc αρosc T c means that αρdt dominates d likewise c α ρ osc c T osc T means that c dt dominates α d. We note that. imlies. once we have T ρ This follows from det = det = v c α ρ c T φ φ T φ φ T T v αv c αv c T c T α c αρ α T c c = det = v c c T αv v v T Argument for 3: Taking dρ = αρdt and ds = c dt, we have T T v T s T t ρ + ρu = t +u ρ + ρ u dρ= αρdt = αρ t +u T + ρ u t +u s ds=c T = d c T t +u T 8
Argument for 4: Comuting first dρ = αρdt ρ = ρ ex T T αdt we write = ρ g + = ρ g + and hence ρ T + g = ex ρ T αdt {{ α osc T {{ αt T Argument for 5: In 4 we wrote = ρ g + which imlies osc ρ gh + osc Hence if we have osc ρ gh then as desired osc ρ gh gh osc ρc c α osc T Why osc ρ gh? Because we exect balance between which imlies dimensionally hence as desired ρ αt T g hρ osc α osc T g g osc ρ gh. 9
.3 Conservation Laws and Dissiation Going back to full system without dissiation t ρ + ρu = t s + u s = t u + u u = ρ. We have three conservation laws for the system. mass t ρ + ρu = entroy t ρs + ρsu = energy t ρ u + ε + g + ρ u + ε+gu + u =. In order to incororate dissiation into our model, we have to take inner friction & heat conduction into account. Our hilosohy is keeing mass and energy conserved, but giving u on entroy conservation. For the dynamics we write an additional force er volume t u + u u = ρ + g + σ ρ with stress tensor σ to be secified. Then based on mass conservation and force equation t ρ u + ε + g + ρ u + ε + gu + u u σ = σ : u + T t ρs + ρsu Hence we want T t ρs + ρsu + q = σ : u Then t ρ u + ε + g + ρ u + ε + gu + u u σ + q = and t ρs + ρsu + T q = T T q + T σ : u 3. The second law of thermodynamics states roughly that in a thermodynamical rocess the total entroy is increasing. Hence it follows for our system that the dissiation term is ositive, i.e. inner friction and heat conduction cannot create energy locally. For the stream rate e = u + u, we have σ = η trace-free art of e + ζ comlement
η and ζ dynamic viscosities Then = η u + u u id + ζ u id 3 σ : u = η u + u + ζ u With the thermal conductivity κ, we write Then Hence q = κ T T q = κ T t ρ + ρu = ρ t u + u u + σ = ρg T ρ t s + u s + q = σ : u This follows from the identity: u σ = u σ u : σ conservation of mass balance law for kinetic energy force balance t ρ u + ρ u u + u + u σ = u σ : u conservation of mass balance law for internal energy ε t ρε + g + ρε + gu = u + ρ t s + u s s v = tρs+ ρsu =T.4 Boussinesq & Dissiation. Recall the simlification from the Boussinesq aroximation t ρ + ρu = ρ u = t s + u s = c t T + u T = Hence ρ + g ρ αt T g for = gρ + ρ u = ρ t u + u u + c ρ t T + u T + = η u+ u {{ σ = ρ αt T g q = κ T c ρ tt T +u T T κ T T = σ : u = η u+ u
. In Boussinesq regime, σ : u is negligible. More recisely, σ : u is negligible w.r.t. ρ c T T u Argument: force balance kinetic energy balance t ρ u + ρ u u + u u σ = σ : u ρ αt T u g Hence exect in sace-time averaged sense and our claim reads Dimensionally, this is true for which follows from 3. Freeing in η and κ: η u + u = σ : u ρ αt T u g ρ αt T u g negligible w.r.t. ρ c T T u gh c αgh c αg c h α osc T αt & c α T c α t gh c αt c gh α osc T ρ u = ρ t u + u u + η u = ρ αt T g c ρ t T + u T κ T =.5 Nondimensional Parameters and Nondimensionaliation For our system it turns out that rescaling the equations is of advantage, i.e. working with quantities relative to some reference unit. Therefore we write t u + u u + ρ η ρ t T T + u T T κ ρ c =χ =ν u = u = αt T g T T = with ν kinematic viscosity and χ thermometric conductivity. Furthermore we introduce a material number, the Prandtl number Pr, Pr.= ν χ
Nondimensionaliation: Then where = h, T = T =, T = T x = hˆx t = h χ ˆt u = χ hû T = T + T T ˆT value at =h = ρ χ h ˆ ˆ û = Pr ˆt û + û ˆ û + ˆ ˆ ˆ û = Ra ˆT ê ˆt ˆT + û ˆ ˆT ˆ ˆT = Ra = αgt T h 3 χν 3 Sketch of the Proof Rigorous results for Ra : Constantin & Doering 96 [3]: Nu Ra for all Pr Constantin & Doering 99 []: Nu Ra 3 log 3 Ra for Pr = + Doering & O. & Renikoff 6 [4]: Seis & O. [5]: Nu Ra log Ra 3 for Pr = + Nu Ra log 5 Ra Xiaoming Wang 7 [6]: Nu Ra 3 log 3 Ra for Pr Ra 3
Seis & O. [5]: Nu Ra 3 log 3 log Ra for Pr = + Choffrut & Nobili & O. 4 []: Nu { Ra log Ra 3 for Pr Ra log Ra 3 Ra log Ra for Pr Ra log Ra 3 Pr 3. Identities and Inequalities Involving Nu and Ra From u =, T = t T + u T T = u = u =, T = { t T + T u T = u =. T t= T via maximum rincile. Horiontal average = L d...dx u = u = vanishes at =, t T + T u T e = osc t bounded Nu = 3. On the one hand Nu = T u T e d T u T e d T u d = t = lim su t + Helful because of no-sli boundary conditions T u T e dt t t t T u T e dt d T u d + T = u d + at = u =, u = u = 4. On the other hand Nu = T u T e = = T u T = = T = 4
Test with T: t T + T u T + T = t T + T u T + T = T T t T d T = + T d = {{ bounded Nu = T d From u = Pr t u + u u u + = Ra T e u = u = Energy inequality test with u, Pr u d {{ neglect + d u d Pr u t= d + Ra T u d = T u T e d Get u d Ra T u T e d = RaNu 5
3. The Constantin-Doering 96 Bound Nu T u d + T d u d + Cauchy-Schwar π Poincaré π t T d T d T d Nu RaNu + π = π Nu RaNu 4 π Nu Ra 4 π Ra Poincaré π u d t u d u d + 3.3 The Constantin-Doering 99 Bound Pr = u = u + = Ra T e = f u = u = 3.3. Ignoring Logarithms Princile of maximum regularity, alied to the stationary Stokes equation For any well-behaved norm in sace u + f Unfortunately, f = su << f is not a well-behaved norm. Suose it were. Then su u Ra << 6
No-sli boundary condition & Poincaré inequality u su u << d 6 su << u Ra Hence Nu u d + Ra + Otimie in = Ra 3 Nu Ra 3 3.3. Rigorous Argument [à la Choffrut-Nobili-O 4] u = u + = f u = u = Suose f is horiontally banded, i.e. F f = unless R k with F horiontal Fourier transform F fk, = L d fx, e ik x dx [,L d and the horiontal length scale R. Then indeed the solution for Stokes CNO su << u su << f Now given r, and N N, R = N r, we introduce the oerators rojections via F P < f = χ R ε <F f low ass filter P <, P,..., P N, P >, F P > f = χ r ρ >F f high ass filter F P j f = χ < j R = N j r 7 k <F f
By construction, N P < + P j + P > = id j= P <, P j, P > symmetric w.r.t. Hence T u d = T P < u d N + T P j u d + P > T u d j= Middle length scales P j u solves Stokes equation with r.h.s. P j Ra T e = RaP j T e Hence T Pj u Pj u su P j u P j u su RaP j T e =Ra P j T Ra Pj T Ra T and thus N j= T P j u d N Ra Small length scales P > T u d r P> T d r T r T d T d u d π u d u d 8
Large length scales T P < u d T d π T d P< u d {{ π P <u d Since P < u solves Stokes equation with r.h.s. Ra P < T e we have in articular and thus Hence Altogether: = P < u = P < u + P < u P < u d = T P < u d R Nu T u d + T P < u d r + R = T d P < u d R u π R + {{ Nu N j= r + Ra Nu +N Ra + R T d + u d u d T P j u d u d {{ RaNu P > T u d + N Ra + + Otimie for R by Then r = N, R = N so that R = N r Nu N Ra Nu +N Ra + 9
Otimie in N N by so that ln Ra 3 ln N ln Ra 3 ln + N Ra 6 and N ln Ra Then Nu Ra 3 Nu + Ra ln Ra + Otimie in < by = Ra ln Ra 3 Then Nu ln Ra 3 Nu +Ra ln Ra 3 and thus Nu Ra ln Ra 3 For later use, even the large scale cut-off R is small R = N Ra 6 Ra ln Ra 3 = Ra 6 ln Ra 3 3.4 Our New Bound 3.4. Ignoring Logarithms Starting oint Nu u d + For estimate on u, slit u into two solutions to nonstationary Stokes and Pr t u CD u CD + CD = f CD.= Ra T e u CD = Pr t u NL u NL + NL = f NL.= Pr u u u NL =
Princile of maximal regularity, alied to nonstationary Stokes. For any well-behaved norm in sace & time w.l.o.g. no initial data Unfortunately, Pr t u + u + f f CD = su f << f NL = f d where here f = t f dt t are not. Why do we want to work with these norms? f CD CD = Ra su T Ra << f NL NL = Pr Pr d u u = d + d u d + u d u d 5 Now we use Hardy s inequality. = d d u d = u d 4 u ud u d d u d Hence 5 cont d. 4 Pr 4 Nu Ra Pr in u d t u d 4 Pr u d Suose CD and NL were well-behaved norms. Then u NL d su Ra << Nu Pr Ra in limit t
Then by no-sli boundary conditions for u CD and u NL no-sli Hence in the limit t : Otimie in u d u CD d + u NL d u NL = Ra + Nu Pr Young with exonents 3, 3 : su u CD + su su u CD + Ra + Nu Pr Nu Ra + Nu Pr 3 Nu Ra + Nu 3 Ra Pr Nu Ra 3 + Ra Pr u NL d u NL d in limit t + 3 + Ra Pr 3 Nu 3 { Ra 3 for Pr Ra 3 Ra Pr for Pr Ra 3 3.4. Rigorous Argument Proosition CNO. Consider the nonstationary Stokes equation Suose f is horiontally banded, i.e. Pr t u u + = f u = u = {, u = t = F f = unless R k 4 for some R. Then Pr t u u + u + Pr t u + u + u + f,
where f = inf f=f +f { su f << = f CD + f d {{ = f NL. Of the roosition, we just need u f On the one hand, we have u d su u su u << << On the other hand, we have f min u d =f +f su f + << su << f, f d d f d f Hence the roosition yields under horiontal bandedness f d u Corollary. u d min su << f, d f As in 3.3. we need rojections given small scale cut-off r large scale cut-off R number of dyadic intervals N This time, we need a smooth equiartition in k-sace related by R = N r F P < f = η < F f, F P j f = η j F f, F P > f = η > F f 3
As before T u d = T P < u d N + T P j u d + P > T u d j= Middle length scales: We note P j u solves nonstationary Stokes equation with r.h.s P j Ra T e Pr u u = RaP j T e Pr P ju u Hence from Corollary P j u d Ra su P j T + << Pr T Ra su << Ra + Pr T + Pr u d P j u u u u d d u u d see 3.4. u For large and small scales, we roceed as in 3.3.. Hence we obtain T u d r + T d R for, we need the smooth artition of unity 4 u d + N Ra + Pr u d
and for t Nu r + Ra Nu + N Ra + Pr R Nu Ra + Otimie in rr by r = N, R = N as 3.3. Nu N Ra Nu + N Ra + Pr Nu Ra + Otimie in N by N = [ ln Ra 3 ln + ] as 3.3. Nu Ra 3 Nu + + Nu Ra ln Ra + Pr Otimie in < by = + Nu Pr Ra ln Ra 3 as 3.3. & 3.4. Nu + Nu ln Ra Pr {{ Nu + + Nu 3 Ra ln Ra Pr absorb Nu Ra ln Ra 3 + Nu Ra ln Ra 3 Pr Young as 3.4. Recall Proosition Nu Ra ln Ra 3 Ra ln Ra + Pr { Ra ln Ra 3 Pr Ra ln Ra 3 Ra ln Ra Pr Ra ln Ra 3 Pr Pr t u u + = f, u =, u = {, u = t = Suose f is horiontally banded in sense of F f = unless R k 4 for some R. Then Pr t u + u + u + Pr t u + u + f 5
where f = inf f=f +f { su f + << f d with = t t L d [,L d dx dt Rehrase as L d [,L d dx dt Sketch of roof of Proosition. Reduction Pr = time rescaling, stri half sace only here, we need R, only u : i.e. if and horiontally banded, then t u u + = f > u = = t u + u f where f = inf f=f +f su > f + f d. Slitting of f u a φ = f > φ b t v = f φ f > v = = 6
c + u = v > u = = Argument for : Stokes = f = + = +.=φ Stokes + Stokes Hence t + u + + v φ t v = φ f + f = f φ f = f + f 3. For f, u horiontally banded, satisfying t u u + = f > u = = we have u f. a For f, u horiontally banded, satisfying we have u = f > u u f b For f, u horiontally banded, satisfying t u = f > u = = we have t u + u + u f 7
c For f, u horiontally banded, satisfying + u = f > u = = we have u f Argument for 3: Here we use horiontal bandedness, which imlies f R f R f and thus f R f R f 6 Rewrite a as Hence by 3a φ f = f + f φ f f + f and thus in articular by -inequality φ f + f + f and hence by 6 φ f. 7 Aly 3b to b: t v + v + v f φ + f In articular v f φ + f and hence by 6 v f + φ 8 We finally aly 3c to c: u v 9 The combination of 6, 8 and 9 yields as desired u f 8
4. Suose t u = f > u = = F f = unless R k 4 for some R, Then t u + u + u f Argument for 4: Slits into t u + u + u { min f d, su f more interesting Slit into f d a t u D = f > u D = = t ud + u D d f d b very similar to 4a t u N = f > u N = = u N d f d c retty obvious t u C = > u C = g = su u C su g d retty obvious t u D = f > u D = = u D = f d Argument for 4b: By Duhamel, enough to treat initial value roblem t u N = >, t > u N = = u N = f t = By even reflection t u N = R, t > u N = f t = u N d dt f d 9
Get reresentation with heat kernel Gt,, u N t, = R Gt, f d = Gt, + + Gt, f d We slit into small/large terms u N t, d t 3 f d t R f f d R f { min t R, R f d t 3 This imlies the result because of { min t R, R dt = t 3 min { R Both estimates follow from the reresentation: u N t, = u N t, = u N t, = t, R Gt, + + Gt, f d t 3 dt R Gt, + + Gt, f d = G t, ++ G t, G d t, f G t, + + G t, { G d t, f G d t, f u N t, G t, + + G t, =.K t { f t f u N t, K t + K t =. Kt, { f t f d d d 3
u N t, d su K t, d R = R = t t t K t d + su R G t, = t u D = >, t > u D = =, t > u D = f t = t R G, f d f d K t d + su G t, R = t 3 G, t 3 t G, d + su G, Argument for 4d: By Duhamel, enough to treat initial value roblem u D = dt By odd reflection, we obtain the reresentation and thus = u D t = u D t, = Gt, + + Gt, f d Gt, f d = f d G t, G d t, f d Again, we slit into small/large terms = u D t Indeed, = u D t = t 3 { G t, f d f d t R R t 3 f d f d G d t, f G d t, f d 3
= u D su t, { G t, f t f G t, t = t G, t t su G, d f d f d 5. For f and u satisfying u = f > and being banded, we have u f Argument for 5: Because of the equation, it is enough to show We slit the statement into max u u f. u max f f d By Duhamel s rincile in, we have the reresentation u = u d, where u solves the boundary roblem u = <, u = f = In fact, u is the harmonic extension of f onto { <. From the latter, via the reresentation with the Poisson kernel, we infer the estimates f u f f. 3 By bandedness, this turns into u min { R,, R f. 4 3
Inserting 4 into yields u { min R,, R f d 5 Since { min R, R dw = w min {, R dw w R =, from 5, we immediately deduce and the unweighted estimate u d f d 6 We now turn to. By the -inequality, it is enough to consider the case of for some H, and to show in this case f = unless H < 4H 7 su u H f d 8 and H u d f d 9 so that the slitting in the definition of u is u = I H u+i > H u. We first address 8 and fix a H: u 5 7 4H H f d f d 4H H f d H f d. 33
We turn to 9: H u d H 6 H 7 = H u d 4H H f d f d f d 6. For f, u banded and related by + u = f > u = = we have u f. Argument for 6: Since by alying to we obtain + u = f and since f = f so that by bandedness f f, we have by the triangle inequality u u + f. Hence it is enough to show u + u f. Using again u u we see from alied to, + u = f and the triangle inequality that it is enough to show u f which reduces to showing which slits into the two estimates su u f u su f 34
and u d f d In order to establish and, we use Duhamel s rincile in : where u u = solves the boundary roblem + u = >, u = f =. u d 3 In fact, u is the harmonic extension of f onto { >. From the Poisson kernel reresentation we thus learn { u f f Together with bandedness, we obtain Using this in 3 yields u min u { R, R f { min R, R f d. 4 From this estimate, and are easily derived: On the one hand, we have On the other hand, so that u 4 u 4 u d In both cases, the result relies on { min R, R d { min R, R d su f, { min R, R f d { min R, R d = f d { min R, R d min {, R dw w R. 35
References [] Antoine Choffrut, Camilla Nobili, and Felix Otto. Uer bounds on Nusselt number at finite Prandtl number. Submitted to the Journal of Differential Equations. [] Peter Constantin and Charles R Doering. Infinite Prandtl number convection. Journal of Statistical Physics, 94-:59 7, 999. [3] Charles R Doering and Peter Constantin. Variational bounds on energy dissiation in incomressible flows. III. Convection. Physical Review E, 536:5957, 996. [4] Charles R Doering, Felix Otto, and Maria G Renikoff. Bounds on vertical heat transort for infinite-prandtl-number Rayleigh-Bénard convection. Journal of Fluid Mechanics, 56:9 4, 6. [5] Felix Otto and Christian Seis. Rayleigh Bénard convection: imroved bounds on the Nusselt number. Journal of Mathematical Physics, 58:837,. [6] Xiaoming Wang. Asymtotic behavior of the global attractors to the Boussinesq system for Rayleigh-Bénard convection at large Prandtl number. Communications on Pure and Alied Mathematics, 69:93 38, 7. 36