Interaction of the Longwave and Finite-Wavelength Instability Modes of Convection in a Horizontal Fluid Layer Conﬁned Between Two Fluid-Saturated Porous Layers

fluids

Article Interaction of the Longwave and Finite-Wavelength Instability Modes of Convection in a Horizontal Fluid Layer Conﬁned between Two Fluid-Saturated Porous Layers

Tatyana P. Lyubimova 1,2,* and Igor D. Muratov 2

1 Computational Fluid Dynamics Laboratory, Institute of Continuous Media Mechanics UB RAS, Ak. Koroleva str. 1, 614013 Perm, Russia 2 Theoretical Physics Department, Perm State University, Bukireva str. 15, 614990 Perm, Russia; [email protected] * Correspondence: [email protected]; Tel.: +7-342-2396646

Received: 2 May 2017; Accepted: 14 July 2017; Published: 16 July 2017

Abstract: The onset of convection in a three-layer system consisting of two fluid-saturated porous layers separated by a homogeneous fluid layer is studied. It is shown that both a longwave convective regime developing in the whole system and a finite-wavelength regime of convection concentrated in the homogeneous fluid layer are possible. Due to the hydraulic resistance of the porous matrix, the flow intensity in the longwave convective regime is much lower than that in the finite-wavelength regime. Moreover, it grows at a much slower pace with the increase of the Grashof number. Because of that, the long-wave convective regime becomes unstable at small supercriticalities and is replaced by a finite-wavelength regime.

Keywords: thermal convection; porous medium; multilayer system; longwave and ﬁnite-wavelength instability modes; non-linear regimes

1. Introduction At the onset of convection in layers, the scale of convective cells is usually close to the transversal size of the channel. The original situation is discussed in Reference [1], where the onset of convection in a three-layer system consisting of two fluid-saturated porous layers separated by a pure fluid layer is studied. It was found that, there is a range of parameters where the neutral curves are bimodal, i.e., both, a longwave convection, taking place mainly in porous medium, and a finite-wavelength convection, concentrated mainly in a fluid with a characteristic horizontal scale close to the thickness of a layer, may coexist. Later on, the same phenomena were discovered and analyzed for two-layer systems made up of a superposed pure fluid layer and a fluid-saturated porous layer [2–6]. In Reference [7], it is shown that high frequency transversal vibrations cause a stabilizing effect on the disturbances with any wave numbers, but affects finite-wavelength disturbances much more strongly than longwave ones. The present work is devoted to the investigation of finite-wavelength and longwave instability modes interaction in the supercritical parameter range for the three-layer configuration considered in Reference [1]. The studies of the onset of convection in multi-layer systems consisting of one fluid layer and two fluid-saturated porous layers are important, due to their application to the directional solidification of binary alloys where a two-phase porous zone called the mushy zone is formed between the melt and the crystal. As shown in Reference [8], there are two types of instability in a two-layer system consisting of the melt zone and the mushy zone: short-wave instability and long-wave instability.

Fluids 2017, 2, 39; doi:10.3390/ﬂuids2030039 www.mdpi.com/journal/ﬂuids FluidsFluids2017 2017,,2 2,, 39 39 22 ofof 88

2. Governing Equations and Boundary Conditions 2. Governing Equations and Boundary Conditions The problem of convection in a system which consists of an infinite horizontal layer of an The problem of convection in a system which consists of an infinite horizontal layer of incompressible viscous fluid with the thickness 2d embedded between two plane layers of porous an incompressible viscous fluid with the thickness 2d embedded between two plane layers of medium saturated with the same fluid, each with a thickness of hm , is considered (Figure 1). The porous medium saturated with the same fluid, each with a thickness of hm, is considered (Figure1). temperature of the external boundaries is fixed, and the temperature of the bottom boundary is The temperature of the external boundaries is fixed, and the temperature of the bottom boundary is higher than that of the top (heating from below). The origin of the coordinate system is set in the higher than that of the top (heating from below). The origin of the coordinate system is set in the middle of the fluid layer. middle of the fluid layer.

2d x

Figure 1. Problem conﬁguration. Figure 1. Problem configuration.

EquationsEquations forfor thermal buoyancy convection in in fluid fluid in in the the Boussinesq Boussinesq approximation approximation look look like like [9]: [9 ]:  → ∂ ∂ v →v1+⋅∇=−∇ → 1 +Δ+νβγ→  → + v · ∇vvv = − ∇PPgTf f+ ν∆ vv + gβT γ ∂t ∂t ρρ0 ∂T → 0 + v · ∇T = χ f ∆T (1) ∂t → ∂T  (1) div v = 0+⋅∇=v TTχ Δ ∂t f  = For the description of convectivediv v filtration 0 in a porous medium, we use the Darcy-Boussinesq modelFor [10 the]: description of convective filtration in a porous medium, we use the Darcy-Boussinesq 1 ν → → model [10]: − ρ ∇Pm − K u + gβϑγ = 0 0 → ∂ϑ + · ∇ = (2) ρ0cp1m ∂t νρ0cp f u  ϑ κm∆ϑ −∇−→ Pug +βϑγ =0 ρ= m divu 0 0 K → → ∂ϑ  (2) Here, v and u are the convective()ρρccu flow velocity+⋅∇=Δ() in fluidϑκ and the ϑ convective filtration velocity in 00ppmmf∂t a porous medium, respectively; T and ϑ are the temperature deviations in fluid and in the porous divu = 0 medium from a constant average temperature; m is the porosity coefficient; and K is the permeability   coefficientHere, ofv porous and u medium. are the convective Other designations flow velocity are usual.in fluid The and quantities the convective marked filtration by the indexvelocityf concernin a porous the fluid, medium, and thoserespectively; marked byT theand index ϑ arem concernthe temperature the porous deviations medium. in fluid and in the porousLet medium us discuss from the a constant boundary average conditions. temperature; The externalm is the boundariesporosity coefficient; are considered and K tois the be impenetrable,permeability coefficient i.e., the vertical of porous component medium. of convective Other designations filtration velocity are usual. vanishes The quantities at these boundaries.marked by Atthe the index same f time concern the horizontal the fluid, componentsand those marked of the by filtration the index velocity m concern on the externalthe porous boundaries medium. are generallyLet us non-zero, discuss such the thatboundary the filtration conditions. of fluid The in theexternal porous boundaries medium along are considered the impenetrable to be boundaryimpenetrable, is possible. i.e., the The vertical temperature component of the externalof conv boundariesective filtration is supposed velocity to bevanishes fixed. Thus,at these the conditionsboundaries. at At the the external same boundariestime the horizontal are as follows: components of the filtration velocity on the external boundaries are generally non-zero, such that the filtration of fluid in the porous medium along the z = (d + hm) : uz = 0, ϑ = −Θ, impenetrable boundary is possible. The temperature of the external boundaries is supposed to (3)be z = −(d + hm) : uz = 0, ϑ = +Θ fixed. Thus, the conditions at the external boundaries are as follows: Different types of boundary conditions=+ at the interface =ϑ of =−Θ pure fluid and fluid-saturated porous zdh()mz:0,, u medium were suggested (see Reference [10]). The conditions proposed by Beavers and Joseph [11(3)] zdhu=−() +:0, =ϑ =+Θ determine the jump of the tangential componentsmz of the velocity at the interface while the normal Fluids 2017, 2, 39 3 of 8 component of the velocity and pressure are assumed to be continuous. The conditions obtained by Ochoa-Tapia and Whitaker [12,13] by the direct averaging of the Navier–Stokes equations at the pore-scale level impose the continuity of flow velocities and normal stresses, while the tangential viscous stresses have a jump across the interface, describing the resistance of the porous matrix in the boundary layer, which has a thickness to of the order of K1/2. These conditions, as well as the Beavers-Joseph conditions, include an empirical parameter — the stress jump coefficient. The influence of the stress jump coefficient on the onset of thermal buoyancy convection in horizontal stratified fluid/porous layers was studied in Reference [14]. It was shown that the increase of the stress jump coefficient strongly influences the fluid mode, inducing a more unstable situation at large wave numbers, whereas the porous mode remains unchanged. In Reference [15], the onset of thermal buoyancy convection in a system consisting of a fluid layer overlying a homogeneous porous medium was studied in the framework of the model including the Brinkman term, in a two-domain approach. A comparison of neutral curves with those obtained in the one-domain approach and in the framework of the Darcy model with the two-domain approach shows that the inclusion of the Brinkman term plays a secondary role in the stability results. At zero value of this parameter, the tangential stress jump condition is reduced to the tangential stress continuity condition proposed in Reference [1]. In References [16,17], the improvement of the approach by Ochoa-Tapia and Whitaker was suggested. In our work, we use the conditions at the interface of pure fluid and fluid-saturated porous medium suggested by Lyubimov and Muratov in Reference [1], which include the continuity of temperature, heat flux, normal component of velocity and pressure, and the condition of vanishing of the tangential components of fluid velocity:

∂T ∂ϑ z = ±d : T = ϑ, κ = κ , v = u , P = P , v = v = 0 (4) f ∂z m ∂z z z m f x y

The application of the boundary condition vx = vy = 0 is justified by the fact that the velocities of the convective filtration in a porous medium are generally small because of the small typical values of the porous medium permeability. In this case, the condition of the normal stress balance is reduced to pressure continuity. The advantages of the boundary conditions (Equation (4)) are their simplicity and the fact that they do not include any empirical parameters. It is possible to show that, just as in the case of a homogeneous fluid, the condition for a conductive state of fluid in the described system is the linear dependence of temperature on vertical coordinate:

→ ∇T0 = −A f γ → (5) ∇ϑ0 = −Am γ

Then, from the condition of heat ﬂux continuity follows:

κ f A f = κm Am (6) and the condition: ϑ = Θ at z = −(d + hm) gives:

A f d + Amhm = Θ (7)

Further consideration is limited to the case when the thermal properties of the ﬂuid and the “skeleton” are identical, that is: κ f = κm ≡ κ, χ f = χm ≡ χ Fluids 2017, 2, 39 4 of 8

It is convenient to present the condition of pressure continuity on the interface of the porous medium and ﬂuid in the terms of velocities. Using horizontal projections of momentum and continuity equations, we obtain this condition in the form:

∂v 1 ∂u ∆ z = − z (8) ∂z K ∂z The analysis was restricted to the case of two-dimensional flows. In this case, it is convenient to introduce the stream function and vorticity for the description of the flow in the fluid layer and the stream function for the flow in the fluid-saturated porous layers. Equations rewritten in terms of stream function and vorticity in the dimensionless form are:

∂ϕ ∂ϕ ∂ψf ∂ϕ ∂ψf ∂T ∂t + ∂x ∂z − ∂z ∂x = ∆ϕ − Gr ∂x ,

∂T ∂T ∂ψf ∂T ∂ψf 1 ∆ψf = ϕ, ∂t + ∂x ∂z − ∂z ∂x = Pr ∆T,

∂ϑ ∂ϑ ∂ϑ ∂ψm ∂ϑ ∂ψm 1 (9) ∆ψm = −GrDa ∂x , ∂t + ∂x ∂z − ∂z ∂x = Pr ∆ϑ,

∂ψm z = ±(1 + h) : ψm = 0, ∂z = 0, ϑ = ∓1, ∂3ψ z = ± = T = ∂T = ∂ϑ f = − 1 ∂ψm 1 : ψf ψm, ϑ, ∂z ∂z , ∂z3 Da ∂z

∂ψf ∂ψf → ∂ψm ∂ψm where vx = ∂z , vz = − ∂x , ϕ = curly v , ux = ∂z , uz = − ∂x , and the last condition at z = ±1 is obtained from Equation (8), taking into account the smallness of the Darcy number. The problem is characterized by four dimensionless parameters: the ratio of the porous layer thickness to the pure ﬂuid layer thickness, the Darcy number, the Prandtl number, and the Grashof number: h K ν gβΘd3 h = m , Da = , Pr = , Gr = d d2 χ ν2

3. Numerical Results The problem was solved numerically by the finite difference method. The explicit finite difference scheme of the second order of approximation for spatial variables was used. The grid with the spatial step h = 1/20 was chosen for the main calculations, after the test calculations confirmed the convergence of the numerical results with the decrease of h. As shown in Reference [1], the coexistence of finite-wavelength and longwave instability takes place at sufficiently large values of h in some range of√Da values, which are extended while h increases. In Figure2, the neutral curves C(k) where C = Gr are plotted for h = 10 and different values of Da. It can be seen that at Da = 10−3 and Da = 2 · 10−3 the neutral curves are bimodal; moreover, at Da = 10−3, the finite-wavelength and longwave minima of the neutral curve correspond to nearly the same values of the Grashof number (the longwave minimum is just slightly lower). On the basis of these results, the parameter values h = 10, Da = 10−3, Pr = 1 were chosen for the nonlinear calculations. The length of the computational domain was found to be equal to 10π, that is, approximately equal to the wavelength of the most dangerous longwave disturbances. On the vertical boundaries of the area, the conditions of periodicity were imposed. Such conditions allow not only longwave solutions but also finite-wavelength ones, with the wavelength equal to 10π/n, where n is an integer value. Fluids 2017, 2, 39 5 of 8 Fluids 2017, 2, 39 5 of 8 On the vertical boundaries of the area, the conditions of periodicity were imposed. Such On the vertical boundaries of the area, the conditions of periodicity were imposed. Such conditions allow not only longwave solutions but also finite-wavelength ones, with the wavelength conditionsFluids 2017, 2 ,allow 39 not only longwave solutions but also finite-wavelength ones, with the wavelength5 of 8 equal to 10π / n , where n is an integer value. equal to 10π / n , where n is an integer value.

Figure 2. Neutral curves for h = 10 and different values of Da . Figure 2. Neutral curves for h = 10 and different values of Da . Figure 2. Neutral curves for h = 10 and different values of Da. As follows from the results of the calculations obtained in the framework of linear theory, critical As follows from the results of the calculations obtained in the framework of linear theory, critical values of the Grashof number for disturbances with n = 4,5,6,7 are close to those for longwave valuesAs of follows the Grashof from the number results for of thedisturbances calculations with obtained n = 4,5,6,7 in the framework are close to of linearthose theory,for longwave critical disturbancesvalues of the with Grashof n =1 number, although for they disturbances are a little with bit higher.n = 4, 5, 6, 7 are close to those for longwave disturbances with n =1 , although they are a little bit higher. disturbancesNaturally, with it nis= easier1, although to obtain they arefinite-wavelength a little bit higher. solutions in calculations with short Naturally, it is easier to obtain finite-wavelength solutions in calculations with short computationalNaturally, domain. it is easier However, to obtain finite-wavelengthin this case we lose solutions an opportunity in calculations to look with after short their computational interaction computational domain. However, in this case we lose an opportunity to look after their interaction withdomain. longwave However, disturbances. in this case As we the lose symmetry an opportunity of longwave to look and after finite-wavelength their interaction solutions with longwave is the with longwave disturbances. As the symmetry of longwave and finite-wavelength solutions is the same,disturbances. such an interaction As the symmetry may be sufficiently of longwave strong. and finite-wavelength solutions is the same, such same, such an interaction may be sufficiently strong. an interactionThe numerical may becalculations sufficiently strong.show that, as expected, finite-wavelength stationary solutions The numerical calculations show that, as expected, finite-wavelength stationary solutions describeThe the numerical convective calculations flow concentrated show that, as mainly expected, in the finite-wavelength fluid (Figure 3), stationary and longwave solutions solutions describe describe the convective flow concentrated mainly in the fluid (Figure 3), and longwave solutions describethe convective the flow flow spreading concentrated both mainlythrough in the fluidlayer (Figureof homogeneous3), and longwave fluid and solutions the porous describe layers the describe the flow spreading both through the layer of homogeneous fluid and the porous layers (Figureflow spreading 4). both through the layer of homogeneous fluid and the porous layers (Figure4). (Figure 4).

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-10.00 -10.00 -15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 -15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 FigureFigure 3. 3. StreamlinesStreamlines of of the the finite-wavelength ﬁnite-wavelength regime. regime. Figure 3. Streamlines of the finite-wavelength regime. Fluids 2017, 2, 39 6 of 8 FluidsFluids 2017 2017, ,2 2, ,39 39 66 of of 8 8

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FigureFigure 4. 4. Streamlines Streamlines of of the the longwave longwave regime regime (only (only(only the thethe upper upperupper part partpart of ofof the thethe physical physicalphysical domain domaindomain is isis shown). shown).shown).

TheTheThe intensity intensity of of finite-wavelengthfinite-wavelength finite-wavelength solutions solutions solutions near near near the the the threshold threshold threshold grows grows grows according according according to theto to the squarethe square square root rootlaw,root law, i.e.,law, i.e., the i.e., excitationthe the excitation excitation of short-waveof of short-wave short-wave convection convecti convectionon occurs occurs occurs through through through supercritical supercritical supercritical bifurcation bifurcation bifurcation (Figure(Figure (Figure5 5). ).5). TheTheThe calculations calculationscalculations show showshow that thatthat in inin the thethe longwave longwave mode mode,mode, ,where wherewhere the thethe flow flowflow covers coverscovers not notnot only onlyonly the the fluid fluidfluid layer layer butbutbut also alsoalso the thethe porous porousporous medium, medium,medium, the thethe flow flowflow intensity intensityintensity is isis lower lowerlower than thanthan in inin the thethe finite-wavelength finite-wavelengthfinite-wavelength mode, mode,mode, where wherewhere thethethe flow flowflow is isis concentrated concentratedconcentrated in inin the thethe fluid. fluid.fluid. This ThisThis is isis caus causedcauseded by by the the hydraulic hydraulic resistance resistance of ofof the the porous porous medium mediummedium skeleton.skeleton.skeleton. Moreover, Moreover, the thethe intensity intensityintensity of ofof longwave longwavelongwave soluti solutionssolutionsons grows grows very very slowly slowlyslowly with with the the increase increaseincrease of ofof the thethe GrashofGrashofGrashof number. number.number. ThisThis is is why why already alreadyalready at atat small smallsmall supe supercriticalitiessupercriticalitiesrcriticalities with withwith respect respectrespect to toto finite-wavelength finite-wavelengthfinite-wavelength disturbances,disturbances,disturbances, the the intensity intensity of of the the latter latter surpasses surpasses th thethee intensity intensity of of the the longwave longwave solution solution (Figure (Figure 5). 55).). Apparently,Apparently,Apparently, the the fast fast loss loss of of stabilitystability of of the the longwave longwave solution solution is is related related to to this this circumstance. circumstance.

ΨΨm m a ax x 11

00 6060 8080 100100 120120 140140 160160 180180 GGr r

Figure 5. The intensity of the ﬁnite-wavelength (curve 1) and longwave (curve 2) regimes. FigureFigure 5. 5. The The intensity intensity of of the the finite-wavelength finite-wavelength (cur (curveve 1) 1) and and longwave longwave (curve (curve 2) 2) regimes. regimes.

LetLetLet us usus discuss discussdiscuss the thethe interaction interactioninteraction of ofof longwave longwavelongwave and andand finite-wavelength finite-wavelengthfinite-wavelength disturbances disturbancesdisturbances at atat small smallsmall supercriticalities.supercriticalities.supercriticalities. It It turns Itturns turns out out outthat that thatthe the critical critical the critical value value of valueof the the Grashof Grashof of the Grashofnumber number for numberfor the the longwave longwave for the instability longwaveinstability modeinstabilitymode isis smallersmaller mode is than smallerthan thatthat than forfor that thethe for finite-wavelengthfinite-wavelength the finite-wavelength mode.mode. mode. BecauseBecause Because ofof this, ofthis, this, atat at veryvery very small smallsmall supercriticalitiessupercriticalitiessupercriticalities the thethe finite-wavelength finite-wavelengthfinite-wavelength disturbances disturbadisturbancesnces fade fadefade and theandand longwave thethe longwavelongwave mode is mode realized.mode isis However,realized.realized. However,asHowever, the calculations as as the the calculations calculations show, as soon show, show, as theas as soon Grashofsoon as as the numberthe Grashof Grashof values number number become values values higher become become than thehigher higher critical than than value the the criticalforcritical finite-wavelength value value for for finite-wavelength finite-wavelength disturbances, disturbances, disturbances, the longwave the the mode longwave longwave loses mode mode its stability loses loses its its and stability stability the flow and and isthe the broken flow flow isintois broken broken vortices into into ofvortices vortices the small of of the the size: small small at thesize: size: Grashof at at the the Gras Gras numberhofhof number number values values largervalues thanlarger larger 80 than than after 80 80 the after after transient the the transient transient stage, stage,dependingstage, depending depending on the on on initial the the initial conditions,initial conditions, conditions, a 4, 5, a ora 4, 4, 6 5, wave5, or or 6 6 regimewave wave regime regime is established. is is established. established. TheTheThe process processprocess of ofof the thethe structure structurestructure transformation transformationtransformation emerges emergesemerges very very slowly, slowly, so soso it itit is isis possible possiblepossible to toto observe observeobserve the thethe variousvariousvarious mixed mixedmixed modes modesmodes repres representingrepresentingenting the thethe superposition superpositionsuperposition of ofof finite-w finite-wavelengthfinite-wavelengthavelength and andand longwave longwavelongwave modes. modes.modes. The TheThe exampleexampleexample of ofof such suchsuch a aa situation situationsituation is isis presented presentedpresented in inin Figure FigureFigure 66. 6.. Fluids 2017, 2, 39 7 of 8 Fluids 2017, 2, 39 7 of 8

Fluids 2017, 2, 3910.00 7 of 8

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0.00 -15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 0.00 FigureFigure 6. 6. MixedMixed-15.00 regime: regime: -10.00 the the superposition superposition -5.00 ofof 0.00 fi finite-wavelengthnite-wavelength 5.00 10.00and and longwave longwave 15.00 modes. Figure 6. Mixed regime: the superposition of finite-wavelength and longwave modes. In this figure, one of the phases of the long-term transient process of the merging neighboring In this figure, one of the phases of the long-term transient process of the merging neighboring cells is shown.In this Asfigure, one onecan of see, the in phases the central of the partlong-t oferm a cavity, transient two process vortices of the of themerging same neighboring sign gradually cells is shown. As one can see, in the central part of a cavity, two vortices of the same sign gradually mergecells into is shown.one vortex As one (by can which see, inpoint the central the centra part lof vortex a cavity, has two already vortices practically of the same disappeared). sign gradually merge into one vortex (by which point the central vortex has already practically disappeared). mergeIn Figure into 7,one the vortex dependences (by which ofpoint the the stationary central vortex flow hasintensity already on practically the parameter disappeared). C (square root In Figure7, the dependences of the stationary flow intensity on the parameter C (square root of of the GrashofIn Figure number 7, the dependencesvalue) for the of disturbancesthe stationary flowwith intensity various onvalues the parameter of k are Cdescribed. (square root On the the Grashof number value) for the disturbances with various values of k are described. On the vertical verticalof the axis, Grashof the squared number maximal value) for value the disturbances of the stream with function various invalues the fluid of k is are plotted. described. On the axis,vertical the squared axis, the maximal squared value maximal of the value stream of the function stream function in the fluidin the is fluid plotted. is plotted. 3 3 2 Ψm 2 Ψm

2 2

1 1

0 0 8.5 8.75 9 9.25 9.5 9.75 10 8.5 8.75 9 9.25 9.5 9.75C 10 C FigureFigure 7. The 7. The dependences dependences of of stationary stationary flow flow intensity intensity on on the the parameter parameter C forC variousfor various values values of k : of k: ■— k = 1.0 , ▲— k = 1.2 , ●— k = 0.8 , ♦— k = 1.4 . Figure—k = 7.1.0, TheN dependences—k = 1.2, — ofk =stationary0.8, —k flow= 1.4. intensity on the parameter C for various values of k : ■— k = 1.0 , ▲— k = 1.2 , ●— k = 0.8 , ♦— k = 1.4 . Critical values of the parameter C , obtained by the extrapolation of the amplitude curves on theCritical zero value values of ofthe the stream parameter function,C ,are obtained in good by agreement the extrapolation with the results of the of amplitude linear theory curves [1]. on the zeroCritical value of values the stream of the function, parameter are inC good, obtained agreement by the with extrapolation the results of of the linear amplitude theory [ 1curves]. on the zero4. Conclusions value of the stream function, are in good agreement with the results of linear theory [1]. 4. Conclusions 4. ConclusionsThe calculations show that the finite-wavelength stationary regimes correspond to the convectiveThe calculations flow concentrated show that thein the finite-wavelength fluid layer and stationarythe longwave regimes stationary correspond regimes to of the the convective flow flowoccupyingThe concentrated calculations both inlayers. the show fluidFor the layerthat parameter the and thefinite-wavelength values longwave selected stationary for stationarythe calculations, regimes regimes of the the critical flowcorrespond occupyingvalue of theto boththe convectivelayers.Grashof For theflow number parameter concentrated for the values longwave in selectedthe instabilityfluid for layer the mode calculations, and isthe smaller longwave the than critical thatstationary valuefor the of finite-wavelengthregimes the Grashof of the number flow occupyingfor themode. longwave Becauseboth layers. instabilityof that, For at thevery mode parameter small is supercriticaliti smaller values than selectedes that the finite-wavelength for for the the finite-wavelength calculations, disturbances the critical mode. fade value and Because the of the of Grashofthat,longwave at very number small mode for supercriticalities is therealized. longwave However, theinstability finite-wavelengthas the calculat modeions is show,smaller disturbances as soonthan as fadethat the Grashof andfor the the numberfinite-wavelength longwave values mode is become higher than the critical value for finite-wavelength disturbances, the longwave mode loses mode.realized. Because However, of that, as the at very calculations small supercriticaliti show, as soones as the the finite-wavelength Grashof number valuesdisturbances become fade higher and than the its stability and the flow is broken into vortices of the small size. In our opinion, the reason for that is longwavethe critical mode value is for realized. finite-wavelength However, as disturbances, the calculat theions longwave show, as soon mode as loses the Grashof its stability number and the values flow the following. Due to the hydraulic resistance of the porous matrix, the intensity of flow for the becomeis broken higher into vorticesthan the of critical the small value size. for Infinite our-wavelength opinion, the disturbances, reason for that the is longwave the following. modeDue loses to its stability and the flow is broken into vortices of the small size. In our opinion, the reason for that is the following. Due to the hydraulic resistance of the porous matrix, the intensity of flow for the Fluids 2017, 2, 39 8 of 8 the hydraulic resistance of the porous matrix, the intensity of flow for the longwave mode is much lower than that for the finite-wavelength mode. Moreover, it grows very slowly with the increase of the Grashof number value. Because of that, already at the Rayleigh number value, which is just slightly higher than the instability threshold to finite-wavelength disturbances, the flow intensity of the finite-wavelength mode surpasses the intensity of the longwave mode, which results in the fast loss of stability of the longwave regime.

Acknowledgments: The work was supported by the Russian Science Foundation (grant 14-21-00090). Author Contributions: Tatyana. P. Lyubimova made contribution to the problem formulation, development of the concept and method of investigation, development of the numerical code, analysis and interpretation of the numerical results, preparation of the draft and revision of the paper, she also worked as corresponding author. Igor D. Muratov contributed to the numerical simulation, analysis of numerical results and revision of the paper. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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