Theoretical investigations of the idealized model for the mushy region

In this paper the theoretical analysis of the behaviour of the stream function, temperature, and local solid fraction for the model of ideal mushy layer is presented. In the case of steady free mush convection, explicit lower and upper estimates for the main characteristics of the process are found...

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Veröffentlicht in:	Физика и техника высоких давлений
Datum:	2014
Hauptverfasser:	Namlyeyeva, Yu.V., Taranets, R.M.
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Veröffentlicht:	Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України 2014
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spelling	Namlyeyeva, Yu.V. Taranets, R.M. 2014-10-18T15:05:48Z 2014-10-18T15:05:48Z 2014 Theoretical investigations of the idealized model for the mushy region / Yu.V. Namlyeyeva, R.M. Taranets // Физика и техника высоких давлений. — 2014. — Т. 24, № 1. — С. 35-47. — Бібліогр.: 13 назв. —англ. 0868-5924 PACS: 02.30.Hq, 47.56.+r, 64.70.Dv https://nasplib.isofts.kiev.ua/handle/123456789/69687 In this paper the theoretical analysis of the behaviour of the stream function, temperature, and local solid fraction for the model of ideal mushy layer is presented. In the case of steady free mush convection, explicit lower and upper estimates for the main characteristics of the process are found for the large values of the Rayleigh number. For the unsteady regime the one of explicit forms of these characteristics is obtained. Досліджено поведінку функцій потоку, температури та локальної твердої фракції для ідеальної моделі мішаного шару. У випадку стійкої вільної конвекції знайдено точні нижні та верхні оцінки зверху та знизу для основних функцій, які характеризують процес, що має місце при великих значеннях числа Релея. Для нестаціонарного режиму отримано також явний вид основних характеристик. В данной работе исследуется поведение функции потока, температуры и локальной твердой фракции для идеальной модели смешанного слоя. В случае устойчивой свободной конвекции найдены точные нижние и верхние оценки основных функций, характеризующих процесс, при больших значениях числа Рэлея. Для нестационарного режима также найден явный вид этих основных характеристик. en Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України Физика и техника высоких давлений Theoretical investigations of the idealized model for the mushy region Теоретические исследования идеальной модели «mushy region» Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Theoretical investigations of the idealized model for the mushy region
spellingShingle	Theoretical investigations of the idealized model for the mushy region Namlyeyeva, Yu.V. Taranets, R.M.
title_short	Theoretical investigations of the idealized model for the mushy region
title_full	Theoretical investigations of the idealized model for the mushy region
title_fullStr	Theoretical investigations of the idealized model for the mushy region
title_full_unstemmed	Theoretical investigations of the idealized model for the mushy region
title_sort	theoretical investigations of the idealized model for the mushy region
author	Namlyeyeva, Yu.V. Taranets, R.M.
author_facet	Namlyeyeva, Yu.V. Taranets, R.M.
publishDate	2014
language	English
container_title	Физика и техника высоких давлений
publisher	Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України
format	Article
title_alt	Теоретические исследования идеальной модели «mushy region»
description	In this paper the theoretical analysis of the behaviour of the stream function, temperature, and local solid fraction for the model of ideal mushy layer is presented. In the case of steady free mush convection, explicit lower and upper estimates for the main characteristics of the process are found for the large values of the Rayleigh number. For the unsteady regime the one of explicit forms of these characteristics is obtained. Досліджено поведінку функцій потоку, температури та локальної твердої фракції для ідеальної моделі мішаного шару. У випадку стійкої вільної конвекції знайдено точні нижні та верхні оцінки зверху та знизу для основних функцій, які характеризують процес, що має місце при великих значеннях числа Релея. Для нестаціонарного режиму отримано також явний вид основних характеристик. В данной работе исследуется поведение функции потока, температуры и локальной твердой фракции для идеальной модели смешанного слоя. В случае устойчивой свободной конвекции найдены точные нижние и верхние оценки основных функций, характеризующих процесс, при больших значениях числа Рэлея. Для нестационарного режима также найден явный вид этих основных характеристик.
issn	0868-5924
url	https://nasplib.isofts.kiev.ua/handle/123456789/69687
citation_txt	Theoretical investigations of the idealized model for the mushy region / Yu.V. Namlyeyeva, R.M. Taranets // Физика и техника высоких давлений. — 2014. — Т. 24, № 1. — С. 35-47. — Бібліогр.: 13 назв. —англ.
work_keys_str_mv	AT namlyeyevayuv theoreticalinvestigationsoftheidealizedmodelforthemushyregion AT taranetsrm theoreticalinvestigationsoftheidealizedmodelforthemushyregion AT namlyeyevayuv teoretičeskieissledovaniâidealʹnoimodelimushyregion AT taranetsrm teoretičeskieissledovaniâidealʹnoimodelimushyregion
first_indexed	2025-11-27T00:14:44Z
last_indexed	2025-11-27T00:14:44Z
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fulltext	Физика и техника высоких давлений 2014, том 24, № 1 © Ю.В. Намлеева, Р.М. Таранец, 2014 PACS: 02.30.Hq, 47.56.+r, 64.70.Dv Yu.V. Namlyeyeva, R.M. Taranets THEORETICAL INVESTIGATIONS OF THE IDEALIZED MODEL FOR THE MUSHY REGION Institute of Applied Mathematics and Mechanics of NAS of Ukraine 74, R. Luxemburg Str., 83114, Donetsk, Ukraine Received December 18, 2013 In this paper the theoretical analysis of the behaviour of the stream function, tempera- ture, and local solid fraction for the model of ideal mushy layer is presented. In the case of steady free mush convection, explicit lower and upper estimates for the main charac- teristics of the process are found for the large values of the Rayleigh number. For the unsteady regime the one of explicit forms of these characteristics is obtained. Keywords: mushy layer, steady and unsteady free mush convection, stream function, temperature, local solid and liquid fraction Досліджено поведінку функцій потоку, температури та локальної твердої фракції для ідеальної моделі мішаного шару. У випадку стійкої вільної конвекції знайдено точні нижні та верхні оцінки зверху та знизу для основних функцій, які характери- зують процес, що має місце при великих значеннях числа Релея. Для нестаціонар- ного режиму отримано також явний вид основних характеристик. Ключові слова: мішаний шар, стійка та нестійка конвекція, функція потоку, тем- пература, локальні тверда та рідка фракції 1. Introduction A mushy layer, a two-phase medium of coexisting liquid and solid phases, arises as a result of morphological instability of solidification front, see [5,6]. It can be considered as a porous medium through which the residual liquid can flow [7,13]. Therefore, the permeability structure of the mushy layer has to be calcu- lated simultaneously with solving the coupled equations of heat, mass, and mo- mentum transport [13]. Most theoretical studies of mushy layers consider the process of solidification at horizontal boundaries, see [12] and references therein. However, in many cases the process of solidification takes place at vertical boundaries. For example, in magma chambers various aqueous solutions are cooled and solidified from a sidewall in confined spaces [10,9,8,4]. Физика и техника высоких давлений 2014, том 24, № 1 36 The problem of the lateral solidification of a semi-infinite mushy region influenced by the vertical interstitial melt was investigated in [3]. The authors considered a binary alloy releasing a buoyant residual fluid in the solidification process. The fluid was assumed to be pulled horizontally at the constant speed V past the heat exchanger maintaining the eutectic temperature TE at the fixed vertical plane x = 0, see Figure. The material supplied at x = +∞ had the solute composition C0, and the temperature equals to its liquid temperature TL(C0). A mushy region was considered in the semi-infinite region x > 0 and z > 0. In [3] assuming that the mushy region is ideal, Worster’s model from [13] for description of the evolution of the dimensionless temperature θ and the local solid fraction φ in the domain x > 0 z > 0 is applied: 2=u t x t x ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞− θ+ ⋅∇θ ∇ θ+ ϒ − φ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ , (1) [ ](1 ) = 0u t x ∂ ∂⎛ ⎞− −φ θ+Φφ + ⋅∇θ⎜ ⎟∂ ∂⎝ ⎠ , (2) 2 1 = Ra x ∂θ ∇ ψ − ∇ψ ⋅∇Π − Π Π ∂ , (3) where t is dimensionless time, u is the volume flux (or Darcy velocity), ψ is the stream function defined by u = (–ψz, ψx), and Π is the permeability. In this model the dimensionless constants are the Stefan number = /( )pL c Tϒ Δ , the composi- tional ratio Φ = (Cs – C0)/ΔC, and the mush Rayleigh number Ra = βΔCgΠ0/(νV), where ΔC = C0 – CE, C0, is the initial composition, CE is the eutectic composition, Cs is the composition of the solid phase, L is the specific latent heat, cp is the spe- cific heat capacity, β = β* – Γα, α and β* being the thermal and solutal expan- sion coefficients, g is the gravity acceleration, and ν is the liquid kinematic vis- cosity. Fig. A semi-infinite mushy region of far- field temperature TL(C0) solidifies laterally at fixed speed V to form a solid at the eutectic temperature TE. The release of a buoyant residual is confined to a thermal boundary layer adjacent to the interface. Illustrative streamlines are shown relative to the (moving) solid phase, see [2] Физика и техника высоких давлений 2014, том 24, № 1 37 Equations (1)–(3) are supplemented with the following boundary and initial conditions φ = φ0 at t = 0, φ = φ∞ at x = 0, φ → φ∞ as x → ∞, (4) θ = –1 at x = 0, θ → 0 as x → ∞ (5) for z ≥ 0, t > 0, where the function φ0 = φ0(x, z) matches with φ∞ at x = 0 and x → ∞. Without loss of generality, we will assume that φ∞ = 0. In the case of steady free mush convection, the boundary condition for ψ can be written as ψ = 0 at x = 0, 0 x ∂ψ → ∂ as x → ∞, (6) in the case of unsteady free mush convection, it can be given by ψ = 0 at z = 0, 0 x ∂ψ → ∂ as z → ∞ (7) for х ≥ 0, t > 0. In this paper, we study the processes in a mushy region cooled from one side. In this model the flow occurs in a narrow thermal layer within the mushy region. The main aim of this paper is to study the qualitative asymptotic behaviour of self- similar solutions of the laminar boundary-layer flows in the steady case, describing essential physical properties of the process. We consider the behaviour of the stream function, temperature, and local solid fraction for unsteady situation too. The present paper is organized as follows. In Section 2 we study the situation of steady free mush convection and obtain the explicit lower and upper estimates for a solution of this problem at t = O(Ra–1) In the unsteady case, we find one of the set of explicit solutions of the problem for all t > 0. This result is contained in Section 3. Appendix contains some auxiliary routine calculations connected with Section 2. 2. Steady free mush convection In this section, we consider a particular asymptotic regime where the thermal and flow are steady. This model was considered in [3], where the numerical approach was applied. We use the analytical methods for studying the asymptotic behaviour of the appropriate functions. Using scaling analysis of (1)–(3) (similar to [3]), we consider ϒ , Φ , X, T, Ψ defined by 1/2= Raϒ ϒ , 1/2= RaΦ Φ , x = Ra–1/2X, t = Ra–1/2T, ψ = Ra1/2Ψ. (8) Here ϒ , Φ , X, T and Ψ are assumed to be O(1) as Ra → ∞. Substituting (8) into (1)–(3), taking the limit Ra → ∞ and rearranging, we find that 2 2= z X X z X ∂Ψ ∂θ ∂Ψ ∂θ ∂ θ⎛ ⎞Ω − +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ∂ , (9) Физика и техника высоких давлений 2014, том 24, № 1 38 2 2 1= T X X ∂φ ∂φ ∂ θ − − ∂ ∂ ΩΦ ∂ , (10) 2 2 = XX ∂ Ψ ∂θ − ∂∂ , (11) where 1 /ФΩ = + ϒ . In [3] the boundary-value problem represented by(9), (11), (5) and (6) was considered by using two different approaches: numerical and approximate one. We look for a similarity solution in the form of 1/2 1/2= ( )/z fΨ η Ω , θ = θ(η), φ = φ(η), (12) where η = Ω1/2X/z1/2. Then from (11) and (5), θ is given by ( ) = ( )f ′θ η − η , (13) and from (9) and (6), f satisfies 1( ) ( ) ( ) = 0 2 f f f′′′ ′′η + η η , (14) (0) = 0f , (0) = 1f ′ , ( ) 0f ′ η → as η → ∞. (15) The problem similar to (14), (15) appeared in papers by [2], where it was solved numerically only. Further, we study the behaviour of a solution of the problem (14), (15) and obtain the following proposition (see Appendix for proof): Proposition 1. A solution of the problem (14), (15) satisfies the following estimates: 1.568 < f∞: = f(∞) < 2, (16) 316( ) tanh ( ) min ,2 tanh 28 a f a ∗ ∗ η ⎡ η ⎤⎛ ⎞ ⎛ ⎞≤ η ≤ η ⎜ ⎟⎜ ⎟ ⎢ ⎥⎝ ⎠⎝ ⎠ ⎣ ⎦ (17) for all η ≥ 0, where a* ≈ 0.461. From Proposition 1 it follows the qualified estimations of the main parameters of the initial problem (the stream function ψ(x, z, t), the temperature θ(x, z, t), and local solid fraction φ(x, z, t)) at the small time t = O(Ra–1) only (see Appendix for details). Proposition 2. A solution of the system (9)–(11) with the boundary conditions (4), (5) and (6) satisfies the following estimates: 1/2 1/2 * 3 min := 16( ) tanh 8 z Ra x Raa za ⎡ ⎤Ω⎛ ⎞ ⎛ ⎞ψ ⎢ ⎥⎜ ⎟ ⎜ ⎟Ω⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ ≤ ≤ 1/2 1/2 max := 2 tanh 2 z Ra x Ra z ⎡ ⎤Ω⎛ ⎞ ⎛ ⎞ψ ⎢ ⎥⎜ ⎟ ⎜ ⎟Ω⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ , (18) Физика и техника высоких давлений 2014, том 24, № 1 39 1/2 min max:= 1 2 tanh := 2 x Raa z ⎡ ⎤Ω⎛ ⎞θ − + ≤ θ ≤ θ⎢ ⎥⎜ ⎟ ⎝ ⎠⎢ ⎥⎣ ⎦ 1/ 2 3:= 1 64( ) tanh 4 x Raa a z ⎡ ⎤Ω⎛ ⎞− + ⎢ ⎥⎜ ⎟ ⎝ ⎠⎢ ⎥⎣ ⎦ , (19) 1/ 2 1/2 2 min := 1 tanh 2 Ra a x Ra z z ∞ ⎧ ⎫⎡ ⎤⎛ ⎞ Ω⎪ ⎪⎛ ⎞φ − + φ ≤ φ⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟Ω Φ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎪ ⎪⎣ ⎦⎩ ⎭ ≤ 21/ 2 1/2 2 max := tanh 1 4 Ra a x Ra z z ∞ ⎧ ⎫⎡ ⎤⎛ ⎞ Ω⎪ ⎪⎛ ⎞≤ φ − + φ⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟Ω Φ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎪ ⎪⎣ ⎦⎩ ⎭ , (20) at t = O(Ra–1) for all x ≥ 0, z ≥ 0. Here 0 < a ≤ a, a is from Proposition 1. In comparison with the paper [3], our results describe completely the asymptotic behaviour of solution of system (9)–(11), which has the explicit (not numerical) representation. This is very important for concrete physical interests. It is significant that ψ∞(z) is included in estimates (18), (19) and (20). That is, ψ∞(z) has an influence on the estimation from below of the temperature θ(x, z, t) and the estimations from above of the local solid fraction φ(x, z, t) and the stream function ψ(x, z, t). Thus, this influence is essential and cannot be ignored. 3. Unsteady free mush convection In this section, we look for solutions of unsteady equations (1)–(3) for any Ra and t. As far as we concerned, this interesting situation was not considered before. We are succeeded in finding the explicit solution of system (1)–(3) (perhaps not unique). However, this solution characterizes the real physical behaviour of the mushy layer. In fact, there is obtained a family of solutions of the problem. Since the convection into the mushy region is directed along the axis x then it seems very natural to seek for solution of system (1)–(3) in the form of a travelling wave. Let ξ = x + t. We will seek this solution of the problem in the view θ = θ(ξ, z), ψ = ψ(ξ, z) and φ = φ(ξ, z). Then we arrive at the following system: , 0zξΔ θ = , (20) 0 z z ∂ψ ∂θ ∂ψ ∂θ − + = ∂ ∂ξ ∂ξ ∂ , (21) , =z Raξ ∂θ Δ ψ − ∂ξ , (22) with boundary conditions on the flow and thermal fields: ( ), 1zθ ξ = − at ξ = 1, ( ), 0zθ ξ → as ξ → ∞, z ≥ 0, (23) Физика и техника высоких давлений 2014, том 24, № 1 40 ( , ) = 0zψ ξ at z = 0, ξ ≥ 0, ( , ) 0z∂ψ ξ → ∂ξ as ξ → ∞, z ≥ 0. (24) Proposition 3. System (1)–(3) with the boundary conditions (4), (5) and (7) has the explicit solution: 21 arctan x z ⎛ ⎞θ = − + ⎜ ⎟π ⎝ ⎠ , (25) 2 21 2 2= ln( ) 2 Ra C z Raz z z x z x ψ + − + π π+ , (26) ( ) ( )0, , ,x z t x t zφ = φ + (27) for all x, z, t such that 2 2 2 2=z x C+ , where 0( , ,0) = ( , )x z x zφ φ due to condition (4), and 1 iC ∈ . Here, (27) means that the solid fraction is transmitted to the solid state. The Proposition 3 has clear physical meaning, namely, the local solid fraction into a mushy region decreases in time under the temperature and the stream func- tion which do not change in time. Below we show that Proposition 3 holds. In- deed, it is easy to check that the function ( ) 2, 1 arctan tz z ξ −⎛ ⎞θ ξ = − + ⎜ ⎟π ⎝ ⎠ (28) is an explicit solution of the boundary problem (20), (23). First we derive an equation for the function ψ(ξ, z). Let cosr tξ = ϕ+ , sinz r= ϕ , then from equation (22) with ( , ) ( , )z rξ ϕ we get 2 2 1 1 2 2sinr Ra r r r rr ∂ ∂ψ ∂ ψ ϕ⎛ ⎞ + = −⎜ ⎟∂ ∂ ∂ϕ π⎝ ⎠ . We are looking for solutions of this equation in the following form ( ) ( ), sin .r rψ ϕ = κ ϕ After simple computation we obtain the equation for function f(r): 2 2( ) ( ) ( ) Rar r r r r r′′ ′κ + κ − κ = − π . Solving this equation we find 12 1( ) ln i C Rar C r r r C r κ = + − ∀ ∈ π . Then ( )2 2 12 1 2 2( , ) = ln ( ) 2( ) i C z Raz C z z z t C z t ψ ξ + − + ξ− ∀ ∈ π+ ξ− R , (29) Физика и техника высоких давлений 2014, том 24, № 1 41 and the conditions (24) were satisfied. Substituting (28) and (29) i. e. ( ) ( ) 2 2 2 2 2 1 2 2 2 22 2 ( )= ln ( ) 2 ( )( ) z t z Ra Ra zC C z t z tz t ξ − − ψ + − + ξ − − π π + ξ −+ ξ − , ( ) 2 2 2 22 2 2 ( ) ( )= ( )( ) z t Ra z tC z tz t ξ ξ − ξ − ψ − − π + ξ −+ ξ − , 2 2 2= ( )z t z t ξ − θ − π + ξ − , 2 2 2= ( ) z z tξθ π + ξ − into (21), we obtain ( ) ( )2 22 1 22 ln ( )C Ra RaC z t z t − − + ξ − = π π+ ξ − . (30) Choosing 1 RaС = π , we find from (30) that ( ) ( )2 2 2 2 2 ( ) ln z ( )C z t t Ra π − = + ξ − + ξ − . (31) As the function Φ(v) = vlnv is monotone then there exists an inverse function Φ–1(.). Using this fact, we obtain from (31) that ( )22 1 222 Cz t C Ra − ⎛ ⎞π + ξ − = Φ − =⎜ ⎟ ⎝ ⎠ , (32) where C is an arbitrary constant. Thus, the equality (16) is valid if the variables ξ and z are satisfied to the relation (32). Using changing of variables ( , ) ( , , )z x z tξ in (28), (29) and (32) we obtain the following explicit solution of system (1)–(3) with conditions (4), (5) and (6). Thus, Proposition 3 is proved completely. 4. Convergence to the travelling wave Assume that Π = 1 in (3), and φ ≡ φ0 is a positive constant. Changing variables ( , , ) ( , , )x z t z tξ in (1)–(3), where ξ = x + t, we obtain the following system , ,=t z zξ ξθ + ⋅∇ θ Δ θu , (33) 0 ,(1 ) = 0t zξ− φ θ + ⋅∇ θu , (34) , =z Raξ ξ−Δ ψ θ , (35) where = ( , )z ξ−ψ ψu . We can reduce the system (33)–(35) to the one 0 ,=t zξφ θ Δ θ , (36) Физика и техника высоких давлений 2014, том 24, № 1 42 , =z Raξ ξ−Δ ψ θ (37) in the half-space { }:= 0, 0R z+ ξ ≥ ≥ . Note that if the solid fraction is absent, i.e. φ0 = 0, then we obtain that the solution (25), (26) is unique. Let us denote by st:=v θ−θ . Then from (36) and (20) we obtain for v the following equation 0 ,=t zv vξφ Δ , 0 st( , ,0) = ( , ) := ( , ,0) ( , )v z v z z zξ ξ θ ξ −θ ξ . (38) Problem (38) has the solution ( )2 2 0 0 0 0 0 ( ) ( ) 4( , , ) = ( , )d d 0 4 z z tv z t e v z z t +∞+∞ − φ ξ −ξ + − φ ξ ξ ξ → π ∫ ∫ as t → +∞. (39) From (39) it follows that θ → θst as t → +∞. Hence, it follows from this and (37) that ψ → ψst as t → +∞. Conclusion In this paper we consider situations of steady and unsteady free mush convection. For the steady regime the qualified estimates for the stream function, temperature, and local solid fraction are found for large values of the Rayleigh number and small time. For the unsteady case the precise behavior of these main characteristics is established. At that the behaviour of the temperature and stream function depend on the measured vertical upwards. The local solid fraction decreases under stationary behaviour of the temperature and the stream function. We use the non-similar solution technique giving us possibility to establish the qualified estimates of the main characteristics of the process. A detailed analysis of solidification in mushy region is provided under the assumption that the permeability of the mush is uniform. Appendix Proof of Proposition 1 We consider the following auxiliary Cauchy problem for the problem (14), (15): 1( ) ( ) ( ) = 0 2 f f f′′′ ′′η + η η , (40) (0) = 0f , (0) = 1f ′ , (0) =f a′′ − , (41) where a > 0. Now we show that there exists a parameter a such that the solution of problem (40), (41) satisfy the conditions ( ) 0f ′ η → as η → ∞; ( ) 0f η ≥ is uniformly bounded. (42) By (40) and (41) we deduce that Физика и техника высоких давлений 2014, том 24, № 1 43 0 1( ) = exp ( )d 2 f a f z z η⎧ ⎫⎪ ⎪′′ η − −⎨ ⎬ ⎪ ⎪⎩ ⎭ ∫ , (43) 0 0 1( ) = 1 exp ( )d d 2 y f a f z z y η ⎧ ⎫⎪ ⎪′ η − −⎨ ⎬ ⎪ ⎪⎩ ⎭ ∫ ∫ , (44) 0 0 0 1( ) = exp ( )d d d 2 y v f a f z z v y η ⎧ ⎫⎪ ⎪η η− −⎨ ⎬ ⎪ ⎪⎩ ⎭ ∫∫ ∫ . (45) From (43) it follows that f′ decreases and f is concave. Hence, in view of 1( ) = ( ) ( ) 0 2 f f f′′′ ′′η − η η ≥ , we arrive at ( )f ′ η is convex and 0 ( ) 1 0f ′≤ η ≤ ∀η ≥ . (46) It follows from (45) that ( )f η ≤ η , whence 2 0 0 ( ) exp d d 4 y vf a v y η ⎛ ⎞ η ≤ η− −⎜ ⎟⎜ ⎟ ⎝ ⎠ ∫∫ , (47) and 2 0 0 0 1( ) = 1 exp ( )d d 1 exp d 2 4 y yf a f z z y a y η η⎛ ⎞ ⎛ ⎞ ′ ⎜ ⎟η − − ≤ − −⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ∫ ∫ ∫ . (48) In view of (48), we find that 2 0 0 = ( ) 1 exp d 4 yf a y ∞ ⎛ ⎞ ′ ∞ ≤ − −⎜ ⎟⎜ ⎟ ⎝ ⎠ ∫ , whence it follows that 1 2 0 1 10 = = 0.5641895 exp d 4 a a y y ∞≤ ≤ ≈ ⎛ ⎞ π −⎜ ⎟⎜ ⎟ ⎝ ⎠ ∫ . (49) The last inequality guarantees that conditions (42) is valid for any a satisfying (49). Moreover, using (47), from (45) we deduce 2 2 0 0 0 0 ( ) exp exp d d d d 4 2 4 y v zv a wf a w z v y η ⎧ ⎫⎛ ⎞⎪ ⎪η ≤ η− − + −⎜ ⎟⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭ ∫∫ ∫∫ . (50) Analogously to (49), we find from 2 2 0 0 0 0 = ( ) 1 exp exp d d d 4 2 4 y zy a wf a w z y ∞ ⎧ ⎫⎛ ⎞⎪ ⎪′ ∞ ≤ − − + −⎜ ⎟⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭ ∫ ∫∫ , that Физика и техника высоких давлений 2014, том 24, № 1 44 20 a a≤ ≤ = 0.45342952. (51) Continuing the same iteration procedure, we can find the sharp upper bound a∞ for the desired а, i.e. a a∞≤ . Now we show the estimate from below of the solution to problem (40), (41). It follows from (45) that 2 0 0 ( ) d d = 2 y af a v y η η ≥ η− η− η∫∫ . We denote by f1 the function on the right-hand side of the last inequality: 2 1( ) := 2 af η η− η , 20 x a ≤ ≤ . Taking into account that 1( ) = 1 = 0f aη′ − η for max 1= a η , we see that 1 2 1 1 1 1= = 2 2 af a a aa ⎛ ⎞ −⎜ ⎟ ⎝ ⎠ Hence, 2( ) 2 af η ≥ η− η , 10 x a ≤ ≤ . From the decreasing of f ′ it follows that 1( ) = 2 f A a η ≥ for 1x a ≥ . Using (45) we obtain that 0 0 0 2( ) exp d d = 1 exp d 2 2 y A Af a v v y a y y A η η ⎧ ⎫⎛ ⎞ ⎛ ⎞η ≥ η− − η− − −⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎩ ⎭ ∫∫ ∫ = = 2 2 2 41 exp 1 exp 2 2 A a Aa A A A ⎧ ⎫⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞η− η− − − η ≥ − − η⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎩ ⎭ , where 21 0a A − ≥ . Then A ≥ 2a, 2 1 4 a ≤ , whence 10 2 a≤ ≤ . Let us denote by f2 the following function 3 2( ) = 16 1 exp 4 f a a ⎡ η ⎤⎛ ⎞η − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ . Then 3( ) 16 1 exp 4 f a a ⎡ η ⎤⎛ ⎞η ≥ − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ , 1 a η ≥ . For the continuity we suppose that 1 2 1 1=f f a a ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ : Физика и техника высоких давлений 2014, том 24, № 1 45 3 2 1 1= 16 1 exp 2 4 a a a ⎡ ⎤⎛ ⎞− −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ , 4 5/ 4 2 4 1 1 1exp = 1 > 0 > > 2 324 32 a a a a −⎛ ⎞− − ⇒ ⇒⎜ ⎟ ⎝ ⎠ . Then for 2–5/4 ≈ 0.42045< a ≤ 1/2 we have 2 * 4 14 ln 1 1 = 0 = 0.46106906 32 a a a ⎛ ⎞− + ⇒⎜ ⎟ ⎝ ⎠ . Finally, we derive * 2 * * 3 * * 1, 0 , 2 ( ) 116( ) 1 exp , , 4 a af a a a ⎧ η− η ≤ η ≤⎪ ⎪η ≥ ⎨ ⎡ ⎤η⎛ ⎞⎪ − − η ≥⎜ ⎟⎢ ⎥⎪ ⎝ ⎠⎣ ⎦⎩ where a* = 0.46106906. This means that * 3 * 3 min* ( ) 16( ) 1 exp ( ) := 16( ) tanh 0 4 8 f a f a a a ⎡ ⎤η η⎛ ⎞ ⎛ ⎞η ≥ − − ≥ η ∀ η ≥⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ , (52) where fmin(∞) ≈ 1.568259. Due to the estimate (52), we need that 0 ≤ a ≤ a. As a* > > a∞ then (52) is the lower estimate. Coming back to equation (40) and using (48), we have ( )221 1 1( ) ( ) = ( ) 4 2 2 f f f ′′⎛ ⎞′ ′η + η η ≤⎜ ⎟ ⎝ ⎠ . Integrating this inequality with (41), we deduce that 2 21( ) ( ) 1 4 4 f f aη′ η + η ≤ − η+ . As 2 1 1 4 aη − η+ ≤ for 0 ≤ η ≤ 4a then, solving 21( ) ( ) 1 4 f f′ η + η ≤ with f(0) = 0, (0) = 1f ′ , we find that max( ) ( ) := 2 tanh 2 f f η⎛ ⎞η ≤ η ⎜ ⎟ ⎝ ⎠ (53) for all η:0 ≤ η ≤ 4a. As the graph of the right-hand side of (50) lies under the one of 2 tanh 2 η⎛ ⎞ ⎜ ⎟ ⎝ ⎠ and max (0) = 0f ′′ , then choosing a ≤ a∞ we obtain that the estimate (53) is valid for all η ≥ 0. Физика и техника высоких давлений 2014, том 24, № 1 46 Thus, there is the the parameter point a ∈ (0, a) such that the problem (14), (15) has an unique solution, and the following estimates hold: min maxf f f≤ ≤ . Moreover 1.568 ≤ f(∞) ≤ 2. These estimates provide reliable analytical informa- tion about the behaviour of solution. Proof of Proposition 2 The estimates (18) is a simple corollary from (17) that is, 1/ 2 1/ 2 3 16( ) tanh 2 tanh 28 Raz Raza a η η⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞≤ ψ ≤⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟Ω Ω⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ . (54) From the equalities (44) and (13), in view of estimate (17), we deduce 31 2 tanh ( ) 1 64( ) tanh 2 4 a a aη η⎛ ⎞ ⎛ ⎞− + ≤ θ η ≤ − +⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ . (55) Let us obtain estimations of function φ(η). From (10) we have 1/2 1/2 ( )( ) = z ∞ ′θ η φ η + φ Ω Φ , (56) where φ∞ is defined by (4). Then, taking into account (43), we obtain from (56) that 1/2 0 exp tanh d ( ) 2 a Ra y y z η ∞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟− + φ ≤ φ η⎜ ⎟ ⎜ ⎟⎜ ⎟Φ Ω⎝ ⎠ ⎝ ⎠⎝ ⎠ ∫ ≤ ≤ 1/2 * 3 * 0 exp 8( ) tanh d 8 a Ra ya y z a η ∞ ⎛ ⎞⎛ ⎞⎛ ⎞ ⎜ ⎟− + φ⎜ ⎟ ⎜ ⎟⎜ ⎟Φ Ω⎝ ⎠ ⎝ ⎠⎝ ⎠ ∫ . (57) Inequalities (19) and (20) follow from estimates (55), (57). 1. P. Aussillous, A.J. Sederman, L.F. Gladden, H.E. Huppert, and M.G. Worster, Journal of Fluid Mechanics 552, 99 (2006). 2. P. Cheng, W.J. Minkowycz, J. Geophys. Res. 82, 2040 (1977). 3. P. Guba, M.G. Worster, Journal of Fluid Mechanics 558, 69 (2006). 4. R.A. Jarvise, H.E. Huppert, Journal of Fluid Mechanics 303, 103 (1995). 5. H.E. Huppert, Journal of Fluid Mechanics 212, 209 (1990). 6. W.W. Mullins, R.F. Sekerka, J. Appl. Phys. 35, 444 (1964). 7. O.E. Phillips, Flow and Reactions in Permeable Rocks, Cambridge University Press, Cambridge (1991). 8. R.C. Kerr, S. Tait, Journal of Geophysical Research 91, 3591 (1986). 9. F.J. Spera, D.A. Yum, D.V. Kemp, Nature 310, 764 (1984). 10. J.S. Turner, L.B. Gustafson, J. Volcanol. Geotherm. Res. 11, 93 (1981). Физика и техника высоких давлений 2014, том 24, № 1 47 11. M.G. Worster, NATO ASI Series E219, 113 (1992). 12. M.G. Worster, Journal of Fluid Mechanics 237, 649 (1992). 13. M.G. Worster, Ann. Rev. Fluid Mech. 29, 91 (1997). Ю.В. Намлеева, Р.М. Таранец ТЕОРЕТИЧЕСКИЕ ИССЛЕДОВАНИЯ ИДЕАЛЬНОЙ МОДЕЛИ «MUSHY REGION» В данной работе исследуется поведение функции потока, температуры и ло- кальной твердой фракции для идеальной модели смешанного слоя. В случае устой- чивой свободной конвекции найдены точные нижние и верхние оценки основных функций, характеризующих процесс, при больших значениях числа Рэлея. Для не- стационарного режима также найден явный вид этих основных характеристик. Ключевые слова: смешанный слой, устойчивая и неустойчивая конвекция, функ- ция потока, температура, локальные твердые и жидкие фракции

Theoretical investigations of the idealized model for the mushy region

Institution

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