Vallet, G. Mathematical analysis of thermal transfer in dispersive systems subject to phase transformations. (Analyse mathématique des transferts thermiques dans des systèmes dispersés subissant des transformations de phases.) (French) Zbl 0795.45015 RAIRO, Modélisation Math. Anal. Numér. 27, No. 7, 895-923 (1993). The present paper is devoted to the analytical study of some mathematical models related to crystallization and melting processes within an emulsion, contained in the cylinder \(\Omega=D \times (0,H)\), \(D\) being a disk in \(\mathbb{R}^ 2\) with center at the origin. The basic state equations are the following: \[ \rho \alpha (t,x,u,\varphi) \partial u/ \partial t- \text{div} \bigl[ \kappa (t,x,u,\varphi) \nabla u \bigr] = cJ(u)(1- \varphi) \text{ in } (0,T) \times\Omega:=Q \]\[ \partial \varphi/ \partial t=J (u)(1-\varphi) \text{ in } \Gamma,\;\varphi (0,\cdot)=0 \text{ in } \Omega \] where \(\Gamma=\partial D \times (0,H)\) and \(J\) satisfies: i) \(J(x)=0\) for \(x \in \{0\} \cup [T_ F,+\infty)\); ii) \(| J (x_ 2)- J_ 1(x_ 1) | \leq | x_ 2-x_ 1 |\) \(\forall x_ 1,x_ 2 \in [0,+\infty)\).1st case: crystallization process. Functions \(\alpha\) and \(\kappa\) are assumed to depend on \(u\), only. Replacing \(\varphi\) in terms of \(u\), the problem for \(u\) reduces to the following problem (P1): determine a function \(u \in K=\{v \in H^ 1(Q):\overline \theta \leq u(t,x) \leq T_{\max}\) for a.e. \((t,x) \in Q\}\) satisfying the equations \[ \partial \beta(u)/ \partial t-\Delta \varphi (u)={\mathcal I} (u) \text{ in } Q \]\[ {-\partial \varphi (u) \over \partial \nu} = \lambda (u-u^ \infty) \chi_ \Gamma \text{ on } \partial Q,\quad u(0,\cdot)=u_ 0 \text{ in } \Omega. \] Here \(\overline \theta\) and \(T_{\max}\) are constants, \[ {\mathcal I} (u)(t,x)=cJ \bigl( u(t,x) \bigr) \exp \left( -\int^ t_ 0 J \bigl( u(s,x) \bigr) ds \right) \] and the functions \(\beta (u)=\rho \int^ u_ 0 \alpha (s)ds\), \(\varphi (u)=\int^ u_ 0 \kappa (s)ds\) are assumed to satisfy the following bounds for some positive constants \(m\) and \(M(m \leq M)\): \(m \leq \beta'(u) \leq M\), \(m \leq \varphi' (u) \leq M\) \(\forall u \in [0,T_{\max}]\). Moreover, \(\nu\) and \(\chi_ \Gamma\) denote the outward unit normal vector to \(\Gamma\) and the characteristic function of \(\Gamma\). Finally, the initial datum \(u_ 0\) is assumed to belong to \({\mathcal U} = \{v \in H^ 1(\Omega):\overline \theta \leq v(x) \leq T_{\max}\) for a.e. \(x \in \Omega\), \(\int_ \Omega \nabla \varphi (v) \cdot \nabla w dx \leq c \int_ \Omega J(v)w dx\) \(\forall w \in H^ 1(\Omega)\), \(w \geq 0\) a.e. in \(\Omega\}\) and \(u^ \infty (t,x)=u_ 0(x)+\sigma (t)[\overline \theta-u_ 0(x)]\), where \(\sigma \in H^ s (\mathbb{R}_ +)\) \((s \geq 1)\) satisfies: i) \(\sigma (0)=0\); ii) \(\sigma (t)=0\) \(\forall t \geq \overline t\).Under the previous assumptions the author shows that problem (P1) admits a unique solution continuously depending (with respect to the \(L^ 1 (\Omega)\)-norm) upon the data. Moreover, as \(t\) tends to \(+\infty\), \(u(t)\) converges to \(\overline \theta\), strongly in \(L^ p (\Omega)\) for any \(p \in (1,+ \infty)\) and weakly in \(H^ 1(\Omega)\).Problem (P1) is also studied when the cylinder \(\Omega\) is replaced by the section disk \(D\). In this case we have to substitute everywhere \(D\) and \(\partial D\) for \(\Omega\) and \(\Gamma\), respectively. The author proves a decomposition spectral theorem and shows that radial initial data imply radial solutions. 2nd case: crystallization process. Functions \(\alpha\) and \(\kappa\) are assumed to depend on \(\varphi\), only. We consider the following problem (P2): determine two functions \(u \in K= \{v \in H^ 1(Q)\): \(\overline \theta \leq u(t,x) \leq T_{\max}\) for a.e. \((t,x) \in Q\}\) and \(\varphi\) satisfying the equations \[ g(\varphi) \partial u / \partial t-\text{div} \bigl[ f(\varphi) \nabla u \bigr] = cJ(u)(1-\varphi) \text{ in } Q \]\[ \varphi (t,x)=1-\exp \left( \int^ t_ 0 J \bigl( u(s,x) \bigr) ds \right)\;(t,x) \in Q \]\[ -\partial \varphi (u)/ \partial \nu=\lambda (u- u^ \infty) \chi_ \Gamma \text{ on } \partial Q,\quad u (0,\cdot)=u_ 0 \text{ in } \Omega. \] Here \(f\) and \(g\) are Lipschitz continuous functions, the former being increasing, the latter decreasing, satisfying the following bounds for some positive constants \(m\) and \(M\) \((m \leq M)\): \(m \leq f (\varphi) \leq M\), \(m \leq g (\varphi) \leq M\), \(\forall \varphi \in [0,1]\). The author proves an existence result and a uniqueness one (the latter only when \(\Omega\) is an interval in \(\mathbb{R})\).3rd case: a melting process for a crystallized emulsion. The model leads to dealing with the following problem (P3): determine a function \(u:Q \to \mathbb{R}\) satisfying the equations \[ (\partial u/t)(t,x)-\Delta u(t,x)=(- \partial/ \partial t)H \left( k \int^ t_ 0 u^ +(s,x)ds \right) \]\[ u(0,x)=-1,\;x \in \Omega, \quad -\partial \varphi (u)/ \partial \nu=\lambda (u-u^ \infty) \chi_ \Gamma \text{ on } \partial \Omega \] where \(u^ +=\max (u,0)\) and \(H(x)=\max (x^ +,1)\). Concerning problem (P3) the author proves an existence and uniqueness result. Reviewer: A.Lorenzi (Milano) Cited in 4 Documents MSC: 45K05 Integro-partial differential equations 35K20 Initial-boundary value problems for second-order parabolic equations 35K55 Nonlinear parabolic equations 80A22 Stefan problems, phase changes, etc. Keywords:thermal transfer; phase transformations; dispersive systems; nonlinear integro differential parabolic equations; existence uniqueness; continuous dependence; crystallization; decomposition spectral theorem; radial solutions; melting PDFBibTeX XMLCite \textit{G. Vallet}, RAIRO, Modélisation Math. Anal. Numér. 27, No. 7, 895--923 (1993; Zbl 0795.45015) Full Text: DOI EuDML References: [1] A. ABKARI, Modélisation d’une réaction chimique d’ordre un dans un calorimètre, Thèse, Lyon, 1984. [2] R. DAUTRAY, J. L. LIONS, Analyse mathématique et calcul numérique, Masson, Paris, 1988. [3] J. P. DUMAS, M. STRUB, M. KRICHI, Étude thermique des changements de phases dans une émulsion, Rev. Gén. Therm. Fr., 341, mai 1990, pp. 267-273. [4] G. 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