Symmetry of heat and mass transfer equations in case of dependence of thermal diffusivity coefficient either on temperature or concentration

Степанова, И. В.

doi:10.1002/mma.4811

Cited by 4 publications

(4 citation statements)

References 27 publications

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“…In turn, from the estimates (43), (44), and (45) and from the second identity in (20), it follows that the bilinear form a w,η 1 : V × V → R defined in (39) is continuous and coercive with constant (ν * /2), where ν * is defined in (18). From Lax-Milgram's theorem, it then follows that for any pair (w, η) ∈ V × H 1 0 (Ω), there is a single solution ũ ∈ V of the problem (39), and the following estimate is performed:…”

Section: Global Solvability Of Problemmentioning

confidence: 99%

“…The second group contains works devoted to application of the Li-Ovsyannikov symmetry method to study qualitative properties of solutions of equations of HMT in viscous binary and/or heat-conducting liquids. This group includes a very large quantity of works (see, e.g., [17][18][19][20][21], monographs [22] and reviews [23,24]). Another group of works is that in which mathematical modeling of fluid motion processes takes into account thermodiffusion effects (or Sorét effects) and/or concentration diffusion effects (or Dufort effects).…”

Section: Introduction and Statement Of The Boundary Value Problemmentioning

confidence: 99%

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Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer

Alekseev,

Soboleva

2024

Mathematics

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We consider boundary value problems for a nonlinear mass transfer model, which generalizes the classical Boussinesq approximation, under inhomogeneous Dirichlet boundary conditions for the velocity and the substance’s concentration. It is assumed that the leading coefficients of viscosity and diffusion and the buoyancy force in the model equations depend on concentration. We develop a mathematical apparatus for studying the inhomogeneous boundary value problems under consideration. It is based on using a weak solution of the boundary value problem and on the construction of liftings of the inhomogeneous boundary data. They remove the inhomogeneity of the data and reduce initial problems to equivalent homogeneous boundary value problems. Based on this apparatus we will prove the theorem of the global existence of a weak solution to the boundary value problem under study and establish important properties of the solution. In particular, we will prove the validity of the maximum principle for the substance’s concentration. We will also establish sufficient conditions for the problem data, ensuring the local uniqueness of weak solutions.

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Section: Global Solvability Of Problemmentioning

confidence: 99%

Section: Introduction and Statement Of The Boundary Value Problemmentioning

confidence: 99%

Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer

Alekseev,

Soboleva

2024

Mathematics

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“…Semenov, who construct exact solutions for equations of the considered type that are not diffusion waves [42], and I.V. Stepanova, who deals with modeling heat and mass transfer using nonlinear parabolic systems [43,44].…”

Section: Introductionmentioning

confidence: 99%

Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type

Казаков

2021

Symmetry

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The article deals with nonlinear second-order evolutionary partial differential equations (PDEs) of the parabolic type with a reasonably general form. We consider the case of PDE degeneration when the unknown function vanishes. Similar equations in various forms arise in continuum mechanics to describe some diffusion and filtration processes as well as to model heat propagation in the case when the properties of the process depend significantly on the unknown function (concentration, temperature, etc.). One of the exciting and meaningful classes of solutions to these equations is diffusion (heat) waves, which describe the propagation of perturbations over a stationary (zero) background with a finite velocity. It is known that such effects are atypical for parabolic equations; they arise as a consequence of the degeneration mentioned above. We prove the existence theorem of piecewise analytical solutions of the considered type and construct exact solutions (ansatz). Their search reduces to the integration of Cauchy problems for second-order ODEs with a singularity in the term multiplying the highest derivative. In some special cases, the construction is brought to explicit formulas that allow us to study the properties of solutions. The case of the generalized porous medium equation turns out to be especially interesting as the constructed solution has the form of a soliton moving at a constant velocity.

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“…Note that in the literature, you can find systems of a slightly different form, which are also based on second-order parabolic equations with power nonlinearity. In particular, reaction-diffusion systems where the functions f and p also depend on independent variables are considered in [27,28]; systems describing heat and mass transfer are studied in [29]. The objective of this research is the problem of constructing HDW-type solutions of the system in Equation ( 2) that has a known law of front motion (a problem with a given diffusion front).…”

Section: Introductionmentioning

confidence: 99%

Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type

2020

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The paper deals with a system of two nonlinear second-order parabolic equations. Similar systems, also known as reaction-diffusion systems, describe different chemical processes. In particular, two unknown functions can represent concentrations of effectors (the activator and the inhibitor respectively), which participate in the reaction. Diffusion waves propagating over zero background with finite velocity form an essential class of solutions of these systems. The existence of such solutions is possible because the parabolic type of equations degenerates if unknown functions are equal to zero. We study the analytic solvability of a boundary value problem with the degeneration for the reaction-diffusion system. The diffusion wave front is known. We prove the theorem of existence of the analytic solution in the general case. We construct a solution in the form of power series and suggest recurrent formulas for coefficients. Since, generally speaking, the solution is not unique, we consider some cases not covered by the proved theorem and present the example similar to the classic example of S.V. Kovalevskaya.

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Symmetry of heat and mass transfer equations in case of dependence of thermal diffusivity coefficient either on temperature or concentration

Cited by 4 publications

References 27 publications

Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer

Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer

Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type

Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type

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