The analytical model of unsteady flow of incompressible monodisperse gas suspension with one-dimensional flow geometry 

Dmitriy A. Tukmakov, Candidate of physical and mathematical sciences, researcher, Federal Research Center “Kazan Scientific Center of the Russian Academy of Sciences” (2/31 Lobachevskogo street, Kazan, Russia), E-mail: tukmakovda@imm.knc.ru 

533.2, 51-72 

10.21685/2072-3040-2021-3-5 

Background. One of the developing branches of modern fluid and gas mechanics is the mechanics of multiphase and multicomponent media. Experimental research of many dynamics processes of inhomogeneous media is difficult, therefore, mathematical modeling is of great importance. Moreover, many models are essentially nonlinear, for this reason, numerical methods are used to integrate such models. The purpose of this research is to obtain an exact solution for one of the special cases, assuming a number of simplifications – flow’s one-dimensionality, incompressibility of the carrier and dispersed components, and the linear nature of inter-component momentum exchange. At the same time, to obtain an exact solution, a model was used that implements the continual approach of the mechanics of multiphase media - taking into account the intercomponent exchange of momentum and heat transfer. Materials and methods. The research presents a mathematical model of a onedimensional unsteady flow of an incompressible two-component medium. The equations of dynamics are derived from the equations of the dynamics of the flow of a two-component inhomogeneous medium, taking into account the inter-component exchange of momentum and heat. In the model under consideration, the intercomponent force interaction took into account the Stokes force. The system of partial differential equations, due to the condition of incompressibility of the flow, is reduced to a nonlinear system of ordinary differential equations. Results. The system of nonlinear differential equations is reduced to the sequential solution of three linear systems of ordinary differential equations with respect to six unknown functions. The analytical solution of the mathematical model of the gas suspension flow is implemented in the form of a computer program. Conclusions. The exact solution for the continuous model of aerosol dynamics can be used to test numerical models of aerosol dynamics. 

hydrodynamics, heterogeneous media, continual model, one-dimensional flow, incompressible medium, Euler’s equation 

1. Nigmatulin R.I. Osnovy mekhaniki geterogennykh sred = Fundamentals of mechanics of heterogeneous media. Moscow: Nauka, 1978:336. (In Russ.)
2. Sternin L.E. Dvukhfaznye mono- i polidispersnye techeniya gaza s chastitsami = Twophase mono- and polydisperse gas flows with particles. Moscow: Mashinostroenie, 1980:176. (In Russ.)
3. Nigmatulin R.I. Dinamika mnogofaznykh sred. Ch. I. = Dynamics of multiphase media. Part 1. Moscow: Nauka. Gl. red. fiz.-mat. lit., 1987:464. (In Russ.)
4. Kutushev A.G. Matematicheskoe modelirovanie volnovykh protsessov v aerodispersnykh i poroshkoobraznykh sredakh = Mathematical modeling of wave processes in aerodispersed and powdery media. Saint Petersburg: Nedra, 2003:284. (In Russ.)
5. Fedorov A.V., Fomin V.M., Khmel' T.A. Volnovye protsessy v gazovzvesyakh chastits metallov = Wave processes in gas suspensions of metal particles. Novosibirsk: Parallel', 2015:301. (In Russ.)
6. Gubaydullin D.A. Dinamika dvukhfaznykh parogazokapel'nykh sred = Dynamics of two-phase vapor-gas-droplet media. Kazan: Izd-vo Kazanskogo matematicheskogo obshchestva, 1998:153. (In Russ.)
7. Emel'yanov V.N., Yakimov I.V. Near-nozzle two-phase flows. Khimicheskaya fizika i mezoskopiya = Chemical physics and mesoscopy. 2006;(3):287–294. (In Russ.)
8. Shapovalov A.V., Shapovalov V.A., Ryazanov V.I. Mathematical model of the propagation of impurities in the near zone during the operation of rocket engines. Nauka. Innovatsii. Tekhnologii = Science. Innovation. Technologies. 2017;(2):87–96. (In Russ.)
9. Surov V.S. The hyperbolic model of a single-velocity multicomponent heat-conductive medium. Teplofizika vysokikh temperature = Thermal physics at high temperature. 2009;47(6):905–913. (In Russ.)
10. Kutushev A.G., Rodionov S.P. An interaction of weak shock waves with a layer of a powdery medium. Fizika goreniya i vzryva = Combustion and explosion physics. 2000;(3):131–140. (In Russ.)
11. Shagapov V.Sh., Galimzyanov M.N., Agisheva U.O. Solitary waves in a gas-liquid bubble mixture. Izvestiya Saratovskogo universiteta. Novaya seriya. Ser. Matematika. Mekhanika. Informatika = Proceedings of Saratov University. New series. Series: Mathematics, Mechanics. Informatics. 2020;20(2):232–240. (In Russ.)
12. Landau L.D., Lifshits E.V. Teoreticheskaya fizika. Gidrodinamika = Theoretical physics. Hydrodynamics. Moscow: Nauka, 1986:736. (In Russ.)
13. Loytsyanskiy L.G. Mekhanika zhidkosti i gaza = Mechanics of liquid and gas. Moscow: Drofa, 2003:784. (In Russ.)
14. Fletcher K. Vychislitel'nye metody v dinamike zhidkostey: v 2 t.; per. s angl. = Computational methods in fluid dynamics: in 2 volumes; translated from English. Moscow: Mir, 1991;2:552. (In Russ.)
15. Brykov N.A., Teterina I.V., Sakhin V.V. On the numerical solution of gas dynamics problems in the zero-dimensional formulation. Sistemnyy analiz i analitika = System analysis and analytics. 2017;(3):22–29. (In Russ.)
16. Tukmakov A.L., Tukmakov D.A. Application of an Implicit Finite-Difference Scheme with Weights for Modeling Gas Oscillations in an Acoustic Resonator. Vestnik Kazanskogo gosudarstvennogo tekhnicheskogo universiteta im. A.N. Tupoleva = Bulletin of Kazan State Technical University named after A. N. Tupolev. 2011;(4):119–127. (In Russ.)
17. Tukmakov D.A. Increasing the intensity of gas oscillations in an acoustic resonator. Inzhenerno-fizicheskiy zhurnal = Engineering physics journal. 2015;88(3):638–641. (In Russ.)
18. Tukmakov A.L., Bayanov R.I., Tukmakov D.A. A flow of a polydisperse gas suspension in a channel accompanied by coagulation in a nonlinear wave field. Teplo-fizika i aeromekhanika = Heat physics and aeromechanics. 2015;22(3):319–325. (In Russ.)
19. Tukmakov A.L., Tukmakov D.A. Dynamics of a charged with an initial spatially nonuniform distribution of the average density of a dispersed phase during the transition to an equilibrium state. Teplofizika vysokikh temperature = Thermal physics at high temperature. 2017;(4):509–512. (In Russ.)
20. Tukmakov D.A. Finite-difference model of the dynamics of a homogeneous mixture as applied to the study of propagation and reflection of a shock wave of high intensity in a hydrogen-air medium. Modeli, sistemy, seti v ekonomike, tekhnike, prirode i obshchestve = Models, systems, networks in economics, technology, nature and society. 2020;(1):86–97. (In Russ.)
21. Imas O.N, Pakhomova E.G., Rozhkova S.V., Ustinova I.G. Lektsii po differentsial'nym uravneniyam = Lectures on differential equations. Tomsk: Izd-vo Tomskogo politekhnicheskogo universiteta, 2012:193. (In Russ.)
22. Matrosov A.V. Maple 6. Reshenie zadach vysshey matematiki i mekhaniki = Maple 6. Solving problems of higher mathematics and mechanics. Saint Petersburg, 2001:528. (In Russ.)