Symmetry

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20 pages, 364 KiB

Open AccessArticle

Theoretical Analysis of Boundary Value Problems for Generalized Boussinesq Model of Mass Transfer with Variable Coefficients

by Gennadii Alekseev and Roman Brizitskii

Symmetry 2022, 14(12), 2580; https://doi.org/10.3390/sym14122580 - 6 Dec 2022

Cited by 12 | Viewed by 1367

Abstract

A boundary value problem is formulated for a stationary model of mass transfer, which generalizes the Boussinesq approximation in the case when the coefficients in the model equations can depend on the concentration of a substance or on spatial variables. The global existence [...] Read more.

A boundary value problem is formulated for a stationary model of mass transfer, which generalizes the Boussinesq approximation in the case when the coefficients in the model equations can depend on the concentration of a substance or on spatial variables. The global existence of a weak solution of this boundary value problem is proved. Some fundamental properties of its solutions are established. In particular, the validity of the maximum principle for the substance’s concentration has been proved. Sufficient conditions on the input data of the boundary value problem under consideration, which ensure the local existence of the strong solution from the space

H^{2}

, and conditions that ensure the conditional uniqueness of the weak solution with additional property of smoothness for the substance’s concentration are established. Full article

(This article belongs to the Special Issue Mathematical Fluid Dynamics and Symmetry)

28 pages, 505 KiB

Open AccessFeature PaperArticle

Entropy Correct Spatial Discretizations for 1D Regularized Systems of Equations for Gas Mixture Dynamics

by Alexander Zlotnik, Anna Fedchenko and Timofey Lomonosov

Symmetry 2022, 14(10), 2171; https://doi.org/10.3390/sym14102171 - 17 Oct 2022

Cited by 4 | Viewed by 1139

Abstract

One-dimensional regularized systems of equations for the general (multi-velocity and multi-temperature) and one-velocity and one-temperature compressible multicomponent gas mixture dynamics are considered in the absence of chemical reactions. Two types of the regularization are taken. For the latter system, diffusion fluxes between the [...] Read more.

One-dimensional regularized systems of equations for the general (multi-velocity and multi-temperature) and one-velocity and one-temperature compressible multicomponent gas mixture dynamics are considered in the absence of chemical reactions. Two types of the regularization are taken. For the latter system, diffusion fluxes between the components of the mixture are taken into account. For both the systems, the important mixture entropy balance equations with non-negative entropy productions are valid. By generalizing a discretization constructed previously in the case of a single-component gas, we suggest new nonstandard symmetric three-point spatial discretizations for both the systems which are not only conservative in mass, momentum, and total energy but also satisfy semi-discrete counterparts of the mentioned entropy balance equations with non-negative entropy productions. Importantly, the basic discretization in the one-velocity and one-temperature case is not constructed directly but by aggregation of the discretization in the case of general mixture, and that is a new approach. In this case, the results of numerical experiments are also presented for contact problems between two different gases for initial pressure jumps up to 2500. Full article

(This article belongs to the Special Issue Mathematical Fluid Dynamics and Symmetry)

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14 pages, 1403 KiB

Open AccessFeature PaperArticle

On an Important Remark Concerning Some MHD Motions of Second-Grade Fluids through Porous Media and Its Applications

by Constantin Fetecau and Dumitru Vieru

Symmetry 2022, 14(9), 1921; https://doi.org/10.3390/sym14091921 - 14 Sep 2022

Cited by 5 | Viewed by 2586

Abstract

In this work it is proven that the governing equations for the fluid velocity and non-trivial shear stress corresponding to some isothermal MHD unidirectional motions of incompressible second-grade fluids through a porous medium have identical forms. This important remark is used to provide [...] Read more.

In this work it is proven that the governing equations for the fluid velocity and non-trivial shear stress corresponding to some isothermal MHD unidirectional motions of incompressible second-grade fluids through a porous medium have identical forms. This important remark is used to provide exact steady-state solutions for motions with shear stress on the boundary when similar solutions of some motions with velocity on the boundary are known. Closed-form expressions are provided both for the fluid velocity and the corresponding shear stress and Darcy’s resistance. As a check of the results that are obtained here, the solutions corresponding to motions over an infinite flat plate are presented in different forms whose equivalence is graphically proven. In the case of the motions between infinite parallel plates, the fluid behavior is symmetric with respect to the median plane due to the boundary conditions. Full article

(This article belongs to the Special Issue Mathematical Fluid Dynamics and Symmetry)

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17 pages, 394 KiB

Open AccessArticle

Regularity and Travelling Wave Profiles for a Porous Eyring–Powell Fluid with Darcy–Forchheimer Law

by José Luis Díaz Palencia, Saeed ur Rahman, Antonio Naranjo Redondo and Julián Roa González

Symmetry 2022, 14(7), 1451; https://doi.org/10.3390/sym14071451 - 15 Jul 2022

Cited by 1 | Viewed by 1547

Abstract

The goal of this study is to provide analytical and numerical assessments to a fluid flow based on an Eyring–Powell viscosity term and a Darcy–Forchheimer law in a porous media. The analysis provides results about regularity, existence and uniqueness of solutions. Travelling wave [...] Read more.

The goal of this study is to provide analytical and numerical assessments to a fluid flow based on an Eyring–Powell viscosity term and a Darcy–Forchheimer law in a porous media. The analysis provides results about regularity, existence and uniqueness of solutions. Travelling wave solutions are explored, supported by the Geometric Perturbation Theory to build profiles in the proximity of the equation critical points. Finally, a numerical routine is provided as a baseline for the validity of the analytical approach presented for low Reynolds numbers typical in a porous medium. Full article

(This article belongs to the Special Issue Mathematical Fluid Dynamics and Symmetry)

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17 pages, 4157 KiB

Open AccessArticle

On the Modified Stokes Second Problem for Maxwell Fluids with Linear Dependence of Viscosity on the Pressure

by Constantin Fetecau, Tahir Mushtaq Qureshi, Abdul Rauf and Dumitru Vieru

Symmetry 2022, 14(2), 219; https://doi.org/10.3390/sym14020219 - 24 Jan 2022

Cited by 1 | Viewed by 2770

Abstract

The modified Stokes second problem for incompressible upper-convected Maxwell (UCM) fluids with linear dependence of viscosity on the pressure is analytically and numerically investigated. The fluid motion, between infinite horizontal parallel plates, is generated by the lower wall, which oscillates in its plane. [...] Read more.

The modified Stokes second problem for incompressible upper-convected Maxwell (UCM) fluids with linear dependence of viscosity on the pressure is analytically and numerically investigated. The fluid motion, between infinite horizontal parallel plates, is generated by the lower wall, which oscillates in its plane. The movement region of the fluid is symmetric with respect to the median plane, but its motion is asymmetric due to the boundary conditions. Closed-form expressions are found for the steady-state components of start-up solutions for non-dimensional velocity and the corresponding non-trivial shear and normal stresses. Similar solutions for the simple Couette flow are obtained as limiting cases of the solutions corresponding to the motion due to cosine oscillations of the wall. For validation, it is graphically proved that the start-up solutions (numerical solutions) converge to their steady-state components. Solutions for motions of ordinary incompressible UCM fluids performing the same motions are obtained as special cases of present results using asymptotic approximations of standard Bessel functions. The time needed to reach the permanent or steady state is also determined. This time is higher for motions of ordinary fluids, compared with motions of liquids with pressure-dependent viscosity. The impact of physical parameters on the fluid motion and the spatial–temporal distribution of start-up solutions are graphically investigated and discussed. Ordinary fluids move slower than fluids with pressure-dependent viscosity. Full article

(This article belongs to the Special Issue Mathematical Fluid Dynamics and Symmetry)

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22 pages, 397 KiB

Open AccessArticle

Control Problem Related to 2D Stokes Equations with Variable Density and Viscosity

by Evgenii S. Baranovskii, Eber Lenes, Exequiel Mallea-Zepeda, Jonnathan Rodríguez and Lautaro Vásquez

Symmetry 2021, 13(11), 2050; https://doi.org/10.3390/sym13112050 - 31 Oct 2021

Cited by 9 | Viewed by 1734

Abstract

We study an optimal control problem for the stationary Stokes equations with variable density and viscosity in a 2D bounded domain under mixed boundary conditions. On in-flow and out-flow parts of the boundary, nonhomogeneous Dirichlet boundary conditions are used, while on the solid [...] Read more.

We study an optimal control problem for the stationary Stokes equations with variable density and viscosity in a 2D bounded domain under mixed boundary conditions. On in-flow and out-flow parts of the boundary, nonhomogeneous Dirichlet boundary conditions are used, while on the solid walls of the flow domain, the impermeability condition and the Navier slip condition are provided. We control the system by the external forces (distributed control) as well as the velocity boundary control acting on a fixed part of the boundary. We prove the existence of weak solutions of the state equations, by firstly expressing the fluid density in terms of the stream function (Frolov formulation). Then, we analyze the control problem and prove the existence of global optimal solutions. Using a Lagrange multipliers theorem in Banach spaces, we derive an optimality system. We also establish a second-order sufficient optimality condition and show that the marginal function of this control system is lower semi-continuous. Full article

(This article belongs to the Special Issue Mathematical Fluid Dynamics and Symmetry)

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12 pages, 296 KiB

Open AccessArticle

Exact Solutions to the Navier–Stokes Equations with Couple Stresses

by Evgenii S. Baranovskii, Natalya V. Burmasheva and Evgenii Yu. Prosviryakov

Symmetry 2021, 13(8), 1355; https://doi.org/10.3390/sym13081355 - 26 Jul 2021

Cited by 21 | Viewed by 2939

Abstract

This article discusses the possibility of using the Lin–Sidorov–Aristov class of exact solutions and its modifications to describe the flows of a fluid with microstructure (with couple stresses). The presence of couple shear stresses is a consequence of taking into account the rotational [...] Read more.

This article discusses the possibility of using the Lin–Sidorov–Aristov class of exact solutions and its modifications to describe the flows of a fluid with microstructure (with couple stresses). The presence of couple shear stresses is a consequence of taking into account the rotational degrees of freedom for an elementary volume of a micropolar liquid. Thus, the Cauchy stress tensor is not symmetric. The article presents exact solutions for describing unidirectional (layered), shear and three-dimensional flows of a micropolar viscous incompressible fluid. New statements of boundary value problems are formulated to describe generalized classical Couette, Stokes and Poiseuille flows. These flows are created by non-uniform shear stresses and velocities. A study of isobaric shear flows of a micropolar viscous incompressible fluid is presented. Isobaric shear flows are described by an overdetermined system of nonlinear partial differential equations (system of Navier–Stokes equations and incompressibility equation). A condition for the solvability of the overdetermined system of equations is provided. A class of nontrivial solutions of an overdetermined system of partial differential equations for describing isobaric fluid flows is constructed. The exact solutions announced in this article are described by polynomials with respect to two coordinates. The coefficients of the polynomials depend on the third coordinate and time. Full article

(This article belongs to the Special Issue Mathematical Fluid Dynamics and Symmetry)

14 pages, 325 KiB

Open AccessArticle

A 3D Non-Stationary Micropolar Fluids Equations with Navier Slip Boundary Conditions

by Cristian Duarte-Leiva, Sebastián Lorca and Exequiel Mallea-Zepeda

Symmetry 2021, 13(8), 1348; https://doi.org/10.3390/sym13081348 - 26 Jul 2021

Cited by 1 | Viewed by 1611

Abstract

Micropolar fluids are fluids with microstructure and belong to a class of fluids with asymmetric stress tensor that called Polar fluids, and include, as a special case, the well-established Navier–Stokes model. In this work we study a 3D micropolar fluids model with [...] Read more.

Micropolar fluids are fluids with microstructure and belong to a class of fluids with asymmetric stress tensor that called Polar fluids, and include, as a special case, the well-established Navier–Stokes model. In this work we study a 3D micropolar fluids model with Navier boundary conditions without friction for the velocity field and homogeneous Dirichlet boundary conditions for the angular velocity. Using the Galerkin method, we prove the existence of weak solutions and establish a Prodi–Serrin regularity type result which allow us to obtain global-in-time strong solutions at finite time. Full article

(This article belongs to the Special Issue Mathematical Fluid Dynamics and Symmetry)

9 pages, 280 KiB

Open AccessArticle

Regularity of Weak Solutions to the Inhomogeneous Stationary Navier–Stokes Equations

by Alfonsina Tartaglione

Symmetry 2021, 13(8), 1336; https://doi.org/10.3390/sym13081336 - 24 Jul 2021

Cited by 1 | Viewed by 1477

Abstract

One of the most intriguing issues in the mathematical theory of the stationary Navier–Stokes equations is the regularity of weak solutions. This problem has been deeply investigated for homogeneous fluids. In this paper, the regularity of the solutions in the case of not [...] Read more.

One of the most intriguing issues in the mathematical theory of the stationary Navier–Stokes equations is the regularity of weak solutions. This problem has been deeply investigated for homogeneous fluids. In this paper, the regularity of the solutions in the case of not constant viscosity is analyzed. Precisely, it is proved that for a bounded domain

Ω \subset R^{2}

, a weak solution

u \in W^{1, q} (Ω)

is locally Hölder continuous if

q = 2

, and Hölder continuous around x, if

q \in (1, 2)

and

| μ (x) - μ_{0} |

is suitably small, with

μ_{0}

positive constant; an analogous result holds true for a bounded domain

Ω \subset R^{n} (n > 2)

and weak solutions in

W^{1, n / 2} (Ω)

. Full article

(This article belongs to the Special Issue Mathematical Fluid Dynamics and Symmetry)

15 pages, 394 KiB

Open AccessArticle

Non-Isothermal Creeping Flows in a Pipeline Network: Existence Results

by Evgenii S. Baranovskii, Vyacheslav V. Provotorov, Mikhail A. Artemov and Alexey P. Zhabko

Symmetry 2021, 13(7), 1300; https://doi.org/10.3390/sym13071300 - 19 Jul 2021

Cited by 10 | Viewed by 1970

Abstract

This paper deals with a 3D mathematical model for the non-isothermal steady-state flow of an incompressible fluid with temperature-dependent viscosity in a pipeline network. Using the pressure and heat flux boundary conditions, as well as the conjugation conditions to satisfy the mass balance [...] Read more.

This paper deals with a 3D mathematical model for the non-isothermal steady-state flow of an incompressible fluid with temperature-dependent viscosity in a pipeline network. Using the pressure and heat flux boundary conditions, as well as the conjugation conditions to satisfy the mass balance in interior junctions of the network, we propose the weak formulation of the nonlinear boundary value problem that arises in the framework of this model. The main result of our work is an existence theorem (in the class of weak solutions) for large data. The proof of this theorem is based on a combination of the Galerkin approximation scheme with one result from the field of topological degrees for odd mappings defined on symmetric domains. Full article

(This article belongs to the Special Issue Mathematical Fluid Dynamics and Symmetry)

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13 pages, 1552 KiB

Open AccessArticle

Symmetric and Non-Symmetric Flows of Burgers’ Fluids through Porous Media between Parallel Plates

by Constantin Fetecau and Dumitru Vieru

Symmetry 2021, 13(7), 1109; https://doi.org/10.3390/sym13071109 - 22 Jun 2021

Cited by 6 | Viewed by 2199

Abstract

Unidirectional unsteady flows of the incompressible Burgers’ fluids between two infinite horizontal parallel plates are analytically studied when the magnetic and porous effects are taken into consideration. The fluid motion is induced by the two plates, which move in their planes with time-dependent [...] Read more.

Unidirectional unsteady flows of the incompressible Burgers’ fluids between two infinite horizontal parallel plates are analytically studied when the magnetic and porous effects are taken into consideration. The fluid motion is induced by the two plates, which move in their planes with time-dependent velocities. Exact general expressions are established both for the dimensionless velocity and shear stress fields as well as the corresponding Darcy’s resistance in the channel using the Laplace transform. If both plates move with equal velocities in the same direction, the fluid motion becomes symmetric with respect to the mid-plane between them. Otherwise, its motion is non-symmetric. To bring to light the behavior of the fluid, the dimensionless velocity profiles versus the spatial variable as well as its time evolution are presented both for the symmetric and asymmetric case. Finally, for comparison, similar graphical representations are presented together for the velocities of the incompressible Oldroyd-B and Burgers’ fluids. For large values of the time t, as expected, the behavior of the two different fluids is almost identical. The Darcy’s resistance against y is also graphically represented for the symmetric flow at different values of the time t. The influence of the magnetic field on the fluid motion is graphically revealed and discussed. Full article

(This article belongs to the Special Issue Mathematical Fluid Dynamics and Symmetry)

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13 pages, 321 KiB

Open AccessArticle

Solvability of an Optimization Problem for the Unsteady Plane Flow of a Non-Newtonian Fluid with Memory

by Mikhail A. Artemov

Symmetry 2021, 13(6), 1026; https://doi.org/10.3390/sym13061026 - 7 Jun 2021

Viewed by 1624

Abstract

This paper deals with an optimization problem for a nonlinear integro-differential system that describes the unsteady plane motion of an incompressible viscoelastic fluid of Jeffreys–Oldroyd type within a fixed bounded region subject to the no-slip boundary condition. Control parameters are included in the [...] Read more.

This paper deals with an optimization problem for a nonlinear integro-differential system that describes the unsteady plane motion of an incompressible viscoelastic fluid of Jeffreys–Oldroyd type within a fixed bounded region subject to the no-slip boundary condition. Control parameters are included in the initial condition. The objective of control is to match the velocity field at the final time with a prescribed target field. The control model under consideration is interpreted as a continuous evolution system in an infinite-dimensional Hilbert space. The existence of at least one optimal control is proved under inclusion-type constraints for admissible controls. Full article

(This article belongs to the Special Issue Mathematical Fluid Dynamics and Symmetry)

24 pages, 1174 KiB

Open AccessArticle

An Analytically Derived Shear Stress and Kinetic Energy Equation for One-Equation Modelling of Complex Turbulent Flows

by Ronald M. C. So

Symmetry 2021, 13(4), 576; https://doi.org/10.3390/sym13040576 - 31 Mar 2021

Viewed by 2264

Abstract

The Reynolds stress equations for two-dimensional and axisymmetric turbulent shear flows are simplified by invoking local equilibrium and boundary layer approximations in the near-wall region. These equations are made determinate by appropriately modelling the pressure–velocity correlation and dissipation rate terms and solved analytically [...] Read more.

The Reynolds stress equations for two-dimensional and axisymmetric turbulent shear flows are simplified by invoking local equilibrium and boundary layer approximations in the near-wall region. These equations are made determinate by appropriately modelling the pressure–velocity correlation and dissipation rate terms and solved analytically to give a relation between the turbulent shear stress

τ / ρ

and the kinetic energy of turbulence

(k = q^{2} / 2)

. This is derived without external body force present. The result is identical to that proposed by Nevzgljadov in A Phenomenological Theory of Turbulence, who formulated it through phenomenological arguments based on atmospheric boundary layer measurements. The analytical approach is extended to treat turbulent flows with external body forces. A general relation

τ / ρ = a_{1} [1 - AF ({Ri}_{F})] (q^{2} / 2)

is obtained for these flows, where F

({Ri}_{F})

is a function of the gradient Richardson number

{Ri}_{F}

, and

a_{1}

is found to depend on turbulence models and their assumed constants. One set of constants yields

a_{1} =

0.378, while another gives

a_{1} =

0.328. With no body force, F ≡ 1 and the relation reduces to the Nevzgljadov equation with

a_{1}

determined to be either 0.378 or 0.328, depending on model constants set assumed. The present study suggests that 0.328 is in line with Nevzgljadov’s proposal. Thus, the present approach provides a theoretical base to evaluate the turbulent shear stress for flows with external body forces. The result is used to reduce the

k - ε

model to a one-equation model that solves the k-equation, the shear stress and kinetic energy equation, and an ε evaluated by assuming isotropic eddy viscosity behavior. Full article

(This article belongs to the Special Issue Mathematical Fluid Dynamics and Symmetry)

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