Mathematics

Research

20 pages, 7234 KiB

Open AccessArticle

Predictability of Magnetic Field Reversals

by Daniil Tolmachev, Roman Chertovskih, Simon Ranjith Jeyabalan and Vladislav Zheligovsky

Mathematics 2024, 12(3), 490; https://doi.org/10.3390/math12030490 - 3 Feb 2024

Viewed by 2206

Geomagnetic field measurements indicate that at present we may be on the brink of the Earth’s magnetic field reversal, potentially resulting in all the accompanying negative consequences for the mankind. Mathematical modelling is necessary in order to find precursors for reversals and excursions [...] Read more.

Geomagnetic field measurements indicate that at present we may be on the brink of the Earth’s magnetic field reversal, potentially resulting in all the accompanying negative consequences for the mankind. Mathematical modelling is necessary in order to find precursors for reversals and excursions of the magnetic field. With this purpose in mind, following the Podvigina scenario for the emergence of the reversals, we have studied convective flows not far (in the parameter space) from their onset and the onset of magnetic field generation, and found a flow demonstrating reversals of polarity of some harmonics comprising the magnetic field. We discuss a simulated regime featuring patterns of behaviour that apparently indicate future reversals of certain harmonics of the magnetic field. It remains to be seen whether reversal precursors similar to the observed ones exist and might be applicable for the much more complex geomagnetic dynamo. Full article

(This article belongs to the Special Issue Applications of Mathematics to Fluid Dynamics)

► Show Figures

Figure 1

22 pages, 4709 KiB

Open AccessArticle

Velocity Field due to a Vertical Deformation of the Bottom of a Laminar Free-Surface Fluid Flow

by Rodrigo González and Aldo Tamburrino

Mathematics 2024, 12(3), 394; https://doi.org/10.3390/math12030394 - 25 Jan 2024

Viewed by 575

Abstract

This article investigates the velocity field of a free-surface flow subjected to harmonic deformation of the channel bottom, progressing asymptotically from a flat initial state to a maximum amplitude. Assuming a uniform main flow with the primary velocity component transverse to the bed [...] Read more.

This article investigates the velocity field of a free-surface flow subjected to harmonic deformation of the channel bottom, progressing asymptotically from a flat initial state to a maximum amplitude. Assuming a uniform main flow with the primary velocity component transverse to the bed undulation, analytical solutions are obtained for the three velocity components and free surface distortion using the method of perturbations. The perturbation components of the velocity field, streamlines, and surface deformation depend on a dimensionless parameter that reflects the fluid inertia induced by bed deformation relative to viscous resistance. When viscous effects dominate, a monotonic decay of the perturbations from the bed to the free surface is observed. In contrast, when inertia dominates, the perturbations can exhibit an oscillatory behavior and introduce circulation cells in the plane normal to the main flow. The interplay between inertia and viscosity reveals scenarios where surface and bed deformations are either in or out of phase, influencing vertical velocity components. Figures illustrate these phenomena, providing insights into the complex dynamics of free-surface flows with harmonic bed deformation in the direction normal to the main flow, and amplitude growing with time. The results are limited to small deformations of the channel bottom, as imposed by the linearization of the momentum equations. Even so, to the best of the authors’ knowledge, this problem has not been addressed before. Full article

(This article belongs to the Special Issue Applications of Mathematics to Fluid Dynamics)

► Show Figures

Figure 1

21 pages, 1614 KiB

Open AccessArticle

General Solutions for MHD Motions of Ordinary and Fractional Maxwell Fluids through Porous Medium When Differential Expressions of Shear Stress Are Prescribed on Boundary

by Dumitru Vieru and Constantin Fetecau

Mathematics 2024, 12(2), 357; https://doi.org/10.3390/math12020357 - 22 Jan 2024

Cited by 1 | Viewed by 653

Abstract

Some MHD unidirectional motions of the electrically conducting incompressible Maxwell fluids between infinite horizontal parallel plates incorporated in a porous medium are analytically and graphically investigated when differential expressions of the non-trivial shear stress are prescribed on the boundary. Such boundary conditions are [...] Read more.

Some MHD unidirectional motions of the electrically conducting incompressible Maxwell fluids between infinite horizontal parallel plates incorporated in a porous medium are analytically and graphically investigated when differential expressions of the non-trivial shear stress are prescribed on the boundary. Such boundary conditions are usually necessary in order to formulate well-posed boundary value problems for motions of rate-type fluids. General closed-form expressions are established for the dimensionless fluid velocity, the corresponding shear stress, and Darcy’s resistance. For completion, as well as for comparison, all results are extended to a fractional model of Maxwell fluids in which the time fractional Caputo derivative is used. It is proven for the first time that a large class of unsteady motions of the fractional incompressible Maxwell fluids becomes steady in time. For illustration, three particular motions are considered, and the correctness of the results is graphically proven. They correspond to constant or oscillatory values of the differential expression of shear stress on the boundary. In the first case, the required time to reach the steady state is graphically determined. This time declines for increasing values of the fractional parameter. Consequently, the steady state is reached earlier for motions of the ordinary fluids in comparison with the fractional ones. Finally, the fluid velocity, shear stress, and Darcy’s resistance are graphically represented and discussed for the fractional model. Full article

(This article belongs to the Special Issue Applications of Mathematics to Fluid Dynamics)

► Show Figures

Figure 1

23 pages, 1359 KiB

Open AccessFeature PaperArticle

Numerical Identification of Boundary Conditions for Richards’ Equation

by Miglena N. Koleva and Lubin G. Vulkov

Mathematics 2024, 12(2), 299; https://doi.org/10.3390/math12020299 - 17 Jan 2024

Viewed by 652

Abstract

A time stepping quasilinearization approach to the mixed (or coupled) form of one and two dimensional Richards’ equations is developed. For numerical solution of the linear ordinary differential equation (ODE) for 1D case and elliptic for 2D case, obtained after this semidiscretization, a [...] Read more.

A time stepping quasilinearization approach to the mixed (or coupled) form of one and two dimensional Richards’ equations is developed. For numerical solution of the linear ordinary differential equation (ODE) for 1D case and elliptic for 2D case, obtained after this semidiscretization, a finite volume method is used for direct problems arising on each time level. Next, we propose a version of the decomposition method for the numerical solution of the inverse ODE and 2D elliptic boundary problems. Computational results for some soil types and its related parameters reported in the literature are presented. Full article

(This article belongs to the Special Issue Applications of Mathematics to Fluid Dynamics)

► Show Figures

Figure 1

15 pages, 6030 KiB

Open AccessArticle

Mathematical Modeling of the Hydrodynamic Instability and Chemical Inhibition of Detonation Waves in a Syngas–Air Mixture

by Valeriy Nikitin, Elena Mikhalchenko, Lyuben Stamov, Nickolay Smirnov and Vilen Azatyan

Mathematics 2023, 11(24), 4879; https://doi.org/10.3390/math11244879 - 5 Dec 2023

Cited by 2 | Viewed by 677

Abstract

This paper presents the results of the two-dimensional modeling of the hydrodynamic instability of a detonation wave, which results in the formation of an oscillating cellular structure on the wave front. This cellular structure of the wave, unstable due to its origin, demonstrates [...] Read more.

This paper presents the results of the two-dimensional modeling of the hydrodynamic instability of a detonation wave, which results in the formation of an oscillating cellular structure on the wave front. This cellular structure of the wave, unstable due to its origin, demonstrates the constant statistically averaged characteristics of the cell size. The suppression of detonation propagation in synthesis gas mixtures with air using a combustible inhibitor is studied numerically. Contrary to the majority of inhibitors being either inert substances, which do not take part in the chemical reaction, or take part in chemical reaction but do not contribute to energy release, the suggested inhibitor is also a fuel, which enters into an exothermic reaction with oxygen. The unsaturated hydrocarbon propylene additive is used as an inhibitor. The dependence of the effect of the inhibitor content on the mitigation of detonation for various conditions of detonation initiation is researched. The results make it possible to determine a critical percentage of inhibitor which prevents the occurrence of detonation and the critical percentage of inhibitor which destroys a developed detonation wave. Full article

(This article belongs to the Special Issue Applications of Mathematics to Fluid Dynamics)

► Show Figures

Figure 1

12 pages, 871 KiB

Open AccessArticle

Transient Convective Heat Transfer in Porous Media

by Ruben D’Rose, Mark Willemsz and David Smeulders

Mathematics 2023, 11(21), 4415; https://doi.org/10.3390/math11214415 - 25 Oct 2023

Viewed by 738

Abstract

In this study, several methods to analyze convective heat transfer in a porous medium are presented and discussed. First, the method of Fourier was used to obtain solutions for reduced temperatures

θ_{s}

and

θ_{f}

. The results showed an exponentially decaying [...] Read more.

In this study, several methods to analyze convective heat transfer in a porous medium are presented and discussed. First, the method of Fourier was used to obtain solutions for reduced temperatures

θ_{s}

and

θ_{f}

. The results showed an exponentially decaying propagating temperature front. Then, we discuss the method of integration that was presented earlier by Schumann. This method makes use of a transformation of variables. Thirdly, the system of partial differential equations was directly solved with the Finite Difference method, of which the result showed good agreement with the Fourier solutions. For the chosen

Δ τ

and

Δ ξ

, the maximum error for

θ_{f} = 3.7 %

. The maximum error for

θ_{s}

for the first

ξ

and first

τ

is large (36%) but decays rapidly. The problem was extended by adding a linear heat source term to the solid. Again, making use of the change in variables, analytical solutions were derived for the solid and fluid phases, and corrections to the previous literature were suggested. Finally, results obtained from a numerical model were compared to the analytical solutions, which again showed good agreement (maximum error of 6%). Full article

(This article belongs to the Special Issue Applications of Mathematics to Fluid Dynamics)

► Show Figures

Figure 1

16 pages, 1521 KiB

Open AccessArticle

Mathematical Model of the Flow in a Nanofiber/Microfiber Mixed Aerosol Filter

by Elvina Panina, Renat Mardanov and Shamil Zaripov

Mathematics 2023, 11(16), 3465; https://doi.org/10.3390/math11163465 - 10 Aug 2023

Viewed by 871

Abstract

A new mathematical model of an aerosol fibrous filter, composed of a variety of nano- and microfibers, is developed. The combination of nano- and microfibers in a mixed-type filter provides a higher overall quality factor compared with filters with monodisperse fibers. In this [...] Read more.

A new mathematical model of an aerosol fibrous filter, composed of a variety of nano- and microfibers, is developed. The combination of nano- and microfibers in a mixed-type filter provides a higher overall quality factor compared with filters with monodisperse fibers. In this paper, we propose a mathematical model of the flow of an incompressible fluid in a porous region consisting of a set of cylinders of various diameters in the range of nano- and micrometers to describe a mixed-type aerosol filter. The flow domain is a rectangular periodic cell with one microfiber and many nanofibers. The motion of the carrier medium is described by the boundary value problem in Stokes flow approximation with the no-slip boundary condition for microfibers and the slip condition for nanofibers. The boundary element method taking into account the slip and non-slip conditions is developed. The calculated velocity field, streamlines, vorticity distribution, and drag of separate fibers and the entire periodic cell are presented. Numerical results for the drag force of the porous medium of a mixed-type filter for the various ratios of mass proportion of nano- and microfibers, porosity, and filtration velocity are presented. The obtained results are compared with the analytical formulas based on the approximate theory of filtration of bimodal filters and with known experimental data. It is shown that with an increase in the mass fraction of nanofibers, the total drag force of the cell increases, while the relative contribution of nanofibers to the total drag force tends toward the value that is less than unity. An approximate analytical formula for the drag coefficient of a mixed aerosol filter is derived. The developed flow model and analytical formulas allow for estimating the aerodynamic drag of a mixed filter composed by nano- and microfibers. Full article

(This article belongs to the Special Issue Applications of Mathematics to Fluid Dynamics)

► Show Figures

Figure 1

14 pages, 2085 KiB

Open AccessArticle

The Effects of Heat Transfer through the Ends of a Cylindrical Cavity on Acoustic Streaming and Gas Temperature

by Amir A. Gubaidullin and Anna V. Pyatkova

Mathematics 2023, 11(8), 1840; https://doi.org/10.3390/math11081840 - 12 Apr 2023

Viewed by 1006

Abstract

The longitudinal vibrational motion of a cylindrical cavity with gas, in which the acoustic streaming occurs, is considered. The motion is described by the system of equations for the dynamics and thermal conductivity of a viscous perfect gas, written in a cylindrical coordinate [...] Read more.

The longitudinal vibrational motion of a cylindrical cavity with gas, in which the acoustic streaming occurs, is considered. The motion is described by the system of equations for the dynamics and thermal conductivity of a viscous perfect gas, written in a cylindrical coordinate system associated with the cavity. The system of equations is solved numerically by the finite volume method with an implicit staggered grid scheme, while the convective–diffusion fluxes are approximated by the power law scheme. According to the boundary conditions, the lateral surface of the cavity is maintained at a constant equal initial temperature. The effects of heat transfer through the ends of the cavity are studied. Heat transfer is given by isothermal boundary conditions. The obtained solutions are compared with the solutions under adiabatic boundary conditions. It is shown for the first time that the effects of heat transfer manifest themselves with an increase in the nonlinearity of the process; when the frequency and amplitude of vibration increase, this is also facilitated by an increase in the radius of the cavity. The effects of heat transfer on the period average temperature, on the streaming velocity and on structure are established. Full article

(This article belongs to the Special Issue Applications of Mathematics to Fluid Dynamics)

► Show Figures

Figure 1

10 pages, 387 KiB

Open AccessFeature PaperArticle

Mathematical Modeling of the Wave Dynamics of an Encapsulated Perfluorocarbon Droplet in a Viscoelastic Liquid

by Damir A. Gubaidullin, Dilya D. Gubaidullina and Yuri V. Fedorov

Mathematics 2023, 11(5), 1083; https://doi.org/10.3390/math11051083 - 21 Feb 2023

Cited by 7 | Viewed by 1179

Abstract

A mathematical model has been developed and a numerical study of vapor bubble growth as a result of acoustic evaporation of an encapsulated perfluorocarbon droplet in a viscoelastic liquid is presented. The viscoelasticity of the droplet shell and the carrier liquid is taken [...] Read more.

A mathematical model has been developed and a numerical study of vapor bubble growth as a result of acoustic evaporation of an encapsulated perfluorocarbon droplet in a viscoelastic liquid is presented. The viscoelasticity of the droplet shell and the carrier liquid is taken into account according to the Kelvin–Voigt rheological model. The problem is reduced to solving a system of ordinary differential equations for the radius and temperature of the bubble, the radius of the droplet and the shell together with the thermal conductivity equation for the internal liquid. Spatial discretization of the thermal conductivity equation is carried out using an implicit finite difference scheme. ODEs are solved by the fifth order Runge–Kutta method with an adaptive computational step. To check the correctness of the numerical calculation in a particular case, the theory has been compared with known experimental data. The influence of the shear modulus of the shell and the carrier liquid, and the shell thickness on the radial dynamics of a vapor bubble inside an encapsulated droplet in an external viscoelastic liquid is demonstrated. Full article

(This article belongs to the Special Issue Applications of Mathematics to Fluid Dynamics)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Applications of Mathematics to Fluid Dynamics

Share This Special Issue

Special Issue Editor

Special Issue Information

Keywords

Published Papers (9 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI