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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E4: Mathematical Physics".

Deadline for manuscript submissions: closed (31 January 2024) | Viewed by 14555

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Special Issue Editors

Prof. Dr. Nikolay M. Zubarev

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Guest Editor

Institute of Electrophysics, UB RAS, 620016 Yekaterinburg, Russia
Interests: potential flows; free surface; ideal incompressible fluid; boundary instabilities; integrability; nonlinear dynamics; formation of singularities; electric and magnetic fields

Dr. Evgeny A. Kochurin

E-Mail Website
Guest Editor

Skolkovo Institute of Science and Technology, 121205 Moscow, Russia
Interests: fluid dynamics; numerical simulations; wave turbulence; finite-time singularities; electro- and magnetohydrodynamics; free-surface flows

Special Issue Information

Dear Colleagues,

The Special Issue of Mathematics MDPI on “Numerical and Analytical Study of Fluid Dynamics” aims to attract theoretical (both numerical and analytical) research and review papers in a wide range of topics of mathematical and computational fluid mechanics, including free surface hydrodynamics, interfacial flows, and vortex and wave turbulence. Studies on fluid dynamics have traditionally stimulated the development of methods of mathematical physics; they largely conditioned the emergence of such a branch of physics as nonlinear dynamics. This Special Issue is focusing on the development and application of new effective methods for both numerical and analytical description of unsteady fluid flows under various conditions, including electrohydrodynamic and magnetohydrodynamic ones. Special attention will be paid to the problem of integrability of the equations of fluid motion and the related problem of flow collapse (finite-time singularities, wave breaking, formation of discontinuities in flows, etc.).

Prof. Dr. Nikolay M. Zubarev
Dr. Evgeny A. Kochurin
Guest Editors

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Keywords

fluid dynamics
free surface hydrodynamics
vortex and wave turbulence
interfacial flows
formation of singularities
wave breaking
integrability
numerical simulation
electrohydrodynamic (EHD) flows magnetohydrodynamic (MHD) flows

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Published Papers (10 papers)

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Research

14 pages, 6315 KiB

Open AccessArticle

Magnetic Filaments: Formation, Stability, and Feedback

by Evgeny A. Kuznetsov and Evgeny A. Mikhailov

Mathematics 2024, 12(5), 677; https://doi.org/10.3390/math12050677 - 26 Feb 2024

Viewed by 881

Abstract

As is well known, magnetic fields in space are distributed very inhomogeneously. Sometimes, field distributions have forms of filaments with high magnetic field values. As many observations show, such a filamentation takes place in convective cells in the Sun and other astrophysical objects. This effect is associated with the frozenness of the magnetic field into a medium with high conductivity that leads to the compression of magnetic field lines and formation of magnetic filaments. We analytically show, based on the general analysis, that the magnetic field intensifies in the regions of downward flows in both two-dimensional and three-dimensional convective cells. These regions of the hyperbolic type in magnetic fields play the role of a specific attractor. This analysis was confirmed by numerical simulations of 2D roll-type convective cells. Without dissipation, the magnetic field grows exponentially in time and does not depend on the aspect ratio between the horizontal and vertical scales of the cell. An increase due to compression in the magnetic field of highly conductive plasma is saturated due to the natural limitation associated with dissipative effects when the maximum magnitude of a magnetic field is of the order of the root of the magnetic Reynolds number Re_m. For the solar convective zone, the mean kinetic energy density exceeds the mean magnetic energy density for at least two orders of magnitude, which allows one to use the kinematic approximation of the MHD induction equation. In this paper, based on the stability analysis, we explain why downward flows influence magnetic filaments, making them flatter with orientation along the interfaces between convective cells. Full article

(This article belongs to the Special Issue Numerical and Analytical Study of Fluid Dynamics)

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24 pages, 886 KiB

Open AccessArticle

The Shape of a Compressible Drop on a Vibrating Solid Plate

by Andrey Ivantsov, Tatyana Lyubimova, Grigoriy Khilko and Dmitry Lyubimov

Mathematics 2023, 11(21), 4527; https://doi.org/10.3390/math11214527 - 3 Nov 2023

Cited by 5 | Viewed by 878

Abstract

The influence of high-frequency vibrations on the shape of a compressible drop placed on an oscillating solid substrate is studied in this paper. Due to the significant difference in characteristic temporal scales, the average and pulsating motions of the drop can be considered separately. For nearly hemispherical drop, the solution to the problem of pulsating motion is found in the form of series in Legendre polynomials. Frequencies of natural sound oscillations of hemispherical axisymmetric drop are obtained. Resonances of the acoustic mode of drop oscillations are found. The problem of forced oscillations of hemispherical drop in the limit of weakly compressible liquid is considered. It is found that drop oscillation amplitude grows with vibration intensity according to quadratic law, which is consistent with the solution of the pulsation problem for finite compressibility assumption. A variational principle for calculation of average drop shape is formulated based on minimization of energy functional for the case, so the compressibility of the liquid should be taken into account. It is shown that the functional (the sum of the kinetic and potential energies of the pulsating flow, the kinetic energy of the averaged flow, and the surface tension energy of the drop) decreases and reaches a minimum value at quasi-equilibrium state, in which the average shape of the drop becomes static. The influence of vibrations on the drop shape is studied for small values of the vibrational parameter. The surface of the drop in the absence of vibrations is assumed to be hemispherical. Calculations showed that under vibrations, drop height decreases, while the area of the base increases. Full article

(This article belongs to the Special Issue Numerical and Analytical Study of Fluid Dynamics)

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13 pages, 1222 KiB

Open AccessArticle

Fully Nonlinear Evolution of Free-Surface Waves with Constant Vorticity under Horizontal Electric Fields

by M. V. Flamarion, E. Kochurin and R. Ribeiro-Jr

Mathematics 2023, 11(21), 4467; https://doi.org/10.3390/math11214467 - 28 Oct 2023

Cited by 3 | Viewed by 999

Abstract

This work presents the results of a direct numerical simulation of the nonlinear free surface evolution of a finite-depth fluid with a linear shear flow under the action of horizontal electric fields. The method of time-dependent conformal transformation for the description of the combined effects of the electric fields and constant vorticity is generalized for the first time. The simulation results show that strong shear flow co-directed in the wave propagation direction leads to the formation of large-amplitude surface waves, and, for some limiting vorticity value, a wave breaking process with the formation of an air bubble in the liquid is possible. The oppositely directed shear flow can cause the retrograde motion of a surface wave (wave propagation in the opposite direction to the linear wave speed). The simulations conducted taking into account the electro-hydrodynamic effects demonstrate that a high enough external horizontal electric field suppresses these strongly nonlinear processes, and the surface waves tend to preserve their shape. Full article

(This article belongs to the Special Issue Numerical and Analytical Study of Fluid Dynamics)

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17 pages, 4031 KiB

Open AccessArticle

Effects of LTNE on Two-Component Convective Instability in a Composite System with Thermal Gradient and Heat Source

by Varalakshmi K. Balaji, Manjunatha Narayanappa, Ramalingam Udhayakumar, Ghada AlNemer, Sumithra Ramakrishna and Gangadharaih Yeliyur Honnappa

Mathematics 2023, 11(20), 4282; https://doi.org/10.3390/math11204282 - 13 Oct 2023

Cited by 3 | Viewed by 979

Abstract

An analytical study is conducted to examine the influence of thermal gradients and heat sources on the onset of two-component Rayleigh–Bènard (TCRB) convection using the Darcy model. The study takes into account the effects of local thermal non-equilibrium (LTNE), thermal profiles, and heat sources. The composite structure is horizontally constrained by adiabatic stiff boundaries, and the resulting solution to the problem is obtained using the perturbation approach. The various physical parameters have been thoroughly examined, revealing that the fluid layer exhibits dominance in the two-layer configuration. It has been observed that the parabolic profile demonstrates greater stability in comparison to the step function. Conversely, in the setup where the porous layer dominates, the step function plays a crucial role in maintaining stability. The porous layer, model (iv), exhibits greater stability in the predominant combined structure, while the linear configuration is characterized by higher instability. Full article

(This article belongs to the Special Issue Numerical and Analytical Study of Fluid Dynamics)

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29 pages, 2445 KiB

Open AccessArticle

Start-Up Multilayer Electro-Osmotic Flow of Maxwell Fluids through an Annular Microchannel under Hydrodynamic Slip Conditions

by Cesar A. Valencia, David A. Torres, Clara G. Hernández, Juan P. Escandón, Juan R. Gómez and René O. Vargas

Mathematics 2023, 11(20), 4231; https://doi.org/10.3390/math11204231 - 10 Oct 2023

Viewed by 1644

Abstract

The present investigation analyzes the transient multilayer electro-osmotic flow through an annular microchannel with hydrophobic walls. The fluids are considered immiscible and viscoelastic, following the Maxwell rheological model. In the problem examined, the linearized Poisson–Boltzmann and Cauchy momentum equations are used to determine the electric potential distribution and the flow field, respectively. Here, different interfacial phenomena are studied through the imposed boundary conditions, such as the hydrodynamic slip and specified zeta potentials at solid–liquid interfaces, the velocity continuity, the electroviscous stresses balance, the potential difference, and the continuity of electrical displacements at the interfaces between fluids. The semi-analytic solution uses the Laplace transform theory. In the results, the velocity profiles and velocity tracking show the oscillatory behavior of flow, which strongly depends on the dimensionless relaxation time. Furthermore, the hydrodynamic slip on the channel walls contributes to the release of energy stored in the fluids due to elastic effects at the start-up of the flow. Similarly, other dimensionless parameters are also investigated. This research aims to predict the parallel flow behavior in microfluidic devices under electro-osmotic effects. Full article

(This article belongs to the Special Issue Numerical and Analytical Study of Fluid Dynamics)

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17 pages, 636 KiB

Open AccessArticle

Semi-Analytical Methods in the Problem of Deformation of a Fluid Strip

by Evgenii Karabut and Elena Zhuravleva

Mathematics 2023, 11(15), 3422; https://doi.org/10.3390/math11153422 - 6 Aug 2023

Viewed by 1041

Abstract

A problem from a class of unsteady plane potential flows with a free boundary is considered. The entire boundary occupied by the liquid is free, and a zero pressure is maintained. There are neither external nor capillary forces. The motion is driven by inertia. The parameters prescribed at the initial time are the velocity field and the domain occupied by the fluid. The task is to determine these parameters at subsequent time instants. The solution is sought in the form of power series, which are then summed up with the use of the Pade approximation. Full article

(This article belongs to the Special Issue Numerical and Analytical Study of Fluid Dynamics)

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16 pages, 1139 KiB

Open AccessArticle

Nonlinear Dynamics of Perturbations of a Plane Potential Fluid Flow: Nonlocal Generalization of the Hopf Equation

by Nikolay M. Zubarev

Mathematics 2023, 11(9), 1999; https://doi.org/10.3390/math11091999 - 23 Apr 2023

Viewed by 1131

Abstract

In this paper, we analytically study the two-dimensional unsteady irrotational flow of an ideal incompressible fluid in a half-plane whose boundary is assumed to be a linear sink. It is shown that the nonlinear evolution of perturbations of the initial uniform flow is described by a one-dimensional integro-differential equation, which can be considered as a nonlocal generalization of the Hopf equation. This equation can be reduced to a system of ordinary differential equations (ODEs) in the cases of spatially localized or spatially periodic perturbations of the velocity field. In the first case, ODEs describe the motion of a system of interacting virtual point vortex-sinks/sources outside the flow domain. In the second case, ODEs describe the evolution of a finite number of harmonics of the velocity field distribution; this is possible due to the revealed property of the new equation that the interaction of initial harmonics does not lead to generation of new ones. The revealed reductions made it possible to effectively study the nonlinear evolution of the system, in particular, to describe the effect of nonlinearity on the relaxation of velocity field perturbations. It is shown that nonlinearity can significantly reduce the relaxation rate by more than 1.5 times. Full article

(This article belongs to the Special Issue Numerical and Analytical Study of Fluid Dynamics)

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15 pages, 719 KiB

Open AccessArticle

Significance of Nanoparticle Radius and Gravity Modulation on Dynamics of Nanofluid over Stretched Surface via Finite Element Simulation: The Case of Water-Based Copper Nanoparticles

by Bagh Ali, Anum Shafiq, Meznah M. Alanazi, Awatif A. Hendi, Ahmed Kadhim Hussein and Nehad Ali Shah

Mathematics 2023, 11(5), 1266; https://doi.org/10.3390/math11051266 - 6 Mar 2023

Cited by 9 | Viewed by 2134

Abstract

This communication studies the importance of varying the radius

D_{p}

This communication studies the importance of varying the radius

D_{p}

of Copper nanoparticles for microgravity-modulated mixed convection in micropolar nanofluid flux under an inclined surface subject magnetic field and heat source. In the current era, extremely pervasive modernized technical implementations have drawn attention to free convection governed by g-jitter force connected with microgravity. Therefore, fixed inter-spacing of nanoparticles and effects of g-jitter on the inclined surface are taken into consideration. A mathematical formulation based on conservation principles was non-dimensionalized by enforcement of similarity transformation, yielding a related set of ODEs. The convective non-linearity and coupling, an FE discretization, was implemented and executed on the Matlab platform. The numerical process’ credibility was ensured for its acceptable adoption with the defined outcomes. Then, the computational endeavor was continued to elucidate the impacts of various inputs of

D_{p}

, the amplitude of modulation

ϵ

, material parameter

β

, mixed convection parameter

λ

, inclination angle

γ

, and magnetic parameter M. The enlarging size of nanoparticles accelerated the nanofluid flow due to the depreciation of viscosity and receded the fluid temperature by reducing the surface area for heat transportation. The modulated Nusselt number, couple stress, and skin friction coefficient are significantly affected by the variation of

D_{p}

, M,

β

λ

, and

ϵ

. These results would benefit experts dealing with upper space transportation and materials’ performance, such as the effectualness of chemical catalytic reactors and crystals. Full article

(This article belongs to the Special Issue Numerical and Analytical Study of Fluid Dynamics)

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15 pages, 3947 KiB

Open AccessArticle

Electro-Thermo-Convection of a Dielectric Liquid in the External DC and AC Electric Fields

by Oleg Nekrasov and Boris Smorodin

Mathematics 2023, 11(5), 1188; https://doi.org/10.3390/math11051188 - 28 Feb 2023

Cited by 2 | Viewed by 1517

Abstract

The electro-thermo-convection of a dielectric liquid in a horizontal capacitor is investigated under the autonomous charge injection from the cathode and heating from above. In the case of a DC electric field, the linear stability analysis is carried out, and the thresholds of monotonic and oscillatory instability are determined. The finite difference method is used for the numerical simulation of the nonlinear behavior of electro-thermo-convective patterns: stationary convection and traveling waves. In the case of AC, electric field transient and permanent oscillations are analyzed. Two types of stable solutions are found. The modulated traveling waves are characterized by the quasiperiodic oscillations of convective characteristics. Another solution is modulated electroconvection (MEC). The patterns of MEC oscillate around some average flow synchronously with the external AC field and do not move laterally. The average intensity of convective mixing in modulated traveling waves is several times less than in modulated electroconvection. The spatiotemporal evolution of the stream function, temperature, and charge distributions for different types of transient and permanent solutions are analyzed. Full article

(This article belongs to the Special Issue Numerical and Analytical Study of Fluid Dynamics)

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16 pages, 872 KiB

Open AccessArticle

Significance of Ternary Hybrid Nanoparticles on the Dynamics of Nanofluids over a Stretched Surface Subject to Gravity Modulation

by Meznah M. Alanazi, Awatif Ahmed Hendi, N. Ameer Ahammad, Bagh Ali, Sonia Majeed and Nehad Ali Shah

Mathematics 2023, 11(4), 809; https://doi.org/10.3390/math11040809 - 5 Feb 2023

Cited by 24 | Viewed by 2329

Abstract

Boosting the heat transfer rate in a base fluid is of interest to researchers; many traditional methods have been utilized to do this. One significant way is using nanofluid to boost thermal performance. This investigation sought to improve the transmission of a thermal [...] Read more.

B_{0}

along the microgravity

g * (τ) = g_{0} (1 + a cos (π ω t))

. The G-jitter impacts were analyzed for three colloidal fluids flow; the mono micropolar nanofluid (alumina/water), micropolar hybrid nanofluid (alumina–titanium)/water, and micropolar trihybrid nanofluid (alumina–titanium–silicon)/water. Using suitable transformation, the governing formulation was changed into an ordinary differential equation. In a Matlab script, a computational code was composed to evaluate the impacts of the involved parameters on fluid dynamics. The fluid flow motion and thermal performance for the trihybrid case were greater than the mono and hybrid nanofluid cases subject to a microgravity environment. The fluid velocity and microrotation function decreased in opposition to the magnetic parameter’s increasing strength, but with an increasing trend in the fluid temperature function. Fluctuations in the velocity gradient and heat flow gradient increased as the modulation amplitude increased. Full article

(This article belongs to the Special Issue Numerical and Analytical Study of Fluid Dynamics)

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