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32 pages, 13256 KiB

Open AccessArticle

Brinkman–Bénard Convection with Rough Boundaries and Third-Type Thermal Boundary Conditions

by Pradeep G. Siddheshwar, Mahesha Narayana, David Laroze and C. Kanchana

Symmetry 2023, 15(8), 1506; https://doi.org/10.3390/sym15081506 - 28 Jul 2023

Viewed by 1472

Abstract

The Brinkman–Bénard convection problem is chosen for investigation, along with very general boundary conditions. Using the Maclaurin series, in this paper, we show that it is possible to perform a relatively exact linear stability analysis, as well as a weakly nonlinear stability analysis, [...] Read more.

The Brinkman–Bénard convection problem is chosen for investigation, along with very general boundary conditions. Using the Maclaurin series, in this paper, we show that it is possible to perform a relatively exact linear stability analysis, as well as a weakly nonlinear stability analysis, as normally performed in the case of a classical free isothermal/free isothermal boundary combination. Starting from a classical linear stability analysis, we ultimately study the chaos in such systems, all conducted with great accuracy. The principle of exchange of stabilities is proven, and the critical Rayleigh number,

R a_{c}

, and the wave number,

a_{c}

, are obtained in closed form. An asymptotic analysis is performed, to obtain

R a_{c}

for the case of adiabatic boundaries, for which

a_{c} ≃ 0

. A seemingly minimal representation yields a generalized Lorenz model for the general boundary condition used. The symmetry in the three Lorenz equations, their dissipative nature, energy-conserving nature, and bounded solution are observed for the considered general boundary condition. Thus, one may infer that, to obtain the results of various related problems, they can be handled in an integrated manner, and results can be obtained with great accuracy. The effect of increasing the values of the Biot numbers and/or slip Darcy numbers is to increase, not only the value of the critical Rayleigh number, but also the critical wave number. Extreme values of zero and infinity, when assigned to the Biot number, yield the results of an adiabatic and an isothermal boundary, respectively. Likewise, these extreme values assigned to the slip Darcy number yield the effects of free and rigid boundary conditions, respectively. Intermediate values of the Biot and slip Darcy numbers bridge the gap between the extreme cases. The effects of the Biot and slip Darcy numbers on the Hopf–Rayleigh number are, however, opposite to each other. In view of the known analogy between Bénard convection and Taylor–Couette flow in the linear regime, it is imperative that the results of the latter problem, viz., Brinkman–Taylor–Couette flow, become as well known. Full article

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

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17 pages, 3427 KiB

Open AccessArticle

Semi-Analytical Approach in BiER4BP for Exploring the Stable Positioning of the Elements of a Dyson Sphere

by Sergey Ershkov, Dmytro Leshchenko and Evgeniy Yu. Prosviryakov

Symmetry 2023, 15(2), 326; https://doi.org/10.3390/sym15020326 - 24 Jan 2023

Cited by 5 | Viewed by 2029

Abstract

In this study, we present a new approach with semi-analytical and numerical findings for solving equations of motion of small orbiter m, which is moving under the combined gravitational attraction of three primaries,

M_{1}

,

M_{2}

, and

M_{3}

[...] Read more.

In this study, we present a new approach with semi-analytical and numerical findings for solving equations of motion of small orbiter m, which is moving under the combined gravitational attraction of three primaries,

M_{1}

,

M_{2}

, and

M_{3}

, in case of the bi-elliptic restricted problem of four bodies (BiER4BP), where three such primaries,

M_{1}

,

M_{2}

, and

M_{3}

, are moving on elliptic orbits with hierarchical configuration

M_{3}

<<

M_{2}

<<

M_{1}

within one plane as follows: third primary body

M_{3}

is moving on elliptical orbit around second

M_{2}

, and second primary

M_{2}

is moving on elliptical orbit around first

M_{1}

. Our aim for constructing the aforementioned quasi-planar motion of planetoid m is obtaining its coordinates supporting its orbit in a regime of close motion to the plane of orbiting the main bodies

M_{1}

,

M_{2}

, and

M_{3}

. Meanwhile, the system of equations of motion was successfully numerically explored with respect to the existence and stable positioning of approximate solution for a Dyson sphere. As a result, the concept of the Dyson sphere for possible orbiting variety of solar energy absorbers was transformed to the elongated Dyson space net with respect to their trajectories for the successful process of absorbing the energy from the Sun; this can be recognized as symmetry reduction. We obtain the following: (1) the solution for coordinates {x, y} is described by the simplified system of two nonlinear ordinary differential equations of second order, depending on true anomaly f; (2) the expression for coordinate z is given by an equation of Riccati-type where small orbiter that quasi-oscillates close to the fixed plane

{x, y, 0}

. Full article

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

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7 pages, 2562 KiB

Open AccessArticle

Flow of a Viscous Incompressible Fluid from a Moving Point Source

by Sergey V. Ershkov, Evgeniy Yu. Prosviryakov and Dmytro D. Leshchenko

Symmetry 2022, 14(10), 2156; https://doi.org/10.3390/sym14102156 - 14 Oct 2022

Cited by 1 | Viewed by 1507

Abstract

The flow of a viscous incompressible fluid outflowing from a uniformly moving point source is considered. An exact solution to the problem is found in the way that the velocity decreases inversely with the radial coordinate. It is shown that a spherical volume [...] Read more.

The flow of a viscous incompressible fluid outflowing from a uniformly moving point source is considered. An exact solution to the problem is found in the way that the velocity decreases inversely with the radial coordinate. It is shown that a spherical volume of fluid is carried away by the source, the radius of which is inversely proportional with respect to the velocity of motion. In this case, a cylindrical discontinuity arises in the region of forming a wake behind the body, the dimensions of which are determined by the magnitude of the external pressure and do not depend on the velocity of the source. The obtained solutions are governed by hydrodynamical fields of flows which can be recognized as special invariants at symmetry reduction. Full article

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

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16 pages, 5062 KiB

Open AccessArticle

Magneto-Hydrodynamic Flow above Exponentially Stretchable Surface with Chemical Reaction

by Mubashar Arshad, Azad Hussain, Ashraf Elfasakhany, Soumaya Gouadria, Jan Awrejcewicz, Witold Pawłowski, Mohamed Abdelghany Elkotb and Fahad M. Alharbi

Symmetry 2022, 14(8), 1688; https://doi.org/10.3390/sym14081688 - 15 Aug 2022

Cited by 18 | Viewed by 1624

Abstract

This article is focused on investigating the convective magneto-hydrodynamic single-phase flow for comparative analysis of two different base fluids above an exponentially stretchable porous surface under the effect of the chemical reaction. The Buongiorno fluid model is incorporated to observe the Thermophoresis and [...] Read more.

This article is focused on investigating the convective magneto-hydrodynamic single-phase flow for comparative analysis of two different base fluids above an exponentially stretchable porous surface under the effect of the chemical reaction. The Buongiorno fluid model is incorporated to observe the Thermophoresis and Brownian diffusion in this study. Boussinesq approximation for temperature and concentration are accounted for flow to be naturally convective. In this study, water and ethanol are assumed for comparative analysis. Additionally, to achieve the outcomes of the designed three-dimensional flow boundary value, problem technique is employed to simulate the problem in MATLAB. Increase in the magnetic field, thermophoresis diffusion, temperature exponent, and Prandtl number expand thermal boundary, whereas contraction is observed with an increase in porosity. Shear stress rates in respective directions have decreased with an increase in the stretching ratio of the surface. Moreover, through comparison, reasonably enhanced Nusselt number is observed for water under influence of study parameters while the Nusselt number abruptly decreases for ethanol. High mass coefficients are observed for both examined fluids. Full article

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

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10 pages, 3626 KiB

Open AccessArticle

Capture in Regime of a Trapped Motion with Further Inelastic Collision for Finite-Sized Asteroid in ER3BP

by Sergey Ershkov, Dmytro Leshchenko and Alla Rachinskaya

Symmetry 2022, 14(8), 1548; https://doi.org/10.3390/sym14081548 - 28 Jul 2022

Cited by 8 | Viewed by 1611

Abstract

The application of a modern solving algorithm or method of resolving dynamical equations for small projectile of finite sizes orbiting to be captured in a trapped zigzaging oscillations on orbit around the another large asteroid and in a further inelastic colliding scenario with [...] Read more.

The application of a modern solving algorithm or method of resolving dynamical equations for small projectile of finite sizes orbiting to be captured in a trapped zigzaging oscillations on orbit around the another large asteroid and in a further inelastic colliding scenario with him (using a formulation of the elliptic restricted three-body problem, ER3BP) is studied semi-analytically. Herein, two primaries M_Sun and m_p (m_p < M_Sun) revolve around their barycenter on Keplerian orbits with low eccentricities. A smaller body (projectile for attacking a large asteroid) is supposed to be a solid, almost symmetric ellipsoid, having the gravitational potential of the MacCullagh type. Our aim is to develop a previously introduced solving procedure and to investigate the updated dynamics of the projectile captured to a trapped dynamical resonance, thereby having the inelastic collision of a small projectile orbiting on quasi-stable elliptic orbits around the large asteroid, m_p. Full article

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

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11 pages, 681 KiB

Open AccessArticle

The Fate of Molecular Species in Water Layers in the Light of Power-Law Time-Dependent Diffusion Coefficient

by Mohamed Mokhtar Hefny and Ashraf M. Tawfik

Symmetry 2022, 14(6), 1146; https://doi.org/10.3390/sym14061146 - 2 Jun 2022

Cited by 5 | Viewed by 1810

Abstract

In the present paper, we propose two methods for tracking molecular species in water layers via two approaches of the diffusion equation with a power-law time-dependent diffusion coefficient. The first approach shows the species densities and the growth of different species via numerical [...] Read more.

In the present paper, we propose two methods for tracking molecular species in water layers via two approaches of the diffusion equation with a power-law time-dependent diffusion coefficient. The first approach shows the species densities and the growth of different species via numerical simulation. At the same time, the second approach is built on the fractional diffusion equation with a time-dependent diffusion coefficient in the sense of regularised Caputo fractional derivative. As an illustration, we present here the species densities profiles and track the normal and anomalous growth of five molecular species OH, H₂O₂, HO₂, NO₃^-, and NO₂^- via the calculation of the mean square displacement using the two methods. Full article

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

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16 pages, 2345 KiB

Open AccessArticle

Combined Pressure-Driven and Electroosmotic Slip Flow through Elliptic Cylindrical Microchannels: The Effect of the Eccentricity of the Channel Cross-Section

by Pearanat Chuchard and Nattakarn Numpanviwat

Symmetry 2022, 14(5), 999; https://doi.org/10.3390/sym14050999 - 13 May 2022

Cited by 1 | Viewed by 1776

Abstract

Electroosmotic force has been used extensively to manipulate fluid flow in a microfluidic system with various channel shapes, especially an elliptic cylinder. However, developing a computational domain and simulating fluid flow for a system involving an elliptic channel consumes a large amount of [...] Read more.

Electroosmotic force has been used extensively to manipulate fluid flow in a microfluidic system with various channel shapes, especially an elliptic cylinder. However, developing a computational domain and simulating fluid flow for a system involving an elliptic channel consumes a large amount of time. Moreover, the mathematical expression for the fluid velocity of electroosmotic flow in an elliptic channel may be given in the form of the Mathieu functions that have difficulty in achieving the numerical result. In addition, there is clear scientific evidence that confirms the slippage of fluid at the solid-fluid interface in a microscale system. In this study, we present the mathematical model of combined pressure-driven and electroosmotic flow through elliptic microchannels under the slip-fluid condition. From the practical point of view in fluidics, the effect of the eccentricity of the channel cross-section is investigated on the volumetric flow rate to overcome the difficulty. The results show that the substitution of the equivalent circular channel for an elliptic channel provides a valid flow rate under the situation that the areas of both channel cross-sections are equal and the eccentricity of the elliptic cross-section is less than 0.5. Additionally, the flow rate obtained from the substitution is more accurate when the slip length increases or the pressure-gradient-to-external-electric-field ratio decreases. Full article

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

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Review

Jump to: Research

25 pages, 8504 KiB

Open AccessReview

Analysis of Mixing Efficiency in a Stirred Reactor Using Computational Fluid Dynamics

by Adan Ramírez-López

Symmetry 2024, 16(2), 237; https://doi.org/10.3390/sym16020237 - 15 Feb 2024

Cited by 4 | Viewed by 1816

Abstract

Lead recycling is very important for reducing environmental pollution risks and damages. Liquid lead is recovered from exhaust batteries inside stirred batch reactors; the process requires melting to be cleaned. Nevertheless, it is necessary to establish parameters for evaluating mixing to improve the [...] Read more.

Lead recycling is very important for reducing environmental pollution risks and damages. Liquid lead is recovered from exhaust batteries inside stirred batch reactors; the process requires melting to be cleaned. Nevertheless, it is necessary to establish parameters for evaluating mixing to improve the efficiency of the industrial practices. Computational fluid dynamics (CFD) has become a powerful tool to analyze industrial processes for reducing operating costs, avoiding potential damages, and improving the equipment’s performance. Thus, the present work is focused on simulating the fluid hydrodynamics inside a lead-stirred reactor monitoring the distribution of an injected tracer in order to find the best injection point. Then, different injected points are placed on a control plane for evaluation; these are evaluated one by one by monitoring the tracer concentration at a group of points inside the batch. The analyzed reactor is a symmetrical, vertical batch reactor with two geometrical sections: one cylindrical body and a semi-spherical bottom. Here, one impeller with four flat blades in a shaft is used for lead stirring. The tracer concentration on the monitoring points is measured and averaged for evaluating the efficiency inside the tank reactor. Hydrodynamics theory and a comparison between the concentration profiles and distribution of tracer curves are used to demonstrate both methods’ similarities. Then, the invariability of the tracer concentration on the monitoring points is adopted as the main parameter to evaluate the mixing, and the best injection point is found as a function of the shortest mixing time. Additionally, the influence of the impeller rotation speed is analyzed as an additional control parameter to improve industrial practices. Full article

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

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