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19 pages, 902 KiB

Open AccessArticle

Mathematical Analysis of the Poiseuille Flow of a Fluid with Temperature-Dependent Properties

by Evgenii S. Baranovskii, Anastasia A. Domnich and Mikhail A. Artemov

Mathematics 2024, 12(21), 3337; https://doi.org/10.3390/math12213337 - 24 Oct 2024

Viewed by 470

This article is devoted to the mathematical analysis of a heat and mass transfer model for the pressure-induced flow of a viscous fluid through a plane channel subject to Navier’s slip conditions on the channel walls. The important feature of our work is [...] Read more.

This article is devoted to the mathematical analysis of a heat and mass transfer model for the pressure-induced flow of a viscous fluid through a plane channel subject to Navier’s slip conditions on the channel walls. The important feature of our work is that the used model takes into account the effects of variable viscosity, thermal conductivity, and slip length, under the assumption that these quantities depend on temperature. Therefore, we arrive at a boundary value problem for strongly nonlinear ordinary differential equations. The existence and uniqueness of a solution to this problem is analyzed. Namely, using the Galerkin scheme, the generalized Borsuk theorem, and the compactness method, we proved the existence theorem for both weak and strong solutions in Sobolev spaces and derive some of their properties. Under extra conditions on the model data, the uniqueness of a solution is established. Moreover, we considered our model subject to some explicit formulae for temperature dependence of viscosity, which are applied in practice, and constructed corresponding exact solutions. Using these solutions, we successfully performed an extra verification of the algorithm for finding solutions that was applied by us to prove the existence theorem. Full article

(This article belongs to the Special Issue Mathematical Dynamic Flow Models)

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10 pages, 837 KiB

Open AccessArticle

Numerical Solution of Transition to Turbulence over Compressible Ramp at Hypersonic Velocity

by Jiří Holman

Mathematics 2023, 11(17), 3684; https://doi.org/10.3390/math11173684 - 26 Aug 2023

Viewed by 902

Abstract

This work deals with the numerical solution of hypersonic flow of viscous fluid over a compressible ramp. The solved case involves very important and complicated phenomena such as the interaction of the shock wave with the boundary layer or the transition from a [...] Read more.

This work deals with the numerical solution of hypersonic flow of viscous fluid over a compressible ramp. The solved case involves very important and complicated phenomena such as the interaction of the shock wave with the boundary layer or the transition from a laminar to a turbulent state. This type of problem is very important as it is often found on re-entry vehicles, engine intakes, system and sub-system junctions, etc. Turbulent flow is modeled by the system of averaged Navier–Stokes equations, which is completed by the explicit algebraic model of Reynolds stresses (EARSM model) and further enhanced by the algebraic model of bypass transition. The numerical solution is obtained by the finite volume method based on the rotated-hybrid Riemann solver and explicit multistage Runge–Kutta method. The numerical solution is then compared with the results of a direct numerical simulation. Full article

(This article belongs to the Special Issue Mathematical Dynamic Flow Models)

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16 pages, 325 KiB

Open AccessArticle

Remarks on the “Onsager Singularity Theorem” for Leray–Hopf Weak Solutions: The Hölder Continuous Case

by Luigi C. Berselli

Mathematics 2023, 11(4), 1062; https://doi.org/10.3390/math11041062 - 20 Feb 2023

Cited by 3 | Viewed by 1298

Abstract

In this paper, we first present an overview of the results related to energy conservation in spaces of Hölder-continuous functions for weak solutions to the Euler and Navier–Stokes equations. We then consider families of weak solutions to the Navier–Stokes equations with Hölder-continuous velocities [...] Read more.

In this paper, we first present an overview of the results related to energy conservation in spaces of Hölder-continuous functions for weak solutions to the Euler and Navier–Stokes equations. We then consider families of weak solutions to the Navier–Stokes equations with Hölder-continuous velocities with norms uniformly bound in terms of viscosity. We finally provide the proofs of our original results that extend the range of allowed exponents for inviscid limits producing solutions to the Euler equations satisfying the energy equality, and improve the so-called “Onsager singularity” theorem. Full article

(This article belongs to the Special Issue Mathematical Dynamic Flow Models)

22 pages, 15078 KiB

Open AccessArticle

Analysis of Drag Coefficients around Objects Created Using Log-Aesthetic Curves

by Mei Seen Wo, R.U. Gobithaasan, Kenjiro T. Miura, Kak Choon Loy and Fatimah Noor Harun

Mathematics 2023, 11(1), 103; https://doi.org/10.3390/math11010103 - 26 Dec 2022

Cited by 1 | Viewed by 1917

Abstract

A fair curve with exceptional properties, called the log-aesthetic curves (LAC) has been extensively studied for aesthetic design implementations. However, its implementation in terms of functional design, particularly hydrodynamic design, remains mostly unexplored. This study examines the effect of the shape parameter

α

[...] Read more.

A fair curve with exceptional properties, called the log-aesthetic curves (LAC) has been extensively studied for aesthetic design implementations. However, its implementation in terms of functional design, particularly hydrodynamic design, remains mostly unexplored. This study examines the effect of the shape parameter

α

of LAC on the drag generated in an incompressible fluid flow, simulated using a semi-implicit backward difference formula coupled with P

_{2}

−P

_{1}

Taylor–Hood finite elements. An algorithm was developed to create LAC hydrofoils that were used in this study. We analyzed the drag coefficients of 47 LAC hydrofoils of three sizes with various shapes in fluid flows with Reynolds numbers of 30, 40, and 100, respectively. We found that streamlined LAC shapes with negative

α

values, of which curvature with respect to turning angle are almost linear, produce the lowest drag in the incompressible flow simulations. It also found that the thickness of LAC objects can be varied to obtain similar drag coefficients for different Reynolds numbers. Via cluster analysis, it is found that the distribution of drag coefficients does not rely solely on the Reynolds number, but also on the thickness of the hydrofoil. Full article

(This article belongs to the Special Issue Mathematical Dynamic Flow Models)

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14 pages, 565 KiB

Open AccessArticle

Numerical Study of a Confined Vesicle in Shear Flow at Finite Temperature

by Antonio Lamura

Mathematics 2022, 10(19), 3570; https://doi.org/10.3390/math10193570 - 30 Sep 2022

Cited by 3 | Viewed by 1319

Abstract

The dynamics and rheology of a vesicle confined in a channel under shear flow are studied at finite temperature. The effect of finite temperature on vesicle motion and system viscosity is investigated. A two-dimensional numerical model, which includes thermal fluctuations and is based [...] Read more.

The dynamics and rheology of a vesicle confined in a channel under shear flow are studied at finite temperature. The effect of finite temperature on vesicle motion and system viscosity is investigated. A two-dimensional numerical model, which includes thermal fluctuations and is based on a combination of molecular dynamics and mesoscopic hydrodynamics, is used to perform a detailed analysis in a wide range of the Peclet numbers (the ratio of the shear rate to the rotational diffusion coefficient). The suspension viscosity is found to be a monotonous increasing function of the viscosity contrast (the ratio of the viscosity of the encapsulated fluid to that of the surrounding fluid) both in the tank-treading and the tumbling regime due to the interplay of different temperature-depending mechanisms. Thermal effects induce shape and inclination fluctuations of the vesicle which also experiences Brownian diffusion across the channel increasing the viscosity. These effects reduce when increasing the Peclet number. Full article

(This article belongs to the Special Issue Mathematical Dynamic Flow Models)

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19 pages, 7799 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Finite Gradient Models with Enriched RBF-Based Interpolation

by Pedro Areias, Rui Melicio, Fernando Carapau and José Carrilho Lopes

Mathematics 2022, 10(16), 2876; https://doi.org/10.3390/math10162876 - 11 Aug 2022

Cited by 4 | Viewed by 1566

Abstract

A finite strain gradient model for the 3D analysis of materials containing spherical voids is presented. A two-scale approach is proposed: a least-squares methodology for RVE analysis with quadratic displacements and a full high-order continuum with both fourth-order and sixth-order elasticity tensors. A [...] Read more.

A finite strain gradient model for the 3D analysis of materials containing spherical voids is presented. A two-scale approach is proposed: a least-squares methodology for RVE analysis with quadratic displacements and a full high-order continuum with both fourth-order and sixth-order elasticity tensors. A meshless method is adopted using radial basis function interpolation with polynomial enrichment. Both the first and second derivatives of the resulting shape functions are described in detail. Complete expressions for the deformation gradient

F

and its gradient

\nabla F

are derived and a consistent linearization is performed to ensure the Newton solution. A total of seven constitutive properties is required. The classical Lamé parameters corresponding to the pristine material are considered constant. From RVE homogenization, seven properties are obtained, two homogenized Lamé parameters plus five gradient-related properties. Two validation 3D numerical examples are presented. The first example exhibits the size effect (i.e., the stiffening of smaller specimens) and the second example shows the absence of stress singularity and hence the convergence of the discretization method. Full article

(This article belongs to the Special Issue Mathematical Dynamic Flow Models)

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17 pages, 284 KiB

Open AccessArticle

Liouville-Type Results for a Two-Dimensional Stretching Eyring–Powell Fluid Flowing along the z-Axis

by José L. Díaz, Saeed ur Rahman and Muhammad Nouman

Mathematics 2022, 10(4), 631; https://doi.org/10.3390/math10040631 - 18 Feb 2022

Cited by 1 | Viewed by 1499

Abstract

The purpose of this study is to establish Liouville-type results for a three-dimensional incompressible, unsteady flow described by the Eyring–Powell fluid equations. The fluid is studied in a plane

Ω_{p}

while it moves along the z-axis. Therefore the main functions to [...] Read more.

The purpose of this study is to establish Liouville-type results for a three-dimensional incompressible, unsteady flow described by the Eyring–Powell fluid equations. The fluid is studied in a plane

Ω_{p}

while it moves along the z-axis. Therefore the main functions to analyze are given by

u (x, y, z, t)

and

v (x, y, z, t)

, belonging to

Ω_{p}

. The results are obtained for globally bounded initial data as well as their corresponding derivatives, and the variations in velocity along the z-axis belong to the space

L^{2}

and

B M O

. Under such conditions, Liouville-type results are obtained and extended to

L^{p},

p > 2 .

Full article

(This article belongs to the Special Issue Mathematical Dynamic Flow Models)

20 pages, 3050 KiB

Open AccessArticle

Temporal Artificial Stress Diffusion for Numerical Simulations of Oldroyd-B Fluid Flow

by Marília Pires and Tomáš Bodnár

Mathematics 2022, 10(3), 404; https://doi.org/10.3390/math10030404 - 27 Jan 2022

Cited by 2 | Viewed by 2145

Abstract

This paper presents a numerical evaluation of two different artificial stress diffusion techniques for the stabilization of viscoelastic Oldroyd-B fluid flows at high Weissenberg numbers. The standard artificial diffusion in the form of a Laplacian of the extra stress tensor is compared with [...] Read more.

This paper presents a numerical evaluation of two different artificial stress diffusion techniques for the stabilization of viscoelastic Oldroyd-B fluid flows at high Weissenberg numbers. The standard artificial diffusion in the form of a Laplacian of the extra stress tensor is compared with a newly proposed approach using a discrete time derivative of the Laplacian of the extra stress tensor. Both methods are implemented in a finite element code and demonstrated in the solution of a viscoelastic fluid flow in a two-dimensional corrugated channel for a range of Weissenberg numbers. The numerical simulations have shown that this new temporal stress diffusion not only efficiently stabilizes numerical simulations, but also vanishes when the solution reaches a steady state. It is demonstrated that in contrast to the standard tensorial diffusion, the temporal artificial stress diffusion does not affect the final solution. Full article

(This article belongs to the Special Issue Mathematical Dynamic Flow Models)

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