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We discuss the asymptotic stability of stationary solutions to the incompressible Navier–Stokes equations on the whole space in Besov spaces. A critical estimate for the semigroup generated by the Laplacian with a perturbation is the main ingredient of the argument.

We study Boltzmann-Fermi-Dirac equation when quantum effects are taken into account in dilute gas dynamics. By employing new estimates on trilinear terms of collision kernels, we prove the global existence of the classical solution to Boltzmann-Fermi-Dirac equation near equilibrium. Furthermore, the limit from Boltzmann-Fermi-Dirac equation to incompressible Navier-Stokes-Fourier equations is justified rigorously. The corresponding formal analysis was given in the thesis of Zakrevskiy \cite{Zakrevskiy}, Comment: 32 pages

An avalanche dynamic model is proposed for the three-dimensional simulation of avalanche dynamics that adopts the Navier–Stokes equations as the governing framework and introduces μ(I) granular rheology in the constitutive law. Using the finite–volume method as the numerical scheme, a simulation program based on the OpenFOAM open-source platform was developed. The validity of the model as well as the program was verified by numerical simulations of a laboratory-scale granular flow test. An interface developed between the natural terrain data and the calculation program enables the simulation of natural avalanches. The Baige avalanche, a natural event, was reproduced. Comparisons between simulations and real data further verify the feasibility and accuracy of the full avalanche dynamic model. In addition, the evolution of the hydrostatic pressure, shear strain rate, inertial number, and kinematic viscosity over the duration of the avalanche were analyzed to extract the underlying internal mechanism.

An initial-boundary value problem for a coupled chemotaxis-Navier–Stokes model with porous medium type diffusion is considered. Previous related literature has provided profound knowledge in cases when the system is augmented with no-flux/no-flux/no-slip boundary conditions for the density of cells, the chemical concentration and the fluid velocity field, respectively; in particular, available qualitative results strongly indicate that only trivial solution behavior can be expected on large time scales. In line with refined modeling approaches to oxygen evolution near fluid-air interfaces, this study now focuses on situations involving a fixed chemoattractant concentration on the boundary. Despite an apparent loss of mathematically favorable energy structures thereby induced, by means of an alternative variational approach a basic theory of global existence is developed in a natural framework of weak solvability. Beyond this, some additional qualitative information on the large time behavior of these solutions is derived by identifying a certain global relaxation property. Specifically, a second result asserts, within a suitable topological setting, the existence of a bounded set which eventually absorbs each individual of the obtained trajectories, and the diameter of which is bounded only by the physically relevant quantities of total population size and prescribed boundary concentration of the chemical signal.

Scientists and engineers have conducted a considerable lot of experimental and theoretical study to increase the productivity of industrial operations. Experimental evidence from engineers and scientists shows that heat transfer is necessary for excellent multi scale production naturally, continuous phase liquids are used to achieve thermal characteristics. However, due to the limitations of heat transmission, their significance is constrained. We present a mathematical model for unsteady heat and mass transfer in MHD flow of Nano-fluid with buoyancy and variable thermal conductivity. Using self-similar variables, the coupled non-linear system of equations governing the boundary-layer flows of Nano-fluids were converted into a two-point boundary value problem. The dsolve command with the numeric option on a real-valued two-point boundary value problem (BVP) is used to finds a numerical solution for our system of equations. Computation is performed in both hardware precision and arbitrary precision. The output of the computation is plotted using the ode plot generator odeplot and displayed using Maple plot display, and wall rate transfer is extracted. From the result, increase in thermal buoyancy, unsteadiness parameter and suction parameter are seen to increase velocity distributions. Temperature boundary layer is enhanced with increase in unsteadiness parameter, Brownian motion parameter and Eckert number while radiation, thermophoretic diffusion parameter, reactivity parameter, thermal conductivity and suction enhances species concentration. Keywords: Unsteady, Heat and Mass Transfer, Nano-Fluids, Buoyance, Variable Thermal Conduction, thermo-gravitational, Navier Stokes, magneto hydrodynamics. Title: Unsteady Heat and Mass Transfer MHD Flow of Nano-Fluids with Buoyancy and Variable Thermal Conductivity Author: Akindele M. Okedoye, Azeez A. Waheed, Olufemi. A. Akinyemi International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 10, Issue 2, October 2022 - March 2023 Page No: 80-93 Research Publish Journals Website: www.researchpublish.com Published Date: 23-February-2023 DOI: https://doi.org/10.5281/zenodo.7669943 Paper Download Link (Source) https://www.researchpublish.com/papers/unsteady-heat-and-mass-transfer-mhd-flow-of-nano-fluids-with-buoyancy-and-variable-thermal-conductivity, International Journal of Mathematics and Physical Sciences Research, ISSN 2348-5736 (Online), Research Publish Journals, Website: www.researchpublish.com

The Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{rcll} n_t + u\cdot \nabla n &{}=&{} \Delta n - \chi \nabla \cdot (n\nabla c) + \rho n-\mu n^2,\\ c_t + u\cdot \nabla c &{}=&{} \Delta c-c+n, \\ u_t + (u\cdot \nabla )u &{}=&{} \Delta u + \nabla P + n \nabla \phi + f(x,t), \qquad \nabla \cdot u=0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ n t + u · ∇ n = Δ n - χ ∇ · ( n ∇ c ) + ρ n - μ n 2 , c t + u · ∇ c = Δ c - c + n , u t + ( u · ∇ ) u = Δ u + ∇ P + n ∇ ϕ + f ( x , t ) , ∇ · u = 0 , ( ⋆ ) is considered in a smoothly bounded convex domain $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 , with $$\phi \in W^{2,\infty }(\Omega )$$ ϕ ∈ W 2 , ∞ ( Ω ) and $$f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)$$ f ∈ C 1 ( Ω ¯ × [ 0 , ∞ ) ; R 3 ) , and with $$\chi >0, \rho \in \mathbb {R}$$ χ > 0 , ρ ∈ R and $$\mu >0$$ μ > 0 . As recent literature has shown, for all reasonably mild initial data a corresponding no-flux/no-flux/Dirichlet initial-boundary value problem possesses a global generalized solution, but the knowledge on its regularity properties has not yet exceeded some information on fairly basic integrability features. The present study reveals that whenever $$\omega >0$$ ω > 0 , requiring that $$\begin{aligned} \frac{\rho }{\min \{\mu ,\mu ^{\frac{3}{2}+\omega }\}} < \eta \end{aligned}$$ ρ min { μ , μ 3 2 + ω } < η with some $$\eta =\eta (\omega )>0$$ η = η ( ω ) > 0 , and that f satisfies a suitable assumption on ultimate smallness, is sufficient to ensure that each of these generalized solutions becomes eventually smooth and classical. Furthermore, under these hypotheses ($$\star $$ ⋆ ) is seen to admit an absorbing set with respect to the topology in $$L^\infty (\Omega )$$ L ∞ ( Ω ) . By trivially applying to the case when $$\mu >0$$ μ > 0 is arbitrary and $$\rho \le 0$$ ρ ≤ 0 , these results especially assert essentially unconditional statements on eventual regularity in taxis-reaction systems interacting with liquid environments, such as arising in contexts of models for broadcast spawning discussed in recent literature.

In this survey article, we will discuss some regularity criteria for the Navier--Stokes equation that provide geometric constraints on any possible finite-time blowup. We will also discuss the physical significance of such regularity criteria.

In this article, we give a comprehensive characterization of$L^1$-summability for the Navier-Stokes flows in the half space, which is a long-standing problem. The main difficulties are that$L^q-L^r$estimates for the Stokes flow don't work in this end-point case:$q=r=1$; the projection operator$P: L^1\longrightarrow L^1_\sigma$is not bounded any more; useful information on the pressure function is missing, which arises in the net force exerted by the fluid on the noncompact boundary. In order to achieve our aims, by making full use of the special structure of the half space, we decompose the pressure function into two parts. Then the knotty problem of handling the pressure term can be transformed into establishing a crucial and new weighted$L^1$-estimate, which plays a fundamental role. In addition, we overcome the unboundedness of the projection$P$by solving an elliptic problem with homogeneous Neumann boundary condition.

We study the small data global well-posedness and time-decay rates of solutions to the Cauchy problem for three-dimensional compressible Navier–Stokes–Allen–Cahn equations via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, the $\dot {H}^{-s}$($0\leq s) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.

We consider the time periodic solutions to the Keller-Segel (KS) system coupled to Navier-Stokes equation in R N ( N ≥ 4 ) . Based on the theory of real interpolation as Meyer and Yamazaki used, we first consider the existence of time periodic solution in B C ( R , L s , ∞ ( R N ) ) when S ( n ) = n − 1 2 . Next for the case S ( n ) = 0 , we study the mild solution and point out it can become strong solution in B C ( R , L s , ∞ ( R N ) ) if the external force g satisfies a natural condition which comes from the strong solvability of the inhomogeneous Stokes equations.

The Cahn–Hilliard–Navier–Stokes system describes the evolution of two isothermal, incompressible, immiscible fluids in a bounded domain. In this work, we consider the stationary nonlocal Cahn–Hilliard–Navier–Stokes system in two and three dimensions with singular potential. We prove the existence of a weak solution for the system using pseudo-monotonicity arguments and Browder’s theorem. Further, we establish the uniqueness and regularity results for the weak solution of the stationary nonlocal Cahn–Hilliard–Navier–Stokes system for constant mobility parameter and viscosity. Finally, in two dimensions, we establish that the stationary solution is exponentially stable (for convex singular potentials) under suitable conditions on mobility parameter and viscosity.

International audience; The discrete element (roughness) method developed a few decades ago is revisited using the double averaging technique applied to the Navier-Stokes equation. A k − ω based DANS turbulence model is thus derived to be able to account for roughness effects. Several closure relations are proposed to model all terms induced by the use of the double averaging. The momentum and energy equations are considered in their simplified forms adapted to a 1D channel code in accordance with the DNS results used for the validation. To reconcile the discrete element (roughness) method with the double averaged Navier-Stokes equations the notion of representative elementary roughness is introduced. A large validation dataset coming from various DNS configurations is then used to assess the predictions of the proposed DANS model. Yet not fully complete, especially regarding the dispersive terms due to a lack of data, the performed validation already proves the overall excellent behavior of the DANS model and demonstrates the relevance of the present methodology based on the representative elementary roughness.

We study a distributed optimal control problem for a three-dimensional Navier-Stokes-α model. We prove the solvability of the optimal control, derive first-order necessary optimality conditions, using a Lagrange multipliers theorem in Banach spaces, and establish a result concerning to the uniqueness of global optimal solution.

This paper deals with the Keller–Segel–Navier–Stokes model with indirect signal production in a three-dimensional (3D) bounded domain with smooth boundary. When the logistic-type degradation here is weaker than the usual quadratic case, it is proved that for any sufficiently regular initial data, the associated no-flux/no-flux/no-flux/Dirichlet problem possesses at least one globally defined solution in an appropriate generalized sense, and that this solution is uniformly bounded in [Formula: see text] with any [Formula: see text]. Moreover, under an explicit condition on the chemotactic sensitivity, these solutions are shown to stabilize toward the corresponding spatially homogeneous state in the sense of some suitable norms. We underline that the same results were established for the corresponding system with direct signal production in a well-known result if the degradation is quadratic. Our result rigorously confirms that the indirect signal production mechanism genuinely contributes to the global solvability of the 3D Keller–Segel–Navier–Stokes system.

International audience

International audience; In this paper, we study in two and three space dimensions, the a posteriori error estimates for the Large Eddy Simulation applied to the Navier-Stokes system. We begin by introducing the Navier-Stokes and the corresponding Large Eddy Simulation (LES) equations. Then we introduce the corresponding discrete problem based on the finite element method. We establish an a posteriori error estimation with three types of error indicators related to the filter of the LES method, to the discretization and to the linearization. Finally, numerical investigations are shown and discussed.

This paper presents a new method that is constructed by combining the Shehu transform and the residual power series method. Precisely, we provide the application of the proposed technique to investigate fractional-order linear and nonlinear problems. Then, we implemented this new technique to obtain the result of fractional-order Navier-Stokes equations. Finally, we provide three-dimensional figures to help the effect of fractional derivatives on the actions of the achieved profile results on the proposed models.

Квазигидродинамическая система была предложена Шеретовым Ю.В. в 1993 году. Известные точные решения этой системы в подавляющем большинстве случаев удовлетворяют либо уравнениям Навье-Стокса, либо уравнениям Эйлера. В настоящей работе описан новый класс точных решений квазигидродинамической системы, которые не удовлетворяют ни уравнениям Навье-Стокса, ни уравнениям Эйлера. Соответствующие точные решения системы Навье-Стокса получаются из построенных решений предельным переходом при $c_s\to +\infty$, где $c_s$ - скорость звука в жидкости. The quasi-hydrodynamic system was proposed by Sheretov Yu.V. in 1993. The known exact solutions of this system in the overwhelming majority of cases satisfy either the Navier-Stokes equations or the Euler equations. This paper describes a new class of exact solutions of quasi-hydrodynamic system that satisfy neither the Navier-Stokes equations, nor the Euler equations. The corresponding exact solutions of the Navier-Stokes system are obtained from the constructed solutions by passing to the limit at $c_s\to +\infty$, where $c_s$ is the sonic velocity in the fluid.

Blowing a recorder at a low to moderate blowing speed with the toneholes all closed yields the lowest note in the range of the instrument. As the blowing speed is increased, the tone abruptly changes to the tone an octave higher. This "jump" in the frequency of the dominant spectral component of the tone is referred to as "regime change." Interestingly, in conversations with recorder players, several have mentioned that regime change seems to occur at a significantly lower blowing speed for bass recorders than for instruments that sound an octave or more higher. In this paper we study regime change in the recorder and use Navier-Stokes modeling to confirm and study differences in the behavior of different instruments in the recorder family. We show, using modeling, how the regime change threshold in a model of the bass recorder can be increased by changing the geometry in the vicinity of the labium. These results are then confirmed through experimental studies of real recorders with designs inspired by the modeling results. The insights gained from these results suggest new recorder designs that may produce instruments that in some respects are more playable than current instruments.

We study compressible isentropic Navier-Stokes-Poisson equations in ℝ3. With some appropriate assumptions on the density, velocity and potential, we show that the classical solution of the Cauchy problem for compressible unipolar isentropic Navier-Stokes-Poisson equations with attractive forcing will blow up in finite time. The proof is based on a contradiction argument, which relies on proving the conservation of total mass and total momentum.

espanolEl modelo computacional tridimensional idealizado de un aneurisma cerebral sacular es usado para estudiar la hemodinamica vascular, bajo el efecto de campos electromagneticos externos. Los campos analizados son emitidos por resonadores magneticos que tienen diferentes direcciones de aplicacion sobre el paciente en posicion decubito dorsal. El modelo en estado estacionario resuelve las ecuaciones de Navier Stokes teniendo en cuenta el comportamiento no-Newtoniano de la sangre, acoplando las ecuaciones de magneto-hidrodinamica. La simulacion computacional de las dos configuraciones de los resonadores, permite evaluar el efecto de la direccion de aplicacion del campo, en la velocidad de la sangre y en el tiempo de permanencia de particulas inyectadas en el dominio. Se presentan los contornos de velocidad de fluido para comprender el patron de flujo en el aneurisma. Los resultados muestran que bajo la accion del campo magnetico emitido por un resonador abierto, el fluido entra con una mayor velocidad al saco del aneurisma, en comparacion con un resonador cerrado. Se evidencia un aumento significativo en el tiempo de permanencia de las particulas dentro del dominio para el modelo del resonador abierto. El procedimiento seguido permite adquirir una mayor compresion de los fenomenos del campo magnetico en la hemodinamica de patologias vasculares. EnglishThe idealized three-dimensional computational model of a saccular cerebral aneurysm is used to study vascular hemodynamics under the effect of external electromagnetic fields. Magnetic resonators emit the analyzed fields with different application directions on the patient in the dorsal decubitus position. The steady-state model solves the Navier Stokes equations considering the non-Newtonian behavior of the blood by coupling the magneto-hydrodynamic equations. Using computational simulation, the effect of different magnetic field directions on the blood velocity field and the elapsed particles' time is evaluated. Flow velocity contours are presented to understand the flow pattern in the aneurysm, which is significantly altered within the sac in the presence of a magnetic field. The results show that under the magnetic field's action emitted by an open resonator, the fluid enters the sac of the aneurysm with greater speed compared to a closed resonator. Additionally, there is a significant increment in the particles' elapsed time within the domain for the open resonator model. The procedure followed allows acquiring a greater compression of the magnetic field phenomena in the hemodynamics of vascular pathologies.

This work presents a new approach to modelling of free surface non-Newtonian (viscoplastic or viscoelastic) fluid flows on dynamically adapted octree grids. The numerical model is based on the implicit formulation and the staggered location of governing variables. We verify our model by comparing simulations with experimental and numerical results known from the literature.