Nonhomogeneous Navier-Stokes Equations Research Articles

Local solutions for nonhomogeneous Navier–Stokes equations with large flux

The local existence of solutions to nonhomogeneous Navier–Stokes equations in cylindrical domains with arbitrary large flux is demonstrated. The existence is proved by the method of successive approximations. To show the existence with the lowest possible regularity the special Besov spaces called the Sobolev–Slobodetskii spaces are used. The inflow and outflow are prescribed on the parts of the boundary which are perpendicular to the ‐axis. Since the inflow and outflow are positive the crucial point of this paper is to verify that ‐coordinate of velocity is also positive. Finally, we conclude the local existence such that the velocity belongs to , the gradient of pressure to and the density to , where .

Global well-posedness to the 3D Cauchy problem of nonhomogeneous Navier–Stokes equations with density-dependent viscosity and large initial velocity

We are concerned with the global well-posedness of strong solutions to the Cauchy problem of nonhomogeneous Navier–Stokes equations with density-dependent viscosity and vacuum in R3. With the help of energy method, we prove the global existence and uniqueness of strong solutions provided that the initial mass is properly small. In particular, the initial velocity can be arbitrarily large. This improves He, Li, and Lü’s work [Arch. Ration. Mech. Anal. 239, 1809–1835 (2021)]. Moreover, we also extend the result of Liu [Discrete Contin. Dyn. Syst. B 26, 1291–1303 (2021)] to the case that large oscillations of the solutions are allowed.

Inviscid Limit of 3D Nonhomogeneous Navier–Stokes Equations with Slip Boundary Conditions

In this paper, we consider the inviscid limit of a nonhomogeneous incompressible Navier–Stokes system with a slip-without-friction boundary condition. We study the convergence in strong norms for a solution and obtain the convergence rate in space W2,p(Ω) when the boundary is flat. We need to establish the uniform bound of the solution in space W3,p(Ω), and the key of proofs is to obtain a priori estimation of ∂tu in space W1,p(Ω).

Global Well-Posedness and Exponential Decay of 2D Nonhomogeneous Navier–Stokes and Magnetohydrodynamic Equations with Density-Dependent Viscosity and Vacuum

We establish global well-posedness of strong solutions for the nonhomogeneous magnetohydrodynamic equations with density-dependent viscosity and initial density allowing vanish in two-dimensional bounded domains. Applying delicate energy estimates and Desjardins’ interpolation inequality, we derive the global existence of a unique strong solution provided that \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\) is suitably small. Moreover, we also obtain exponential decay rates of the solution. In particular, there is no need to impose some compatibility condition on the initial data despite the presence of vacuum. As a direct application, it is shown that the similar result also holds for the nonhomogeneous Navier–Stokes equations with density-dependent viscosity.

Global well-posedness to some modified 2-D nonhomogeneous Navier-Stokes equations

In this paper, we are concerned with the global well-posedness of modified 2D nonhomogeneous Navier-Stokes equations. Our results improve the results of Zhang [Int. Math. Res. Not. 2008(1), rnn098] to some extent.

Stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel

In this article, we study the local boundary stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel around Poiseuille flow which is a stationary solution for the system under consideration. The feedback control operator we construct has finite dimensional range. The homogeneous Navier–Stokes equations are of parabolic nature and the stabilization result for such system is well studied in the literature. In the present article we prove a stabilization result for non-homogeneous Navier–Stokes equations which involves coupled parabolic and hyperbolic dynamics by using only one boundary control for the parabolic part.

On local strong solutions to the three-dimensional nonhomogeneous Navier–Stokes equations with density-dependent viscosity and vacuum

We study the three-dimensional nonhomogeneous Navier–Stokes equations with density-dependent viscosity and vacuum on Ω⊂R3, which is either a bounded domain or a usual unbounded one such as the whole space R3 and an exterior one. In particular, the initial density can have compact support when Ω is unbounded. For initial data without additional compatibility conditions, we prove that there exists a unique local strong solution to the initial and initial boundary value problems. Moreover, the continuous of strong solutions on the initial data is derived under an additional compatibility condition.

Iterative schemes for the non-homogeneous Navier–Stokes equations based on the finite element approximation

In this paper, we consider the stability and convergence of three iterative schemes for the non-homogeneous steady Navier–Stokes equations. As a nonlinear problem, we will get a nonlinear discrete system if approximating the non-homogeneous Navier–Stokes equations. After proving the stability and error estimates of the finite element method for the non-homogeneous Navier–Stokes equations, three iterative schemes are investigated for solving the resulted nonlinear discrete system. The stability and convergence conditions for these iterative methods are also analyzed, respectively. Furthermore, new results for the stop criterion are proved. Finally, we show some numerical experiments to illustrate the theoretical prediction.

Besov Space Regularity Conditions for Weak Solutions of the Navier–Stokes Equations

Consider a bounded domain $${{\Omega \subseteq \mathbb{R}^3}}$$ with smooth boundary, some initial value $${{u_0 \in L^2_{\sigma}(\Omega )}}$$ , and a weak solution u of the Navier–Stokes system in $${{[0,T) \times\Omega,\,0 < T \le \infty}}$$ . Our aim is to develop regularity and uniqueness conditions for u which are based on the Besov space $$B^{q,s}(\Omega ):=\left\{v\in L^2_{\sigma}(\Omega ); \|v\|_{B^{q,s}(\Omega )} := \left(\int\limits^{\infty}_0 \left\|e^{-\tau A}v\right\|^s_q {\rm d} \tau\right)^{1/s}<\infty \right\}$$ with $${{2 < s < \infty,\,3 < q <\infty,\,\frac2{s}+\frac{3}{q} = 1}}$$ ; here A denotes the Stokes operator. This space, introduced by Farwig et al. (Ann. Univ. Ferrara 55:89–110, 2009 and J. Math. Fluid Mech. 14: 529–540, 2012), is a subspace of the well known Besov space $${{{\mathbb{B}}^{-2/s}_{q,s}(\Omega )}}$$ , see Amann (Nonhomogeneous Navier–Stokes Equations with Integrable Low-Regularity Data. Int. Math. Ser. pp. 1–28. Kluwer/Plenum, New York, 2002). Our main results on the regularity of u exploits a variant of the space $${{B^{q,s}(\Omega )}}$$ in which the integral in time has to be considered only on finite intervals (0, δ ) with $${{\delta \to 0}}$$ . Further we discuss several criteria for uniqueness and local right-hand regularity, in particular, if u satisfies Serrin’s limit condition $${{u\in L^{\infty}_{\text{loc}}([0,T);L^3_{\sigma}(\Omega ))}}$$ . Finally, we obtain a large class of regular weak solutions u defined by a smallness condition $${{\|u_0\|_{B^{q,s}(\Omega )} \le K}}$$ with some constant $${{K=K(\Omega, q)>0}}$$ .

Long time existence of regular solutions to non-homogeneous Navier-Stokes equations

We consider the motion of incompressible viscous non-homogene-ous fluid described by the Navier-Stokes equations in a bounded cylinder $\Omega$ under boundary slip conditions. Assume that the $x_3$-axis is the axis of the cylinder. Let $\varrho$ be the density of the fluid, $v$ -- the velocity and $f$ the external force field. Assuming that quantities $\nabla\varrho(0)$, $\partial_{x_3}v(0)$, $\partial_{x_3}f$, $f_3|_{\partial\Omega}$ are sufficiently small in some norms we prove large time regular solutions such that $v\in H^{2+s,1+s/2}(\Omega\times(0,T))$, $\nabla p\in H^{s,s/2}(\Omega\times(0,T))$, $½ < s < 1$ without any restriction on the existence time $T$. The proof is divided into two parts. First an a priori estimate is shown. Next the existence follows from the Leray-Schauder fixed point theorem.

Influence of Hall effects on characteristics of magnetohydrodynamic converging channel

Influences of Hall effects on the flow characteristics and performance of the magnetohydrodyanmic (MHD) converging channel as an MHD generator are studied numerically in the present study. The numerical algorithm used is composed of an entropy conditioned scheme for solving the non-homogeneous Navier-Stokes equations, in conjunction with an SOR method for solving the elliptic equation governing the electrical potential of the electric field. It is found that Hall effects may lead to distortions in both the flow field and electric field in the MHD converging channel. The electric current lines are distorted and induce eddy currents, which will be mitigated by Hall effects. Comparisons of parameters characterizing the performance of the MHD generator for different geometry also reveal that the Hall effects may deteriorate the performance of MHD generator, especially for converging channel.

Simulation of three-dimensional nonideal MHD flow at low magnetic Reynolds number

A numerical procedure based on a five-wave model associated with non-ideal, low magnetic Reynolds number magnetohydrodynamic (MHD) flows was developed. It is composed of an entropy conditioned scheme for solving the non-homogeneous Navier-Stokes equations, in conjunction with an SOR method for solving the elliptic equation governing the electrical potential of flow field. To validate the developed procedure, two different test cases were used which included MHD Rayleigh problem and MHD Hartmann problem. The simulations were performed under the assumption of low magnetic Reynolds number. The simulated results were found to be in good agreement with the closed form analytical solutions deduced in the present study, showing that the present algorithm could simulate engineering MHD flow at low magnetic Reynolds number effectively. In the end, a flow field between a pair of segmented electrodes in a three dimensional MHD channel was simulated using the present algorithm with and without including Hall effects. Without the introduction of Hall effects, no distortion was observed in the current and potential lines. By taking the Hall effects into account, the potential lines distorted and clustered at the upstream and downstream edges of the cathode and anode, respectively.

Strong solutions for the nonhomogeneous Navier-Stokes equations in unbounded domains

We show the existence of strong solutions for the nonhomogeneous Navier–Stokes equations in three-dimensional domains with boundary uniformly of class C3. Under suitable assumptions, uniqueness is also proved. Copyright © 2009 John Wiley & Sons, Ltd.

Characterization of the three-dimensional supersonic flow for the MHD generator

A numerical procedure based on a five-wave MHD model associated with non-ideal, low magnetic Reynolds number MHD flows was developed in the present study for analyzing the flow fields in the MHD generator of a MHD bypass scramjet. The numerical procedure is composed of an entropy conditioned scheme for solving the non-homogeneous Navier-Stokes equations, in conjunction with an SOR method for solving the elliptic equation governing the electrical potential. It was found that a separation would take place near the downstream edge of the second electrode, where the local adverse pressure gradient is large, and the core of the flow field is characterized as a 2-D flow due to the Hartmann effects along the direction of the magnetic field. The electric current lines would be increasingly distorted as the magnetic interactive parameter increases, and even induce an eddy current. Induced eddy current was also found in the different cross-sections along the axial direction, all of these would definitely deteriorate the performance of the MHD generator. The cross-sectional M-shape velocity profile found along the axial direction between the insulating walls is responsible for the formation of the vortex flow at the corner of the insulator cross-section, which, in turn, induces the corner eddy current at the corner. A numerical parametric study was also performed, and the computed performance parameters for the MHD generator suggest that, in order to enhance the performance of MHD generator, the magnetic interaction parameter should be elevated.

Drag minimization for Navier–Stokes Flow

AbstractThis paper investigates the drag minimization in a two‐dimensional flow which is governed by a nonhomogeneous Navier–Stokes equations. Two approaches are utilized to derive shape gradient of the cost functional. The first one is to use the shape derivative of the fluid state and its associated adjoint state; the second one is to utilize the differentiability of a minimax formulation involving a Lagrange functional with a function space parametrization technique. Finally, a gradient type algorithm is effectively formulated and implemented for the mentioned drag minimization problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

The application of domain derivative of the nonhomogeneous Navier–Stokes equations in shape reconstruction

This paper is concerned with the shape reconstruction of the Newtonian viscous incompressible fluids driven by the stationary nonhomogeneous Navier–Stokes equations. By formulating the domain derivative of the nonhomogeneous Navier–Stokes equations and applying a regularized Gauss–Newton iterative algorithm, the numerical examples are given for recovering the shape. The results show that our theory is useful for practical purpose and the proposed algorithm is feasible.

Another Proof of Stability for Global Weak Solutions of 2D Degenerated Shallow Water Models

We consider non-linear viscous shallow water models with varying topography, extra friction terms and capillary effects, in a two-dimensional framework. Water-depth dependent laminar and turbulent friction coefficients issued from an asymptotic analysis of the three-dimensional free-surface Navier–Stokes equations are considered here. A new proof of stability for global weak solutions is given in periodic domain Ω = T2, adapting the method introduced by J. Simon in [15] for the non-homogeneous Navier–Stokes equations. Existence results for such solutions can be obtained from this stability analysis.

Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spaces

We investigate very weak solutions to the stationary Stokes and Stokes resolvent problem in function spaces with Muckenhoupt weights. The notion used here is similar but even more general than the one used in Amann (Nonhomogeneous Navier–Stokes equations with integrable low-regularity data. Int. Math. Ser., pp. 1–26. Kluwer Academic/Plenum Publishing, New York, 2002) or Galdi et al. (Math. Ann. 331, 41–74, 2005). Consequently the class of solutions is enlarged. To describe boundary conditions we restrict ourselves to more regular data. We introduce a Banach space that admits a restriction operator and that contains the solutions according to such data.

Global Smooth Solutions to the 2-D Nonhomogeneous Navier–Stokes Equations

In two space dimensions and under the assumptions that the initial density ρ 0 has a positive lower bound and the initial data for s>2, we prove the global existence and uniqueness of smooth solutions to a model of the inhomogeneous Navier-Stokes equations with the viscous coefficient depending on the density of the fluid.

Shape optimization for Navier–Stokes flow

This article describes the optimal shape design of the newtonian viscous incompressible fluids driven by the stationary nonhomogeneous Navier–Stokes equations. We use three approaches to derive the structures of shape gradients for some given cost functionals. The first one is to use the Piola transformation and derive the state derivative and its associated adjoint state; the second one is to use the differentiability of a minimax formulation involving a Lagrangian functional with a function space parametrization technique; the last one is to employ the differentiability of a minimax formulation with a function space embedding technique. Finally, we apply a gradient type algorithm to our problem and numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible.