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Navier–Stokes equations
About: Navier–Stokes equations is a research topic. Over the lifetime, 18180 publications have been published within this topic receiving 552555 citations. The topic is also known as: Navier-Stokes equations.
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TL;DR: In this article, the existence of global-in-time unique solutions for the Navier-Stokes equations in the whole n-dimensional space was shown under some smallness assumption on the data.
Abstract: We investigate the Cauchy problem for the inhomogeneous Navier-Stokes equations in the whole n-dimensional space. Under some smallness assumption on the data, we show the existence of global-in-time unique solutions in a critical functional framework. The initial density is required to belong to the multiplier space of \input amssym $\dot {B}^{n/p-1}_{p,1}({\Bbb R}^n)$. In particular, piecewise-constant initial densities are admissible data provided the jump at the interface is small enough and generate global unique solutions with piecewise constant densities. Using Lagrangian coordinates is the key to our results, as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence. © 2012 Wiley Periodicals, Inc.
164 citations
TL;DR: A viscous vortex particle method is presented for computing the fluid dynamics of two-dimensional rigid bodies in motion, and the stability and convergence with respect to numerical parameters are explored in detail, with particular focus on the residual slip velocity.
163 citations
TL;DR: In this article, the consequences of slip at the wall on the flow of a linearly viscous fluid in a channel were investigated, and it was shown that the slip velocity depends on both the shear stress and the normal stress.
Abstract: The assumption that a liquid adheres to a solid boundary (“no-slip” boundary condition) is one of the central tenets of the Navier-Stokes theory. However, there are situations wherein this assumption does not hold. In this paper we investigate the consequences of slip at the wall on the flow of a linearly viscous fluid in a channel. Usually, the slip is assumed to depend on the shear stress at the wall. However, a number of experiments suggests that the slip velocity also depends on the normal stress. Thus, we investigate the flow of a linearly viscous fluid when the slip depends on both the shear stress and the normal stress. In regions where the slip velocity depends strongly on the normal stress, the flow field in a channel is not fully developed and rectilinear flow is not possible. Also, it is shown that, in general, traditional methods such as the Mooney method cannot be used for calculating the slip velocity.
163 citations
TL;DR: A spectral element-Fourier algorithm for simulating incompressible turbulent flows in complex geometries using unstructured quadrilateral meshes and numerical results illustrate the flexibility as well as the exponential convergence of the new algorithm for nonconforming discretizations.
163 citations
TL;DR: In this article, it was shown that an energy decay ∥u(t)∥2 = O(t−µ) for solutions of the Navier-Stokes equations on ℝn, n ≦ 5, implies a decay of the higher order norms.
Abstract: We show that an energy decay ∥u(t)∥2 = O(t−µ) for solutions of the Navier–Stokes equations on ℝn, n ≦ 5, implies a decay of the higher order norms, e.g. ∥Dα u(t)∥2 = O(t−µ −|α|/2) and ∥u(t)|∞ = O(t−µ −n/4).
163 citations