A well-posed problem for the exterior Stokes equations in two and three dimensions

1991 ◽  
Vol 114 (4) ◽  
pp. 313-333 ◽  
Author(s):  
Vivette Girault ◽  
Adelia Sequeira
Keyword(s):  
Stokes Equations ◽  
Three Dimensions ◽  
Well Posed

Singularities

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Stokes Equations ◽  
Negative Answer ◽  
Three Dimensions ◽  
Picard Iteration ◽  
Navier Stokes ◽  
Scale Invariant ◽  
Navier Stokes Equations ◽  
L2 Norm ◽  
Two Dimensional Flow ◽  
Physical Consequences

The initial value problem of the incompressible Navier–Stokes equations is explained. Leray’s classic study of it (using Picard iteration) is simplified and described in the language of physics. The ideas of Lebesgue and Sobolev norms are explained. The L2 norm being the energy, cannot increase. This gives sufficient control to establish existence, regularity and uniqueness in two-dimensional flow. The L3 norm is not guaranteed to decrease, so this strategy fails in three dimensions. Leray’s proof of regularity for a finite time is outlined. His attempts to construct a scale-invariant singular solution, and modern work showing this is impossible, are then explained. The physical consequences of a negative answer to the regularity of Navier–Stokes solutions are explained. This chapter is meant as an introduction, for physicists, to a difficult field of analysis.

scholarly journals A Framework for Approximation of the Stokes Equations in an Axisymmetric Domain

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Niklas Ericsson
Keyword(s):  
Fourier Expansion ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Three Dimensional ◽  
Countable Family ◽  
Angular Variable ◽  
Two Dimensional ◽  
Sobolev Norms ◽  
Well Posed ◽  
Variational Formulations

Abstract We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems.

scholarly journals Numerical simulation of Faraday waves

2009 ◽  
Vol 635 ◽  
pp. 1-26 ◽  
Author(s):  
NICOLAS PÉRINET ◽  
DAMIR JURIC ◽  
LAURETTE S. TUCKERMAN
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Oscillation Period ◽  
Finite Amplitude ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Faraday Waves ◽  
Temporal Behaviour ◽  
Two Fluids ◽  
Front Tracking Method

We simulate numerically the full dynamics of Faraday waves in three dimensions for two incompressible and immiscible viscous fluids. The Navier–Stokes equations are solved using a finite-difference projection method coupled with a front-tracking method for the interface between the two fluids. The critical accelerations and wavenumbers, as well as the temporal behaviour at onset are compared with the results of the linear Floquet analysis of Kumar & Tuckerman (J. Fluid Mech., vol. 279, 1994, p. 49). The finite-amplitude results are compared with the experiments of Kityk et al (Phys. Rev. E, vol. 72, 2005, p. 036209). In particular, we reproduce the detailed spatio-temporal spectrum of both square and hexagonal patterns within experimental uncertainty. We present the first calculations of a three-dimensional velocity field arising from the Faraday instability for a hexagonal pattern as it varies over its oscillation period.

Three-Dimensional Numerical Prediction of the Hydrodynamic Loads and Motions of Offshore Structures

2000 ◽  
Vol 122 (4) ◽  
pp. 294-300 ◽  
Author(s):  
Karl W. Schulz ◽  
Yannis Kallinderis
Keyword(s):  
Flow Structure ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Structural Response ◽  
Offshore Structures ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Hydrodynamic Loads ◽  
Drag Coefficients ◽  
Lift And Drag

A generalized numerical method for solution of the incompressible Navier-Stokes equations in three-dimensions has been developed. This solution methodology allows for the accurate prediction of the hydrodynamic loads on offshore structures, which is then combined with a rigid body structural response to address the flow-structure coupling which is often present in offshore applications. Validation results using this method are first presented for fixed structures which compare the drag coefficients of sphere and cylinder geometries to experimental measurements over a range of subcritical Reynolds numbers. Additional fixed structure results are then presented which explore the influence of aspect ratio effects on the lift and drag coefficients of a bare circular cylinder. Finally, the spanwise flow variations between a fixed and freely vibrating cylindrical structure are compared to demonstrate the ability of the flow-structure method to correctly predict correlation length increases for a vibrating structure. [S0892-7219(00)00904-3]

Direct Simulation of Electrorheological Suspensions

2001 ◽  
Author(s):  
Aijun Wang ◽  
Pushpendra Singh ◽  
Nadine Aubry
Keyword(s):  
Electric Field ◽  
Stokes Equations ◽  
Hydrodynamic Force ◽  
Particle System ◽  
Electrostatic Force ◽  
Body Motion ◽  
Three Dimensions ◽  
Spherical Particles ◽  
Point Dipole ◽  
Direct Simulation

Abstract A new distributed multiplier/fictitious (DLM) domain method is developed for direct simulation of electrorheological (ER) suspensions subjected to spatially uniform electrical fields. The method is implemented both in two and three dimensions. The fluid-particle system is treated implicitly using the combined weak formulation described in [1,2]. The governing Navier-Stokes equations for the fluid are solved everywhere, including the interior of the particles. The flow inside the particles is forced to be a rigid body motion by a distribution of Lagrange multipliers. The electrostatic force acting on the polarized spherical particles is modeled based on the point-dipole approximation. Using our code we have studied the time evolution of particle-scale structures of ER suspensions in channels subjected to the pressure driven flow. In our study, the flow direction is perpendicular to that of the electric field. Simulations show that when the hydrodynamic force is zero, or very small compared to the electrostatic force, the particles form chains that are aligned approximately parallel to the direction of electric field. But, when the magnitude of hydrodynamic force is comparable to that of the electrostatic force the particle chains orient at an angle with the direction of the electric field. The angle between the particle chain and the direction of the electric field depends on the relative strengths of the hydrodynamic and electrostatic forces.

Navier Stokes Derivative Estimates in Three Dimensions with Boundary Values and Body Forces

1991 ◽  
Vol 43 (6) ◽  
pp. 1161-1212 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Stokes Equations ◽  
Finite Energy ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Boundary Values ◽  
Vector Solution ◽  
Navier Stokes Equations ◽  
Body Forces ◽  
Derivatives Of

AbstractFor a vector solution u(x, t) with finite energy of the Navier Stokes equations with body forces and boundary values on a region Ω ⊆ R3 for t > 0, conditions are established on the L6/5(Ω) and L2(Ω) norms of derivatives of the data that ensure the estimates and max , up to any given integer value of the weighted order 2r+s, where r or s = s1 + s2 + s3 > 0 and 0 < T < ∞.

scholarly journals 3D Hydrodynamic Modelling Enhances the Design of Tendaho Dam Spillway, Ethiopia

Water ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 82
Author(s):  
Getnet Kebede Demeke ◽  
Dereje Hailu Asfaw ◽  
Yilma Seleshi Shiferaw
Keyword(s):  
Stokes Equations ◽  
Flow Behavior ◽  
Fluid Flows ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Hydraulic Structures ◽  
Hydrodynamic Modelling ◽  
Navier Stokes Equations ◽  
Pressure Distributions ◽  
Computational Fluid Dynamics Cfd

Hydraulic structures are often complex and in many cases their designs require attention so that the flow behavior around hydraulic structures and their influence on the environment can be predicted accurately. Currently, more efficient computational fluid dynamics (CFD) codes can solve the Navier–Stokes equations in three-dimensions and free surface computation in a significantly improved manner. CFD has evolved into a powerful tool in simulating fluid flows. In addition, CFD with its advantages of lower cost and greater flexibility can reasonably predict the mean characteristics of flows such as velocity distributions, pressure distributions, and water surface profiles of complex problems in hydraulic engineering. In Ethiopia, Tendaho Dam Spillway was constructed recently, and one flood passed over the spillway. Although the flood was below the designed capacity, there was an overflow due to superelevation at the bend. Therefore, design of complex hydraulic structures using the state-of- art of 3D hydrodynamic modelling enhances the safety of the structures. 3D hydrodynamic modelling was used to verify the safety of the spillway using designed data and the result showed that the constructed hydraulic section is not safe unless it is modified.

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