scholarly journals Weak Solutions and Optimal Control of Hemivariational Evolutionary Navier-Stokes Equations under Rauch Condition

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Hicham Mahdioui ◽  
Sultana Ben Aadi ◽  
Khalid Akhlil
Keyword(s):  
Stokes Equations ◽  
Dynamic Pressure ◽  
Normal Component ◽  
Galerkin Approximation ◽  
Stability Result ◽  
Clarke Subdifferential ◽  
Navier Stokes ◽  
Directional Growth ◽  
Navier Stokes Equations ◽  
External Forces

In this paper, we consider the evolutionary Navier-Stokes equations subject to the nonslip boundary condition together with a Clarke subdifferential relation between the dynamic pressure and the normal component of the velocity. Under the Rauch condition, we use the Galerkin approximation method and a weak precompactness criterion to ensure the convergence to a desired solution. Moreover, a control problem associated with such system of equations is studied with the help of a stability result with respect to the external forces. At the end of this paper, a more general condition due to Z. Naniewicz, namely the directional growth condition, is considered and all the results are reexamined.

A simple proof of existence of weak and statistical solutions of Navier–Stokes equations

1992 ◽  
Vol 436 (1896) ◽  
pp. 1-11 ◽  
Keyword(s):  
Weak Solution ◽  
Initial Data ◽  
Natural Number ◽  
Simple Proof ◽  
Stokes Equations ◽  
Galerkin Approximation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Existence Proofs ◽  
Statistical Solutions

The Galerkin approximation to the Navier–Stokes equations in dimension N , where N is an infinite non-standard natural number, is shown to have standard part that is a weak solution. This construction is uniform with respect to non-standard representation of the initial data, and provides easy existence proofs for statistical solutions.

scholarly journals Flow Pressure Behavior Downstream of Ski Jumps

Fluids ◽  
2020 ◽  
Vol 5 (4) ◽  
pp. 168 ◽  
Author(s):  
Agostino Lauria ◽  
Giancarlo Alfonsi ◽  
Ali Tafarojnoruz
Keyword(s):  
High Speed ◽  
Stokes Equations ◽  
Dynamic Pressure ◽  
Pressure Head ◽  
Navier Stokes ◽  
Maximum Pressure ◽  
Navier Stokes Equations ◽  
Free Jet ◽  
Jet Characteristics ◽  
The Impact

Ski jump spillways are frequently implemented to dissipate energy from high-speed flows. The general feature of this structure is to transform the spillway flow into a free jet up to a location where the impact of the jet creates a plunge pool, representing an area for potential erosion phenomena. In the present investigation, several tests with different ski jump bucket angles are executed numerically by means of the OpenFOAM® digital library, taking advantage of the Reynolds-averaged Navier–Stokes equations (RANS) approach. The results are compared to those obtained experimentally by other authors as related to the jet length and shape, obtaining physical insights into the jet characteristics. Particular attention is given to the maximum pressure head at the tailwater. Simple equations are proposed to predict the maximum dynamic pressure head acting on the tailwater, as dependent upon the Froude number, and the maximum pressure head on the bucket. Results of this study provide useful suggestions for the design of ski jump spillways in dam construction.

Compressible Navier–Stokes Computations for Slider Air-Bearings

1991 ◽  
Vol 113 (1) ◽  
pp. 73-79 ◽  
Author(s):  
W. D. Henshaw ◽  
L. G. Reyna ◽  
J. A. Zufiria
Keyword(s):  
Stokes Equations ◽  
Dynamic Pressure ◽  
Temperature Drop ◽  
Navier Stokes ◽  
Back Surface ◽  
Radius Of Curvature ◽  
Two Dimensional ◽  
Air Bearings ◽  
Navier Stokes Equations ◽  
Storage Devices

Two-dimensional computations of the compressible Navier–Stokes equations are used to study the flow at the inlet and outlet of slider air-bearings operating under conditions typical of computer disk-storage devices. Inlet results show a pressure gain of several times the dynamic pressure of the incoming flow, with the actual value depending on the local Reynolds number. Although the outlet of the slider is characterized by a sudden pressure drop, the numerical results show that for sliders operating at small Mach numbers there is only a small temperature drop. The computations show the position of the separating streamline on the back surface of the slider, which is found to depend on the radius of curvature of the outlet corner. Results are also shown from calculations of the global flow underneath and around a two-dimensional slider.

scholarly journals Asymptotically stable attracting sets in the Navier-Stokes equations

1986 ◽  
Vol 34 (1) ◽  
pp. 37-52 ◽  
Author(s):  
P. E. Kloeden
Keyword(s):  
Stokes Equations ◽  
Galerkin Approximation ◽  
Navier Stokes ◽  
Asymptotically Stable ◽  
Compact Sets ◽  
Navier Stokes Equations ◽  
Attracting Set ◽  
Galerkin Approximations ◽  
Steady Solutions ◽  
The Stability

The planar Navier-Stokes equations with periodic boundary conditions are shown to have a nearby asymptotically stable attracting set whenever a Galerkin approximation involving the eigenfunctions of the Stokes operator has such an attracting set, provided the approximation has sufficiently many terms and its attracting set is sufficiently strongly stable. Lyapunov functions are used to characterize the stability of these attracting sets, which are compact sets of arbitrary geometric shape. This generalizes earlier results of Constantin, Foias and Temam and of the author for asymptotically stable steady solutions in the Navier-Stokes equations and such Galerkin approximations.

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