scholarly journals Three-Dimensional Biorthogonal Divergence-Free and Curl-Free Wavelets with Free-Slip Boundary

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Yingchun Jiang ◽  
Qingqing Sun
Keyword(s):  
Boundary Conditions ◽  
Simple Structure ◽  
Three Dimensional ◽  
Unit Cube ◽  
Slip Boundary Conditions ◽  
Slip Boundary ◽  
Divergence Free

This paper deals with the construction of divergence-free and curl-free wavelets on the unit cube, which satisfies the free-slip boundary conditions. First, interval wavelets adapted to our construction are introduced. Then, we provide the biorthogonal divergence-free and curl-free wavelets with free-slip boundary and simple structure, based on the characterization of corresponding spaces. Moreover, the bases are also stable.

ASYMPTOTIC BEHAVIOR OF A VISCOUS FLUID WITH SLIP BOUNDARY CONDITIONS ON A SLIGHTLY ROUGH WALL

2010 ◽  
Vol 20 (01) ◽  
pp. 121-156 ◽  
Author(s):  
J. CASADO-DÍAZ ◽  
M. LUNA-LAYNEZ ◽  
F. J. SUÁREZ-GRAU
Keyword(s):  
Asymptotic Behavior ◽  
Viscous Fluid ◽  
Boundary Conditions ◽  
Critical Size ◽  
Three Dimensional ◽  
Plane Boundary ◽  
Limit Equation ◽  
Slip Boundary Conditions ◽  
Slip Boundary ◽  
Oscillating Boundary

For an oscillating boundary of period and amplitude ε, it is known that the asymptotic behavior when ε tends to zero of a three-dimensional viscous fluid satisfying slip boundary conditions is the same as if we assume no-slip (adherence) boundary conditions. Here we consider the case where the period is still ε but the amplitude is δε with δε/ε converging to zero. We show that if [Formula: see text] tends to infinity, the equivalence between the slip and no-slip conditions still holds. If the limit of [Formula: see text] belongs to (0, +∞) (critical size), then we still have the slip boundary conditions in the limit but with a bigger friction coefficient. In the case where [Formula: see text] tends to zero the boundary behaves as a plane boundary. Besides the limit equation, we also obtain an approximation (corrector result) of the pressure and the velocity in the strong topology of L2 and H1 respectively.

scholarly journals Bifurcations in a quasi-two-dimensional Kolmogorov-like flow

2017 ◽  
Vol 828 ◽  
pp. 837-866 ◽  
Author(s):  
Jeffrey Tithof ◽  
Balachandra Suri ◽  
Ravi Kumar Pallantla ◽  
Roman O. Grigoriev ◽  
Michael F. Schatz
Keyword(s):  
Boundary Conditions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Quantitative Agreement ◽  
Navier Stokes ◽  
Dielectric Fluid ◽  
Two Dimensional ◽  
Slip Boundary Conditions ◽  
Slip Boundary ◽  
2D Model

We present a combined experimental and theoretical study of the primary and secondary instabilities in a Kolmogorov-like flow. The experiment uses electromagnetic forcing with an approximately sinusoidal spatial profile to drive a quasi-two-dimensional (Q2D) shear flow in a thin layer of electrolyte suspended on a thin lubricating layer of a dielectric fluid. Theoretical analysis is based on a two-dimensional (2D) model (Suri et al., Phys. Fluids, vol. 26 (5), 2014, 053601), derived from first principles by depth-averaging the full three-dimensional Navier–Stokes equations. As the strength of the forcing is increased, the Q2D flow in the experiment undergoes a series of bifurcations, which is compared with results from direct numerical simulations of the 2D model. The effects of confinement and the forcing profile are studied by performing simulations that assume spatial periodicity and strictly sinusoidal forcing, as well as simulations with realistic no-slip boundary conditions and an experimentally validated forcing profile. We find that only the simulation subject to physical no-slip boundary conditions and a realistic forcing profile provides close, quantitative agreement with the experiment. Our analysis offers additional validation of the 2D model as well as a demonstration of the importance of properly modelling the forcing and boundary conditions.

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