scholarly journals Analytical solutions for two-dimensional Stokes flow singularities in a no-slip wedge of arbitrary angle

2017 ◽  
Vol 473 (2202) ◽  
pp. 20170134 ◽  
Author(s):  
Darren G. Crowdy ◽  
Samuel J. Brzezicki
Keyword(s):  
Stokes Flow ◽  
Biharmonic Equation ◽  
Elastic Solid ◽  
Two Dimensional ◽  
Slip Boundary Conditions ◽  
Arbitrary Angle ◽  
Full Account ◽  
Slip Boundary ◽  
Singularity Type ◽  
Strain Stress

An analytical method to find the flow generated by the basic singularities of Stokes flow in a wedge of arbitrary angle is presented. Specifically, we solve a biharmonic equation for the stream function of the flow generated by a point stresslet singularity and satisfying no-slip boundary conditions on the two walls of the wedge. The method, which is readily adapted to any other singularity type, takes full account of any transcendental singularities arising at the corner of the wedge. The approach is also applicable to problems of plane strain/stress of an elastic solid where the biharmonic equation also governs the Airy stress function.

Secondary fluid flows driven electromagnetically in a two-dimensional extended duct

2005 ◽  
Vol 461 (2058) ◽  
pp. 1659-1683 ◽  
Author(s):  
Zhi-Min Chen ◽  
W.G Price
Keyword(s):  
Steady State ◽  
Laboratory Experiments ◽  
Fluid Motion ◽  
Fluid Flows ◽  
Two Dimensional ◽  
Slip Boundary Conditions ◽  
Slip Boundary ◽  
Wave Numbers ◽  
Motion Problem ◽  
Secondary Fluid

This study focuses on two-dimensional fluid flows in a straight duct with free-slip boundary conditions applied on the channel walls y =0 and y =2 πN with N >1. In this extended wall-bounded fluid motion problem, secondary fluid flow patterns resulting from steady-state and Hopf bifurcations are examined and shown to be dependent on the choice of longitudinal wave numbers. Some secondary steady-state flows appear at specific wave numbers, whereas at other wave numbers, both secondary steady-state and self-oscillation flows coexist. These results, derived through analytical arguments and truncation series approximation, are confirmed by simple numerical experiments supporting the findings observed from laboratory experiments.

Transition and turbulence in horizontal convection: linear stability analysis

2017 ◽  
Vol 821 ◽  
pp. 31-58 ◽  
Author(s):  
Pierre-Yves Passaggia ◽  
Alberto Scotti ◽  
Brian White
Keyword(s):  
Stability Analysis ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Critical Rayleigh Number ◽  
Base Flow ◽  
Two Dimensional ◽  
Slip Boundary Conditions ◽  
Slip Boundary ◽  
Horizontal Convection

The linear instability mechanisms of horizontal convection in a rectangular cavity forced by a horizontal buoyancy gradient along its surface are investigated using local and global stability analyses for a Prandtl number equal to unity. The results show that the stability of the base flow, a steady circulation characterized by a narrow descending plume and a broad upwelling region, depends on the Rayleigh number, $Ra$. For free-slip boundary conditions at a critical value of $Ra\approx 2\times 10^{7}$, the steady base flow becomes unstable to three-dimensional perturbations, characterized by counter-rotating vortices originating within the plume region. A Wentzel–Kramers–Brillouin (WKB) method applied along closed streamlines demonstrates that this instability is of a Rayleigh–Taylor type and can be used to accurately reconstruct the global instability mode. In the case of no-slip boundary conditions, the base flow also becomes unstable to a self-sustained two-dimensional instability whose critical Rayleigh number is $Ra\approx 1.7\times 10^{8}$. Beyond this critical $Ra$, two-dimensional equilibrium stationary states of the Navier–Stokes equations are computed using the selective frequency damping method. The two-dimensional onset of instability is shown to be characterized by a family of modes also originating within the plume. A local spatio-temporal stability analysis shows that the flow becomes absolutely unstable at the origin of the plume. Taken together, these results illustrate the mechanisms behind the onset of turbulence that has been observed in horizontal convection.

scholarly journals Correlation between No-Slip and Slip Boundary Conditions Associated with A Two-Dimensional Navier-Stokes Flows in a Plane Diffuser

2021 ◽  
Vol 65 (1) ◽  
pp. 1-23
Author(s):  
Ranis Ibragimov ◽  
◽  
Vesselin Vatchev ◽  
Keyword(s):  
Boundary Conditions ◽  
Small Perturbation ◽  
Taylor Series Expansion ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Slip Boundary Conditions ◽  
Viscous Effects ◽  
Stokes Flows ◽  
Slip Boundary ◽  
Two Parameters

We examine the viscous effects of slip boundary conditions for the model describing two-dimensional Navier-Stokes flows in a plane diffuser. It is shown that the velocity profile is related to a half period shifted Weierstrass function with two parameters. This allows to approximate the explicit solution by a Taylor series expansion with two new micro- parameters, that can be measured in physical experiments. It is shown that the assumption for no-slip boundary conditions is stable in the sense that a small perturbation of the boundary values result in a small perturbation in the solutions.

Reduced-Order Modeling of Low Mach Number Unsteady Microchannel Flows

2014 ◽  
Vol 136 (5) ◽  
Author(s):  
Leila Issa ◽  
Issam Lakkis
Keyword(s):  
Mach Number ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Solid Boundary ◽  
Two Dimensional ◽  
Slip Boundary Conditions ◽  
Reduced Order ◽  
Low Mach Number ◽  
First Order ◽  
Slip Boundary

We present reduced-order models of unsteady low-Mach-number ideal gas flows in two-dimensional rectangular microchannels subject to first-order slip-boundary conditions. The pressure and density are related by a polytropic process, allowing for isothermal or isentropic flow assumptions. The Navier–Stokes equations are simplified using low-Mach-number expansions of the pressure and velocity fields. Up to first order, this approximation results in a system that is subject to no-slip condition at the solid boundary. The second-order system satisfies the slip-boundary conditions. The resulting equations and the subsequent pressure-flow-rate relationships enable modeling the flow using analog circuit components. The accuracy of the proposed models is investigated for steady and unsteady flows in a two-dimensional channel for different values of Mach and Knudsen numbers.

scholarly journals Lattice-Gas Simulations of Ternary Amphiphilic Fluid Flow in Porous Media

1998 ◽  
Vol 09 (08) ◽  
pp. 1479-1490 ◽  
Author(s):  
P. V. Coveney ◽  
J.-B. Maillet ◽  
J. L. Wilson ◽  
P. W. Fowler ◽  
O. Al-Mushadani ◽  
...  
Keyword(s):  
Porous Media ◽  
Boundary Conditions ◽  
Single Phase ◽  
Lattice Gas ◽  
Flow In Porous Media ◽  
Two Dimensional ◽  
Slip Boundary Conditions ◽  
Dimensional Lattice ◽  
Slip Boundary ◽  
Saturated Rock

We develop our existing two-dimensional lattice-gas model to simulate the flow of single phase, binary immiscible and ternary amphiphilic fluids. This involves the inclusion of fixed obstacles on the lattice, together with the inclusion of "no-slip" boundary conditions. Here we report on preliminary applications of this model to the flow of such fluids within model porous media. We also construct fluid invasion boundary conditions, and the effects of invading aqueous solutions of surfactant on oil-saturated rock during imbibition and drainage are described.

NAVIER–STOKES SYSTEM ON THE FLAT CYLINDER AND UNIT SQUARE WITH SLIP BOUNDARY CONDITIONS

2010 ◽  
Vol 12 (02) ◽  
pp. 325-349 ◽  
Author(s):  
EFIM DINABURG ◽  
DONG LI ◽  
YAKOV G. SINAI
Keyword(s):  
Boundary Conditions ◽  
Initial Velocity ◽  
Stokes System ◽  
Navier Stokes ◽  
Decay Estimates ◽  
Two Dimensional ◽  
Slip Boundary Conditions ◽  
Unit Square ◽  
Slip Boundary

We study the decay of Fourier modes of solutions to the two-dimensional Navier–Stokes System on a flat cylinder and the unit square with slip boundary conditions. Under some suitable assumptions on the initial velocity, we obtain quantitative decay estimates of the Fourier modes.

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