scholarly journals Exact solutions for hydrodynamic interactions of two squirming spheres

2017 ◽  
Vol 813 ◽  
pp. 618-646 ◽  
Author(s):  
Dario Papavassiliou ◽  
Gareth P. Alexander
Keyword(s):  
Exact Solutions ◽  
Stokes Equations ◽  
Slip Surface ◽  
Rotational Velocity ◽  
Near Field ◽  
Three Dimensions ◽  
Hydrodynamic Interactions ◽  
Reciprocal Theorem ◽  
Approximate Methods ◽  
Stokes Drag

We provide exact solutions of the Stokes equations for a squirming sphere close to a no-slip surface, both planar and spherical, and for the interactions between two squirmers, in three dimensions. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or with each other, to be determined for arbitrary separation and, in particular, in the close proximity regime where approximate methods based on point-singularity descriptions cease to be valid. We give a detailed description of the circular motion of an arbitrary squirmer moving parallel to a no-slip spherical boundary or flat free surface at close separation, finding that the circling generically has opposite sense at free surfaces and at solid boundaries. While the asymptotic interaction is symmetric under head–tail reversal of the swimmer, in the near field, microscopic structure can result in significant asymmetry. We also find the translational velocity towards the surface for a simple model with only the lowest two squirming modes. By comparing these to asymptotic approximations of the interaction we find that the transition from near- to far-field behaviour occurs at a separation of approximately two swimmer diameters. These solutions are for the rotational velocity about the wall normal, or common diameter of two spheres, and the translational speed along that same direction, and are obtained using the Lorentz reciprocal theorem for Stokes flows in conjunction with known solutions for the conjugate Stokes drag problems, the derivations of which are demonstrated here for completeness. The analogous motions in the perpendicular directions, i.e. parallel to the wall, currently cannot be calculated exactly since the relevant Stokes drag solutions needed for the reciprocal theorem are not available.

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.

ANALYSIS OF A HETEROGENEOUS DOMAIN DECOMPOSITION FOR COMPRESSIBLE VISCOUS FLOW

2001 ◽  
Vol 11 (04) ◽  
pp. 565-599 ◽  
Author(s):  
CRISTIAN A. COCLICI ◽  
WOLFGANG L. WENDLAND
Keyword(s):  
Domain Decomposition ◽  
Stokes Equations ◽  
Domain Decomposition Method ◽  
Near Field ◽  
Outer Region ◽  
Navier Stokes ◽  
Far Field ◽  
Computational Domain ◽  
Linearized Euler Equations ◽  
Naca0012 Airfoil

We analyze a nonoverlapping domain decomposition method for the treatment of two-dimensional compressible viscous flows around airfoils. Since at some distance to the given profile the inertial forces are strongly dominant, there the viscosity effects are neglected and the flow is assumed to be inviscid. Accordingly, we consider a decomposition of the original flow field into a bounded computational domain (near field) and a complementary outer region (far field). The compressible Navier–Stokes equations are used close to the profile and are coupled with the linearized Euler equations in the far field by appropriate transmission conditions, according to the physical properties and the mathematical type of the corresponding partial differential equations. We present some results of flow around the NACA0012 airfoil and develop an a posteriori analysis of the approximate solution, showing that conservation of mass, momentum and energy are asymptotically attained with the linear model in the far field.

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.

Strain, tilt, and rotation associated with strong ground motion in the vicinity of earthquake faults

1982 ◽  
Vol 72 (5) ◽  
pp. 1717-1738 ◽  
Author(s):  
Michel Bouchon ◽  
Keiiti Aki
Keyword(s):  
Ground Motion ◽  
Rotational Velocity ◽  
Near Field ◽  
Strike Slip ◽  
Phase Velocities ◽  
Crustal Structures ◽  
Time Histories ◽  
San Fernando ◽  
Earthquake Faults ◽  
Strong Ground

abstract In the absence of near-field records of differential ground motion induced by earthquakes, we simulate the time histories of strain, tilt, and rotation in the vicinity of earthquake faults embedded in layered media. We consider the case of both strike-slip and dip-slip fault models and study the effect of different crustal structures. The maximum rotational motion produced by a buried 30-km-long strike-slip fault with slip of 1 m is of the order of 3 × 10−4 rad while the corresponding rotational velocity is about 1.5 × 10−3 rad/sec. A simulation of the San Fernando earthquake yields maximum longitudinal strain and tilt a few kilometers from the fault of the order of 8 × 10−4 and 7 × 10−4 rad. These values being small compared to the amplitude of ground displacement, the results suggest that most of the damage occurring in earthquakes is caused by translation motions. We also show that strain and tilt are closely related to ground velocity and that the phase velocities associated with strong ground motions are controlled by the rupture velocity and the basement rock shearwave velocity.

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]

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