Erratum: Role of inertia for the rotation of a nearly spherical particle in a general linear flow [Phys. Rev. E 91, 053023 (2015)].

This corrects the article DOI: 10.1103/PhysRevE.91.053023.

Effect of fluid and particle inertia on the rotation of an oblate spheroidal particle suspended in linear shear flow.

This work describes the inertial effects on the rotational behavior of an oblate spheroidal particle confined between two parallel opposite moving walls, which generate a linear shear flow. Numerical results are obtained using the lattice Boltzmann method with an external boundary force. The rotation of the particle depends on the particle Reynolds number, Re(p)=Gd(2)ν(-1) (G is the shear rate, d is the particle diameter, ν is the kinematic viscosity), and the Stokes number, St=αRe(p) (α is the solid-to-fluid density ratio), which are dimensionless quantities connected to fluid and particle inertia, respectively. The results show that two inertial effects give rise to different stable rotational states. For a neutrally buoyant particle (St=Re(p)) at low Re(p), particle inertia was found to dominate, eventually leading to a rotation about the particle's symmetry axis. The symmetry axis is in this case parallel to the vorticity direction; a rotational state called log-rolling. At high Re(p), fluid inertia will dominate and the particle will remain in a steady state, where the particle symmetry axis is perpendicular to the vorticity direction and has a constant angle ϕ(c) to the flow direction. The sequence of transitions between these dynamical states were found to be dependent on density ratio α, particle aspect ratio r(p), and domain size. More specifically, the present study reveals that an inclined rolling state (particle rotates around its symmetry axis, which is not aligned in the vorticity direction) appears through a pitchfork bifurcation due to the influence of periodic boundary conditions when simulated in a small domain. Furthermore, it is also found that a tumbling motion, where the particle symmetry axis rotates in the flow-gradient plane, can be a stable motion for particles with high r(p) and low α.

Role of inertia for the rotation of a nearly spherical particle in a general linear flow.

We analyze the angular dynamics of a neutrally buoyant, nearly spherical particle immersed in a steady general linear flow. The hydrodynamic torque acting on the particle is obtained by means of a reciprocal theorem, a regular perturbation theory exploiting the small eccentricity of the nearly spherical particle, and by assuming that inertial effects are small but finite.

Effect of weak fluid inertia upon Jeffery orbits.

We consider the rotation of small neutrally buoyant axisymmetric particles in a viscous steady shear flow. When inertial effects are negligible the problem exhibits infinitely many periodic solutions, the "Jeffery orbits." We compute how inertial effects lift their degeneracy by perturbatively solving the coupled particle-flow equations. We obtain an equation of motion valid at small shear Reynolds numbers, for spheroidal particles with arbitrary aspect ratios. We analyze how the linear stability of the "log-rolling" orbit depends on particle shape and find it to be unstable for prolate spheroids. This resolves a puzzle in the interpretation of direct numerical simulations of the problem. In general, both unsteady and nonlinear terms in the Navier-Stokes equations are important.

Numerical analysis of the angular motion of a neutrally buoyant spheroid in shear flow at small Reynolds numbers.

We numerically analyze the rotation of a neutrally buoyant spheroid in a shear flow at small shear Reynolds number. Using direct numerical stability analysis of the coupled nonlinear particle-flow problem, we compute the linear stability of the log-rolling orbit at small shear Reynolds number Re(a). As Re(a)→0 and as the box size of the system tends to infinity, we find good agreement between the numerical results and earlier analytical predictions valid to linear order in Re(a) for the case of an unbounded shear. The numerical stability analysis indicates that there are substantial finite-size corrections to the analytical results obtained for the unbounded system. We also compare the analytical results to results of lattice Boltzmann simulations to analyze the stability of the tumbling orbit at shear Reynolds numbers of order unity. Theory for an unbounded system at infinitesimal shear Reynolds number predicts a bifurcation of the tumbling orbit at aspect ratio λ(c)≈0.137 below which tumbling is stable (as well as log rolling). The simulation results show a bifurcation line in the λ-Re(a) plane that reaches λ≈0.1275 at the smallest shear Reynolds number (Re(a)=1) at which we could simulate with the lattice Boltzmann code, in qualitative agreement with the analytical results.

Passive appendages generate drift through symmetry breaking.

Plants and animals use plumes, barbs, tails, feathers, hairs and fins to aid locomotion. Many of these appendages are not actively controlled, instead they have to interact passively with the surrounding fluid to generate motion. Here, we use theory, experiments and numerical simulations to show that an object with a protrusion in a separated flow drifts sideways by exploiting a symmetry-breaking instability similar to the instability of an inverted pendulum. Our model explains why the straight position of an appendage in a fluid flow is unstable and how it stabilizes either to the left or right of the incoming flow direction. It is plausible that organisms with appendages in a separated flow use this newly discovered mechanism for locomotion; examples include the drift of plumed seeds without wind and the passive reorientation of motile animals.

Conjoint psychotherapy of marital pairs.

Many traditional concepts are being challenged in contemporary psychiatric practice, including the classical "one-to-one" relationship of individual psychotherapy. Where the patient's presenting difficulties include significant inability to function or feel happy in the marital role, the technique of conjoint psychotherapy (having both partners treated simultaneously by the same doctor) may be indicated. Conjoint therapy is envisaged as a continuum, embracing a considerable range of situations where it is sound practice to see husband and wife together. The treatment plan has three stages: complaint, clarification, and compromise, each of which presents specific features and pitfalls. Emphasis is placed on dealing with individual psychopathology of each partner, both per se and in relation to the marital situation. Results to date suggest that conjoint treatment represents a promising therapeutic modality. Even in cases where individual psychopathology cannot fully be resolved, certain plateaus of satisfaction may be attained as communication improves.

Further experience in conjoint psychotherapy of marital pairs.

Conjoint therapy of marital partners is a technique that lends itself to the counseling efforts of health professionals. Its growing use has, however, brought with it seemingly inevitable pitfalls, such as inadequate assessment of individual needs and psychopathology, overzealous application and disregard of certain contraindications, management problems and goal definition that may be unclear to patients or more related to the therapist's personality than to an objective view of the marriage dynamics. Despite the difficulties and pitfalls of this relatively new field, conjoint therapy can be the treatment of choice when the primary difficulties are related to the inability to cope in the marital situation, even though functioning in other social roles is adequate.