Nonlinear stability for active suspensions

A novel pseudo-rigid body approach to the non-linear dynamics of soft micro-particles in dilute viscous flow

A temporal discretization scheme to compute the motion of light particles in viscous flows by an immersed boundary method

Experiments were made on the orientation of cylindrical particles in viscous flow. Particles were immersed in a moving fluid between two concentric cylinders. Tests showed two different final orientations: (1) a position perpendicular to the plane of the undisturbed flow, and (2) a position parallel to the velocity direction. The relation between the final orientation and the length-diameter ratio of the particle was found for different speeds. Comparison is made between the experimental results and theoretical predictions.

In this study, we consider the nonlinear stability of a strong viscous contact discontinuity in a free boundary problem for the one-dimensional, full compressible Navier–Stokes equations in half space [0,∞). For the local stability of contact discontinuities, the local stability of a weak viscous contact discontinuity is well established, but for the global stability of an impermeable gas, fewer strong nonlinear wave stability results have been obtained, excluding zero dissipation or a γ→1 gas. Thus, our main aim is to determine the corresponding nonlinear stability result using the elementary energy method. For a certain class of large perturbation, we show that the global stability result can be obtained for a strong viscous contact discontinuity in Navier–Stokes equations.

The rotational diffusivity and the translational diffusivity perpendicular to the rod axis of rigid rodlike (RRL) molecules decreases rapidly with increasing molecular length, during polymerization in semidilute solutions. This can result in slowing of step-growth polymerizations of RRL molecules with reactive groups at the rod ends and a near-collinearity requirement for reaction. Here a theoretical analysis of the rate of RRL polymerization in semidilute solutions, based on Smoluchowski’s approach, and incorporating the rotational and anisotropic translational diffusion of the molecules is presented. The work is an extension of our analysis of polymerization in dilute solutions in which the translational diffusion was assumed to be isotropic [J. Chem. Phys. 96, 7125 (1992)]. The effective second order rate constant for the system is obtained for different parameter values using a numerical finite element method. With reduction in rotational diffusivity, for a fixed translational diffusivity, the effective reaction rate constant is found to decrease to a limiting value determined by only the translational flux of the correctly oriented molecules. Similarly, for a given rotational diffusivity, with reduction in translational diffusivity perpendicular to the rod axis, the reaction rate constant is found to decrease to a limiting value determined by the flux only due to translational diffusion parallel to the rod axis, aided by rotational diffusion. An asymptotic analysis for this case is presented. For low rotational diffusivities, reduction in the translational diffusion perpendicular to the rod axis results in a significant decrease in the effective rate constant, even for reactions with relatively slow intrinsic kinetics. A qualitative comparison of the theoretical predictions with experimental results is presented.

Understanding and controlling the transport of microscopic particles in viscous flows stems from the fundamental question of fluid-structure interactions but has also important implications for separation processes or bacterial contamination. Using recent microfabrication techniques, we produce a variety of microscopic particles and control precisely their shape and material properties. Investigating the transport dynamics of these particles in representative microfluidic flows we demonstrate how shape, mechanical properties or even activity govern particle trajectories. Combining our experimental findings with numerical and theoretical modeling performed by our collaborators we elucidate in detail the role of particle symmetry, chirality or deformability.

Elastic confinements play an important role in many soft matter systems and affect the transport properties of suspended particles in viscous flow. On the basis of low-Reynolds-number hydrodynamics, we present an analytical theory of the axisymmetric flow induced by a point-force singularity (Stokeslet) directed along the symmetry axis of a finite-sized circular elastic membrane endowed with resistance toward shear and bending. The solution for the viscous incompressible flow surrounding the membrane is formulated as a mixed boundary value problem, which is then reduced into a system of dual integral equations on the inner and outer sides of the domain boundary. We show that the solution of the elastohydrodynamic problem can conveniently be expressed in terms of a set of inhomogeneous Fredholm integral equations of the second kind with logarithmic kernel. Basing on the hydrodynamic flow field, we obtain semi-analytical expressions of the hydrodynamic mobility function for the translational motion perpendicular to a circular membrane. The results are valid to leading-order in the ratio of particle radius to the distance separating the particle from the membrane. In the quasi-steady limit, we find that the particle mobility near a finite-sized membrane is always larger than that predicted near a no-slip disk of the same size. We further show that the bending-related contribution to the hydrodynamic mobility increases monotonically upon decreasing the membrane size, whereas the shear-related contribution displays a minimum value when the particle-membrane distance is equal to the membrane radius. Accordingly, the system behavior may be shear or bending dominated, depending on the geometric and elastic properties of the system. Our results may find applications in the field of nanoparticle-based sensing and drug delivery systems near elastic cell membranes.

The spectral distribution of the depolarized component of light scattered from a dilute solution of molecules in dynamic chemical equilibrium between two states differing in optical anisotropy and dynamics is calculated. It is assumed that the molecules undergo isotropic translational and anisotropic rotational diffusion. It is furthermore assumed that the duration of a molecular transformation is much less than the characteristic times for rotational and translational diffusion. It is shown that in the most general case considered, the spectrum consists of 20 superposed lines with widths dependent on the translational diffusion coefficients and components of the rotational diffusion tensors of the two species and the backward and forward chemical rate constants. The relative strengths of the lines depend on the optical anisotropies and rotational diffusion tensor components of the two species as well as on their equilibrium concentrations. When the reaction is so fast that rotational and translational diffusion contributions to the linewidth may be ignored altogether, the spectrum reduces to a single Lorentzian line with half-width independent of scattering angle and proportional to the sum of the backward and forward rate constants for the chemical transformation.

Model systems of cholestane and 5-doxylstearic acid analogue spin probes in lipid bilayer dispersions of dipalmitoylphosphatidylcholine and cholesterol (9:1 w/w) are used to analyze saturation transfer electron paramagnetic resonance spectral behavior for slow rotational diffusion in an anisotropic medium. Measurements are made at both 9 and 35 GHz to provide enhanced spectral resolution for different types of motion. Parameter correlation plots of spectral parameters from different regions of the saturation transfer spectra appear to be potentially useful in characterizing different types of motion. Anisotropic rotational diffusion about a symmetry axis coincident with the nitroxide y principal axis is clearly distinguishable from isotropic rotational diffusion and may be distinguishable from rotational diffusion about the nitroxide z principal axis. Approximate anisotropic rotational diffusion about a symmetry axis coincident with the nitroxide z principal axis is distinguishable from isotropic rotational diffusion under some, but not all, conditions.

This paper is concerned with nonlinear symmetric stability problems. For the moist, adiabatic (saturated) system, the authors utilize the ECM (energy–Casimir method) to establish nonlinear stability criteria, which extends the previous work from the dry atmosphere to the moist case and demonstrates the complexity related to the moist symmetric instability problem. For the nonhydrostatic, Boussinesq equations on an f plane with the northward component of the earth rotation f = 2Ω cosϕ, which has been utilized to show the importance of f term in the mesoscale linear symmetric instability problem, both ECM and the ELM (energy–Lagrange method) are employed to study the “zonal” and“meridional” nonlinear symmetric stability problems. In both cases, the nonlinear stability of the basic states are obtained if the potential vorticity and the vertical component of absolute vorticity of the basic state are positive (for f > 0). In the zonal case, the potential vorticity depends upon f explicitly, and this s...

We measure all nonzero elements of the three-dimensional diffusion tensor D for clusters of colloidal spheres to a precision of 1% or better using digital holographic microscopy. We study both dimers and triangular trimers of spheres, for which no analytical calculations of the diffusion tensor exist. We observe anisotropic rotational and translational diffusion arising from the asymmetries of the clusters. In the case of the three-particle triangular cluster, we also detect a small but statistically significant difference in the rotational diffusion about the two in-plane axes. We attribute this difference to weak breaking of threefold rotational symmetry due to a small amount of particle polydispersity. Our experimental measurements agree well with numerical calculations and show how diffusion constants can be measured under conditions relevant to colloidal self-assembly, where theoretical and even numerical prediction is difficult.

A nonlinear (energy) stability analysis is performed for a magnetized ferrofluid layer heated from below, in the stress-free boundary case. By introducing a suitable generalized energy functional, a rigorous nonlinear stability result is derived for a thermoconvective magnetized ferrofluid. The mathematical emphasis is on how to control the nonlinear terms caused by the magnetic body and inertia forces. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of the linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in the case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter,M3, on subcritical instability region has also been analysed. It is shown that with the increase of magnetic parameter,M3, the subcritical instability region between the two theories decreases quickly. We also demonstrate coupling between the buoyancy and the magnetic forces in the nonlinear energy stability analysis.

A nonlinear (energy) stability analysis is performed for a rotating magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. By introducing a generalized energy functional, a rigorous nonlinear stability result for a thermoconvective rotating magnetized ferrofluid is derived. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M 3, medium permeability, D a , and rotation, $$T_{A_1}$$ , on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter, M 3, and Darcy number, D a , the subcritical instability region between the two theories decreases quickly while with the increase of Taylor number, $$T_{A_1} $$ , the subcritical region expands. We also demonstrate coupling between the buoyancy and magnetic forces in the presence of rotation in nonlinear energy stability analysis as well as in linear instability analysis.

A nonlinear stability analysis for rotating magnetized ferrofluid heated from below

Dynamic light scattering of thin disks: Coupling of diffusive motions