•Preliminary examination of an approach to relaxation and retardation in viscoelasticity.•Relaxation and retardation can be asymptotically equivalent for small Deborah number.•Causal relaxation is ...shown to correspond to ill-posed retardation through this equivalence.
We present a preliminary examination of a new approach to a long-standing problem in non-Newtonian fluid mechanics. First, we summarize how a general implicit functional relation between stress and rate of strain of a continuum with memory is reduced to the well-known linear differential constitutive relations that account for “relaxation” and “retardation.” Then, we show that relaxation and retardation are asymptotically equivalent for small Deborah numbers, whence causal pure relaxation models necessarily correspond to ill-posed pure retardation models. We suggest that this dichotomy could be a possible way to reconcile the discrepancy between the theory of and certain experiments on viscoelastic liquids that are conjectured to exhibit only stress retardation.
We show that the constitutive relation for the thermal flux proposed by Xu & Hu (2011) admits an unconditional instability. We also highlight the difference between mathematical models containing ...delay and those that include relaxation effects.
Micellar solutions of nonionic surfactants Brij 35 and Tween 20 are confined between two surfaces in a colloidal-probe atomic-force microscope (CP-AFM). The experimentally detected oscillatory forces ...due to the layer-by-layer expulsion of the micelles agree very well with the theoretical predictions for hard-sphere fluids. While the experiment gives parts of the stable branches of the force curve, the theoretical model allows reconstruction of the full oscillatory curve. Therewith, the strength and range of the ordering could be determined. The resulting aggregation number from the fits of the force curves for Brij 35 is close to 70 and exhibits a slight tendency to increase with the surfactant concentration. The last layer of micelles cannot be pressed out. The measured force-vs-distance curve has nonequilibrium portions, which represent “jumps” from one to another branch of the respective equilibrium oscillatory curve. In the case of Brij 35, at concentrations <150 mM spherical micelles are present and the oscillation period is close to the micelle diameter, slightly decreasing with the rise of concentration. For elongated micelles (at concentration 200 mM), no harmonic oscillations are observed anymore; instead, the period increases with the decrease of film thickness. In the case of Tween 20, the force oscillations are almost suppressed, which implies that the micelles of this surfactant are labile and are demolished by the hydrodynamic shear stresses due to the colloidal-probe motion. The comparison of the results for the two surfactants demonstrates that in some cases the micelles can be destroyed by the CP-AFM, but in other cases they can be stable and behave as rigid particles. This behavior correlates with the characteristic times of the slow micellar relaxation process for these surfactants.
Cutting and shuffling, a novel approach to mixing, can be used to predict the degree of mixing for three-dimensional (3D) granular flows in rotating tumblers. An idealized prototype, a half-full ...“blinking” spherical tumbler alternating in rotation about each of two horizontal axes in the limit of a vanishingly thin flowing layer, is studied using piecewise isometries (PWIs). Mixing is characterized by the normalized difference between the center of mass of a collection of tracer particles and the centroid of the filled portion of the container. The degree of mixing is highly dependent on the combination of rotation angles about the two axes and the angle between the axes of rotation. Various rotation angle combinations (θ1,θ2) are investigated over 1°≤θi≤89° (in 1° increments). Combinations of rotation angles that can lead to good mixing after only a few iterations of the blinking flow are identified. Protocols with non-orthogonal axes are also studied by varying the angle ϕ between rotation axes over 15°≤ϕ≤90° (in 15° increments). When ϕ=60° or 75°, we observe the highest number of rotation angle pairs that lead to good mixing.
► It is shown cutting and shuffling describes mixing in biaxial granular tumblers. ► Flow in a half-full sphere is characterized using piecewise isometry mappings. ► The degree of mixing is inferred from measuring the center of mass of seed particles. ► Cutting and shuffling is maximized for certain parameters and rotation angles. ► For finite-thickness flowing layers, cutting and shuffling still leads to good mixing.
•Kink collisions in higher-order field-theoretic models (ϕ8,ϕ10,ϕ12) are examined.•The intricacies of collisions between kinks with power-law tails are analyzed.•Three- and more-bounce windows at the ...edges of two- and lower-bounce ones are found.•Different multi-kink initialization scenarios are proposed and compared.
We study collisions of coherent structures in higher-order field-theoretic models, such as the ϕ8,ϕ10 and ϕ12 ones. The main distinguishing feature, of the example models considered herein, is that the collision arises due to the long-range interacting algebraic tails of these solitary waves. We extend the approach to suitably initialize the relevant kinks, in the additional presence of finite initial velocity, in order to minimize the dispersive wave radiation potentially created by their slow spatial decay. We find that, when suitably initialized, these models still feature the multi-bounce resonance windows earlier found in models in which the kinks bear exponential tails, such as the ϕ4 and ϕ6 field theories among others. Also present is the self-similar structure of the associated windows with three- and more-bounce windows at the edges of two- and lower-bounce ones. Moreover, phenomenological but highly accurate (and predictive), scaling relations are derived for the dependence of the time between consecutive collisions and, e.g., the difference in kinetic energy between the incoming one and the critical one for one-bounces. Such scalings are traced extensively over two-bounce collision windows throughout the three models, hinting at the possibility of an analytical theory in this direction.
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