Itzchak Frankel

Stability of a capillary jet with linearly increasing axial velocity (with application to shaped charges)

The stability of a capillary jet of an ideal liquid with a linear variation of axial velocity is investigated. Because of the time dependence in the basic extensional flow the evolution of surface perturbations in the jet is an initial-value problem instead of an eigenvalue one (as in the case of non-stretching jets). The amplification of any given peturbation is found to depend on the elative effects of surface tension and intertia terms associated with the extensional flow as well as on the initial wavenumber and the specific time when the perturbation is introduced in the flow field. The simulation of a shaped-charge jet by the present model is discussed. The esults obtained are found to give a good description of the essential features of the breakup phenomenon of such jets.

On the foundations of generalized Taylor dispersion theory

Generalized Taylor dispersion theory extends the basic long-time, asymptotic scheme of Taylor and Aris greatly beyond the class of rectilinear duct and channel flow dispersion problems originally addressed by them. This feature has rendered it indispensable for studying flow and dispersion phenomena in porous media, chromatographic separation processes, heat transfer in cellular media, sedimentation of non-spherical Brownian particles, and transport of flexible clusters of interacting Brownian particles, to mention just a few examples of the broad class of non-unidirectional transport phenomena encompassed by this scheme. Moreover, generalized Taylor dispersion theory enjoys the attractive feature of conferring a unified paradigmatic structure upon the analysis of such apparently disparate physical problems. For each of the problems thus treated it provides an asymptotic, macroscale description of the original microscale transport process, being based upon a convective-diffusive ‘model’ problem characterized by a set of constant (position- and time-independent) phenomenological coefficients. The present contribution formally substantiates the scheme. This is accomplished by demonstrating that the coarse-grained (macroscale) transport ‘model’ equation leads to a solution which accords asymptotically with the leading-order behaviour of the comparable solution of the exact (microscale) convective–diffusive problem underlying the transport process. It is also shown, contrary to current belief, that no systematic improvement in the asymptotic order of approximation is possible through the incorporation of higher-order gradient terms into the model constitutive equation for the coarse-grained flux. Moreover, the inherent difference between the present rigorous asymptotic scheme and the dispersion models resulting from Gill–Subramanian moment-gradient expansions is illuminated, thereby conclusively resolving a long-standing puzzle in longitudinal dispersion theory.

Symmetry breaking in induced-charge electro-osmosis over polarizable spheroids

We study the induced-charge electro-osmotic flow around a stationary polarizable dielectric spheroid in the presence of a uniform arbitrarily oriented external electric field. A Robin-type condition connecting the respective electric potentials within the dielectric solid and the bulk electro-neutral solution is highlighted in formulating the macroscale description for the limit of thin electric double layers and low potentials. The results illustrate symmetry breaking phenomena in the ensuing flow and demonstrate qualitative differences associated with variations of the dielectric constant. We briefly discuss the potential impact of these differences on the rotation of freely suspended spheroids.

On electro-osmotic flows through microchannel junctions

We study the electro-osmotic flow through a T-junction of microchannels whose dielectric walls are weakly polarizable. The present global analysis thus extends earlier studies in the literature concerning the local flow of an unbounded electrolyte solution around nearly insulated wedges. The velocity field is obtained via superposition of an irrotational part associated with the equilibrium zeta potential and the induced-charge electro-osmotic flow originating from the interaction of the externally applied electric field and the charge cloud it induces owing to field leakage through the polarizable dielectric channel walls. Along the channel walls the latter component gives rise to fluid velocities converging toward the corner which dominate the flow in its immediate vicinity. Recent experimental observations in the literature regarding the appearance and subsequent expansion of flow reversal and vortices downstream (initially) and upstream (subsequently) of the junction, are both rationalized in terms of...

Generalized Taylor dispersion phenomena in unbounded homogeneous shear flows

Generalized Taylor dispersion theory is extended so as to enable the analysis of the transport in unbounded homogeneous shear flows of Brownian particles possessing internal degrees of freedom (e.g. rigid non-spherical particles possessing orientational degrees of freedom, flexible particles possessing conformational degrees of freedom, etc.). Taylor dispersion phenomena originate from the coupling between the dependence of the translational velocity of such particles in physical space upon the internal variables and the stochastic sampling of the internal space resulting from the internal diffusion process. Employing a codeformational reference frame (i.e. one deforming with the sheared fluid) and assuming that the eigenvalues of the (constant) velocity gradient are purely imaginary, we establish the existence of a coarse-grained, purely physical-space description of the more detailed physical-internal space ( microscale ) transport process. This macroscale description takes the form of a convective–diffusive ‘model’ problem occurring exclusively in physical space, one whose formulation and solution are independent of the internal (‘local’-space) degrees of freedom. An Einstein-type diffusion relation is obtained for the long-time limit of the temporal rate of change of the mean-square particle displacement in physical space. Despite the nonlinear (in time) asymptotic behaviour of this displacement, its Oldroyd time derivative (which is the appropriate one in the codeformational view adopted) tends to a constant, time-independent limit which is independent of the initial internal coordinates of the Brownian particle at zero time. The dyadic dispersion-like coefficient representing this asymptotic limit is, in general, not a positive-definite quantity. This apparently paradoxical behaviour arises due to the failure of the growth in particle spread to be monotonic with time as a consequence of the coupling between the Taylor dispersion mechanism and the shear field. As such, a redefinition of the solutes dispersivity dyadic (appearing as a phenomenological coefficient in the coarse-grained model constitutive equation) is proposed. This definition provides additional insight into its physical (Lagrangian) significance as well as rendering this dyadic coefficient positive-definite, thus ensuring that solutions of the convective–diffusive model problem are well behaved. No restrictions are imposed upon the magnitude of the rotary Peclet number, which represents the relative intensities of the respective shear and diffusive effects upon which the solute dispersivity and mean particle sedimentation velocity both depend. The results of the general theory are illustrated by the (relatively) elementary problem of the sedimentation in a homogeneous unbounded shear field of a size-fluctuating porous Brownian sphere (which body serves to model the behaviour of a macromolecular coil). It is demonstrated that the well-known case of the translational diffusion in a homogeneous shear flow of a rigid, non- fluctuating sphere (for which the Taylor mechanism is absent) is a particular case thereof.

Influence of viscosity on the capillary instability of a stretching jet

The hydrodynamic stability of a rapidly elongating, viscous liquid jet such as obtained in shaped charges is presented. The flow field depends on three characteristic timescales associated with the growth of perturbations (due esaentially to the effect of the surface tension), the elongation of the jet, and the inward diffusion of vorticity from the free surface, respectively. The latter process introduces a time lag resulting in the current values of the free-surface perturbation and its time derivative being a function of their past history. Solutions of the integro-differential equation for the evolution of disturbances exhibit a novel dual role played by the viscosity : besides the traditional damping effect it is associated with a destabilizing mechanism in the elongating jet. The wavelength of maximum instability is also a function of time elapsed since the jet formation, longer wavelengths becoming dominant at later stages. Understanding of these instability processes can help in both promoting and delaying instability as required by specific applications.

Macro-scale description of transient electro-kinetic phenomena over polarizable dielectric solids

We have studied the temporal evolution of electro-kinetic flows in the vicinity of polarizable dielectric solids following the application of a ‘weak’ transient electric field. To obtain a macro-scale description in the limit of narrow electric double layers (EDLs), we have derived a pair of effective transient boundary conditions directly connecting the electric potentials across the EDL. Within the framework of the above assumptions, these conditions apply to a general transient electro-kinetic problem involving dielectric solids of arbitrary geometry and relative permittivity. Furthermore, the newly derived scheme is applicable to general transient and spatially non-uniform external fields. We examine the details of the physical mechanisms involved in the relaxation of the induced-charging process of the EDL adjacent to polarizable dielectric solids. It is thus established that the time scale characterizing the electrostatic relaxation increases with the dielectric constant of the solid from the Debye time (for the diffusion across the EDL) through the ‘intermediate’ scale (proportional to the product of the respective Debye- and geometric-length scales). Thus, the present rigorous analysis substantiates earlier results largely obtained by heuristic use of equivalent RC-circuit models. Furthermore, for typical values of ionic diffusivity and kinematic viscosity of the electrolyte solution, the latter time scale is comparable to the time scale of viscous relaxation in problems concerning microfluidic applications or micro-particle dynamics. The analysis is illustrated for spherical micro-particles. Explicit results are thus presented for the temporal evolution of electro-osmosis around a dielectric sphere immersed in unbounded electrolyte solution under the action of a suddenly applied uniform field, combining both induced charge and ‘equilibrium’ (fixed charge) contributions to the zeta potential. It is demonstrated that, owing to the time delay of the induced-EDL charging, the ‘equilibrium’ contribution to fluid motion (which is linear in the electric field) initially dominates the (quadratic) ‘induced’ contribution.

Taylor dispersion of orientable Brownian particles in unbounded homogeneous shear flows

The physical- and orientation-space transport of non-spherical , generally non-neutrally buoyant, Brownian particles in unbounded homogeneous shear flows is analysed with the goal of studying the respective effects of the orientational degrees of freedom of such particles upon their sedimentation and dispersion rates. In particular, the present contribution concentrates on the interaction between the Taylor dispersion mechanism (arising from coupling between the orientational dependence of the particles translational velocity and the stochastic sampling of the orientation space via rotary Brownian diffusion) and the shear velocity field. Making use of a recent extension of generalized Taylor dispersion theory to homogeneous (unbounded) shear flows, the mean transport process in physical space is modelled by a convection–diffusion problem characterized by a pair of constant phenomenological coefficients, provided that the eigenvalues of the (constant) undisturbed velocity gradient are purely imaginary. The latter phenomenological coefficients – namely, U * , the average ‘slip velocity’ vector (of the particles relative to the ambient fluid), and D * , the dispersivity dyadic or, equivalently, the pair of dyadics

On the Rayleigh-Bénard problem in the continuum limit

\bar{\bm M}

The motion of axisymmetric dipolar particles in homogeneous shear flow

and D C (or