Velocity gradient

About: Velocity gradient is a research topic. Over the lifetime, 3013 publications have been published within this topic receiving 77120 citations.

Inertial effects on the rheology of a dilute emulsion

Abstract: The behaviour of an isolated nearly spherical drop in an ambient linear flow is examined analytically at small but finite Reynolds numbers, and thereby the first effects of inertia on the bulk stress in a dilute emulsion of neutrally buoyant drops are calculated. The Reynolds numbers, Re = γa 2 ρ/μ and Re = γa 2 ρ/μ, are the relevant dimensionless measures of inertia in the continuous and disperse (drop) phases, respectively. Here, α is the drop radius, γ is the shear rate, ρ is the common density and μ and μ are, respectively, the viscosities of the drop and the suspending fluid. The assumption of nearly spherical drops implies the dominance of surface tension, and the analysis therefore corresponds to the limit of the capillary number (Ca) based on the viscosity of the suspending fluid being small but finite; in other words, Ca « 1, where Cα = μαγ/T, T being the coefficient of interfacial tension. The bulk stress is determined to O(φRe) via two approaches. The first one is the familiar direct approach based on determining the force density associated with the disturbance velocity field on the surface of the drop; the latter is determined to O(Re) from a regular perturbation analysis. The second approach is based on a novel reciprocal theorem formulation and allows the calculation, to O(Re), of the drop stresslet, and hence the emulsion bulk stress, with knowledge of only the leading-order Stokes fields. The first approach is used to determine the bulk stress for linear flows without vortex stretching, while the reciprocal theorem approach allows one to generalize this result to any linear flow. For the case of simple shear flow, the inertial contributions to the bulk stress lead to normal stress differences (N 1 , N 2 ) at O(φRe), where φ(«1) is the volume fraction of the disperse phase. Inertia leads to negative and positive contributions, respectively, to N 1 and N 2 at O(φRe). The signs of the inertial contributions to the normal stress differences may be related to the O(ReCa) tilting of the drop towards the velocity gradient direction. These signs are, however, opposite to that of the normal stress differences in the creeping flow limit. The latter are O(φCa) and result from an O(Ca 2 ) deformation of the drop acting to tilt it towards the flow axis. As a result, even a modest amount of inertia has a significant effect on the rheology of a dilute emulsion. In particular, both normal stress differences reverse sign at critical Reynolds numbers (Re c ) of O(Ca) in the limit Ca «1. This criterion for the reversal in the signs of N 1 and N 2 is more conveniently expressed in terms of a critical Ohnesorge number (Oh) based on the viscosity of the suspending fluid, where Oh = μ(ρT) 1/2 . The critical Ohnesorge number for a sign reversal in N 1 is found to be lower than that for N 2 , and the precise numerical value is a function of λ. In uniaxial extensional flow, the Trouton viscosity remains unaltered at O(φRe), the first effects of inertia now being restricted to O(φRe 3/2 ). The analytical results for simple shear flow compare favourably with the recent numerical simulations of Li & Sarkar.

Wave‐current Models for Design of Marine Structures

Abstract: Modifications to vertical velocity profiles of coastal currents due to surface gravity waves were experimentally and theoretically examined. Results on these modifications were used to develop guidelines for approximating the form of current velocity profiles to be used in predictive models of wave kinematics. The validity of the guidelines was established by comparing the experimental data of wave particle velocities with the theoretical predictions of two wave‐current models which approximate the shear current by: (1) A constant vorticity current; and (2) a uniform velocity over the water depth. The results show that opposing waves decrease the current mean velocity close to the bottom and increase the mean velocity and the current shear near the free surface. Following waves increase mean velocity and velocity gradient of the current, close to the bottom, and might cause the shear at the surface to be negative. Models for constant vorticity and uniform velocity approximations are found to yield accurat...

Influence of Alfven waves on thermal instability in the interstellar medium

Abstract: The effect of Alfven waves on the thermal instability of the Interstellar Medium (ISM) is investigated both analytically and numerically. A stability analysis of a finite amplitude circularly polarized Alfven wave propagating parallel to an ambient magnetic field in a thermally unstable gas at thermal equilibrium is performed, leading to a dispersion relation that depends on 3 parameters, namely the square ratio of the sonic and Alfven velocities (β), the wave amplitude and the ratio between the wave temporal period and the cooling time. Depending on the values of these 3 parameters, the Alfven waves can stabilize the large-scale perturbations, destabilize those whose wavelength is a few times the Alfven wavelength λ AW , or leave the growth rate of the short scales unchanged. To investigate the non-linear regime, two different numerical experiments are performed in a slab geometry. The first one deals with the development of an initial density perturbation in a thermally unstable gas in the presence of Alfven waves. The second one addresses the influence of those waves on the thermal transition induced by a converging flow. The numerical results confirm the trends inferred from the analytic calculations, i.e. the waves prevent the instability at scales larger than λ AW and trigger the growth of wavelengths close to λ AW , therefore producing a very fragmented cold phase. The second numerical experiments shows that i) the magnetic pressure prevents the merging of the CNM fragments therefore maintaining the complex structure of the flow and organizing it into groups of clouds ii) these groups of CNM clouds have an Alfvenic internal velocity dispersion iii) strong density fluctuations (≃10ρ cnm ) triggered by magnetic compression occur. We note that during this event there is no stiff variation of the longitudinal velocity field. This is unlike the hydrodynamical case for which the clouds are uniform and do not contain significant internal motions except after cloud collisions. In this situation a strong density fluctuation occurs, accompanied by a stationary velocity gradient through the cloud.

Causality in Strong Shear Flows

Abstract: It is well known that the standard transport equations violate causality when gradients are large or when temporal variations are rapid. We derive a modified set of transport equations that satisfy causality. These equations are obtained from the underlying Boltzmann equation. We use a simple model for particle collisions which enables us to derive moment equations non-perturbatively, i.e. without making the usual assumption that the distribution function deviates only slightly from its equilibrium value. We apply the model to two problems: particle diffusion and viscous transport. In both cases we show that signals propagate at a finite speed and therefore that the formalism obeys causality. When the velocity gradient is large on the scale of a mean free path, the viscous shear stress is suppressed relative to the prediction of the standard diffusion approximation. The shear stress reaches a maximum at a finite value of the shear amplitude and then decreases as the velocity gradient increases. In the case of a steady Keplerian accretion disk with hydrodynamic turbulent viscosity, the stress-limit translates to an upper bound on the Shakura-Sunyaev $\alpha$-parameter, namely $\alpha<0.07$. The limit on $\alpha$ is much stronger in narrow boundary layers where the velocity shear is larger than Keplerian.