Values Of Dimensionless Frequency Research Articles

Time-averaged transport in oscillatory squeeze flow of a viscoelastic fluid

Periodically-driven flows are known to generate non-zero, time-averaged fluxes of heat or solute species, due to the interactions of out-of-phase velocity and temperature/concentration fields, respectively. Herein, we investigate such transport (a form of the well-known Taylor--Aris dispersion) in the gap between two parallel plates, one of which oscillates vertically, generating a time-periodic squeeze flow of either a newtonian or Maxwellian fluid. Using the method of multiple time-scale homogenization, the mass/heat balance equation describing transport in this flow is reduced to a one-dimensional advection--diffusion--reaction equation. This result indicates three effective mechanisms in the mass/heat transfer in the system: an effective diffusion that spreads mass/heat along the concentration/temperature gradient, an effective advective flux, and an effective reaction that releases or absorbs mass/heat - in the time-averaged frame. Our results demonstrate that there exist resonant modes under which the velocity peaks when the dimensionless plate oscillation frequency (embodied by the Womersley number, the ratio of the transient inertia to viscous forces) approaches specific values. As a result, transport in this flow is significantly influenced by the dimensionless frequency. On the one hand, the effective, time-averaged dispersion coefficient is always larger than the molecular diffusivity, and is sharply enhanced near resonance. The interaction between fluid elasticity and the oscillatory forcing enhances the efficiency of transport in the system. On the other hand, the identified effective advection and reaction mechanisms may transport mass/heat from regions of high concentration/temperature to those of low concentration/temperature, or vice versa, depending on the value of dimensionless frequency.

On the critical layer of Alfvn waves in the solar wind

Alfvén waves are considered in a radial flow and external magnetic field, which relates to some features of the solar wind near the critical point. The Alfvén wave equation for the velocity perturbation is derived, showing that it has in general two singularities (besides the origin and infinity), namely a critical layer (at real distance), where the Alfvén speed equals the mean flow velocity, and a transition level (at imaginary distance), where the spatial derivative of the flow velocity equals the wave frequency. It is shown that in the case of mean flow velocity varying as a power of radial distance the wave field is specified at all distances by a combination of solutions of the Alfvén wave equation around three singularities: a regular singularity at the center, so that ascending power-series solutions exist, some with logarithmic terms; an irregular singularity at infinity, leading to the non-existence of any solution as an ascending Frobenius–Fuchs series, and the existence of two solutions as ascending–descending Laurent series; the region of validity of the preceding solutions is limited by a regular singularity at a finite, non-zero radial distance, which is the critical layer, where the flow velocity and Alfvén speed are equal. The wave field is singular at the critical layer, and has an amplitude jump, which is illustrated by plotting the wave field in the neighborhood of the critical layer, for several values of dimensionless frequency and Alfvén number, combined into a single parameter. When considering Alfvén waves in the solar wind, at least three kinds of boundary conditions could be applied: (i) an initial condition specifying the wave field at the surface of the Sun; (ii) an asymptotic condition excluding wave sources at infinity, by specifying an outward-propagating wave (radiation condition); (iii) a finiteness condition that the wave field be finite at the critical layer. Since the Alfvén wave equation is of second order, only two conditions can in general be applied. It is shown, for example, that (ii) and (iii) are generally incompatible. If the conditions (i) and (iii) are chosen, i.e. an initial wave field is given and the radiation condition of outward propagation at infinity is met, then (ii) will not in general be met; thus the wave field would be singular at the critical layer, in the absence of dissipation, corresponding to the resonance of a linear undamped system. It is shown that in the presence of dissipation, either by fluid viscosity or Ohmic resistivity, the wave field would be finite at the critical layer, corresponding to the resonance of a linear damped system.

On dissipative Alfvén waves in atmospheres with steep temperature gradients

The present paper concerns (Section 1) dissipative Alfvén waves in an atmosphere with a steep temperature gradient, such that the background temperature varies on a scale shorter than the wavelength; an application is the transition region between the solar chromosphere and corona (Section 7). for which the temperature rises from 104 K to 1.8 × 106 K over a distance of a few hundred kilometers. The starting point is the Alfvén wave equation, in an atmosphere under an uniform vertical magnetic field, allowing (Section 2) the Alfvén speed and viscous and resistive diffusivities to be arbitrary functions of altitude. A three-parameter family of temperature temperature profiles is introduced, allowing independent choice (Section 3) of initial and asymptotic temperature, and initial temperature gradient. The wave equation for the magnetic field perturbation is written with the temperature as independent variable, instead of altitude. This leads to a second-order wave equation, whose coefficients are polynomials of degree ten, and which is solved (Section 4) in terms of power series, allowing the application of boundary, asymptotic, radiation and dissipation conditions (Section 5). The wave fields are plotted, versus altitude (Figs. 4, 5, 7) or temperature (Figs. 3,6,8), for several values of dimensionless frequency and viscous and resistive damping, of the order of magnitude of those relevant (Section 6) to dissipative Alfvén waves in the solar transition region, between the chromosphere and corona.

On cut-off frequencies for magneto-atmospheric waves

We address the topic of the cut-off frequencies for magneto-atmospheric waves, a subject on which there are gaps and inconsistencies in the literature. We consider magneto-acoustic-gravity waves, in an isothermal atmosphere, under a uniform oblique magnetic field, with a wavevector in the plane of the latter and gravity. In the case of vertical waves, it is shown that the acoustic and magnetic cut-off frequencies are the same for the vertical and horizontal velocity perturbations. These two cut-off frequencies, together with the gravity cut-off or Brünt-Vaisala frequency, apply to oblique waves in a vertical magnetic field. The method used to determine the cut-off frequencies in the preceding cases of wave equations with exponential coefficients, is based on two characteristic polynomials, which specify also the exact wave field at all altitudes for any frequency. These solutions can be used to plot the waveforms of the magnetic mode and amplitude and phase of the dynamic mode, for different values of dimensionless frequency and wavenumber, and plasma β. The cut-off frequencies obtained in the present paper are compared with those given in the literature for other cases of magneto-atmospheric waves.