Viscous banding instabilities: non-porous viscous fingering

We demonstrate a novel instability found within unconfined viscous bands/rims, or free-surface flows involving a longitudinal viscosity contrast. Such instabilities may be described as viscous banding instabilities, non-porous viscous fingering instabilities or unconfined viscous fingering instabilities of free-surface flows involving the intrusion of a less viscous fluid into a band of more viscous fluid. A consequence of this work is that viscous fingering instabilities, widely known to occur in porous media following the seminal work of Saffman & Taylor (Proc. R. Soc. Lond. A, vol. 245, 1958, pp. 312–329), also occur in non-porous environments. Although the mechanism of the viscous banding instability is characteristically different from that of the Saffman–Taylor instability, there are important similarities between the two. The main similarity is that a viscosity contrast leads to instability. A distinguishing feature is that confinement, such as the rigid walls of a Hele-Shaw cell, is not necessary for viscous banding instabilities to occur. More precisely, Saffman–Taylor instabilities are driven by a jump in dynamic pressure gradient, whereas viscous banding instabilities, or non-porous viscous fingering instabilities, are driven by a jump in hydrostatic pressure gradient, directly related to a slope discontinuity across the intrusion front. We examine the onset of instability within viscous bands down an inclined plane, determine conditions under which viscous banding instabilities occur and map out a range of behaviours in parameter space in terms of two dimensionless parameters: the viscosity ratio and the volume of fluid ahead of the intrusion front.

Compressibility, Porosity, and Permeability of Shales Involving Stress Shock and Loading/Unloading Hysteresis

Summary This paper presents the theory, formulation, and correlation of the compressibility, porosity, and permeability of shale reservoirs by considering the effects of stress shock causing a slope discontinuity and loading/unloading hysteresis. The slope discontinuity occurs because the relative contributions of the matrix or fracture change at a critical effective stress depending on whether the process is loading or unloading. The hysteresis phenomenon occurs because of partially reversible and irreversible deformations of the various shale rock constituents by various processes during loading and unloading. Two successful modeling approaches are developed for describing the stress dependency of the petrophysical properties of porous rock formations. The first approach implements a kinetic model leading to a modified power–law equation, and the second approach applies an elastic cylindrical pore–shell model leading to a semianalytical equation. The primary advantage of the kinetic model is its applicability to any stress–dependent property, including strain, void ratio, porosity, pore compressibility, and permeability, thus making it a universal method. The semianalytical equation derived from an elastic cylindrical pore–shell model is applicable only for correlation of permeability. Both approaches are shown to yield high–quality correlations of the properties of porous rocks with effective stress by honoring the slope discontinuity observed at a critical effective stress.

Wave Number-Based Technique for Detecting Slope Discontinuity in Simple Beams Using Moving Test Vehicle

The wavelength characteristic is a useful clue for locating and assessing the severity of slope discontinuity in beams. In this study, the slope discontinuity of a beam is represented by an internal hinge restrained by elastic springs, and the wavelength of the beam is calculated indirectly from the vertical response of a test vehicle during its travel over the beam. The key parameters of the problem at hand are first unveiled using an approximate, closed-form solution for the response of the vehicle moving at low speeds over the bridge. Then a two-beam element model with slope discontinuity is formulated for the vehicle–bridge interaction (VBI) system for use in numerical simulation. In the examples, the wavenumber-based response of the test vehicle is used to identify the location and severity of the discontinuity in the beam. It is demonstrated that the wavelength-based technique presented herein by using the moving test vehicle as a moving sensor system offers a promising, alternative approach for damage detection in girder type bridges.

Size and shape of shock waves and slipline for Mach reflection in steady flow

For Mach reflection in steady supersonic flow, the slipline and reflected shock wave from the triple point are disturbed by secondary Mach waves generated over the slipline and by the expansion fan from the rear wedge corner. Analytical expressions for the shape of the curved slipline and reflected shock wave are derived in this paper. It is found that, due to transmitted expansion waves from the expansion fan, the slipline has a slope discontinuity at the turning point, i.e., the intersection point of the slipline and the leading characteristics of the transmitted expansion wave. The hypothetical shock wave calculated by considering this slope discontinuity as flow deflection angle matches a similar wave observed in numerical results by computational fluid dynamics, suggesting the existence of a weak shock wave from this turning point. The effects of the secondary Mach waves upstream of the turning point and of the turning point weak shock wave mutually cancel out approximately so that the transmitted Mach waves can be approximated as straight characteristic lines. This simplification leads to a fast analytical model which can predict the Mach stem height and shape of the slipline and reflected shock wave with increasing accuracy for the decreasing deflection angle of the slipline at the triple point. The slipline slope discontinuity at the turning point and the hypothetical turning point weak shock wave are new phenomena found in this work.

Ice formation within a thin film flowing over a flat plate

We present a model for ice formation in a thin, viscous liquid film driven by a Blasius boundary layer after heating is switched off along part of the flat plate. The flow is assumed to initially be in the Nelson et al. (J. Fluid Mech., vol. 284, 1995, pp. 159–169) steady-state configuration with a constant flux of liquid supplied at the tip of the plate, so that the film thickness grows like $x^{1/4}$ in distance along the plate. Plate cooling is applied downstream of a point, $Lx_{0}$, an $O(L)$-distance from the tip of the plate, where $L$ is much larger than the film thickness. The cooling is assumed to be slow enough that the flow is quasi-steady. We present a thorough asymptotic derivation of the governing equations from the incompressible Navier–Stokes equations in each fluid and the corresponding Stefan problem for ice growth. The problem breaks down into two temporal regimes corresponding to the relative size of the temperature difference across the ice, which are analysed in detail asymptotically and numerically. In each regime, two distinct spatial regions arise, an outer region of the length scale of the plate, and an inner region close to $x_{0}$ in which the film and air are driven over the growing ice layer. Moreover, in the early time regime, there is an additional intermediate region in which the air–water interface propagates a slope discontinuity downstream due to the sudden onset of the ice at the switch-off point. For each regime, we present ice profiles and growth rates, and show that for large times, the film is predicted to rupture in the outer region when the slope discontinuity becomes sufficiently enhanced.