Oblique liquid curtains with a large Froude number

This paper examines two-dimensional liquid curtains ejected at an angle to the horizontal and affected by gravity and surface tension. The flow in the curtain is, generally, sheared. The Froude number based on the injection velocity and the outlet’s width is assumed large; as a result, the streamwise scale of the curtain exceeds its thickness. A set of asymptotic equations for such (slender) curtains is derived and its steady solutions are examined. It is shown that, if the surface tension exceeds a certain threshold, the curtain – quite paradoxically – bends upwards, i.e. against gravity. Once the flow reaches the height where its initial supply of kinetic energy can take it, the curtain presumably breaks up and splashes down.

An ellipsoidal cap model for adhesion of a soft particle on a rigid substrate

Surface energy outside the contact zone, which is not accounted for in the classical Johnson–Kendall–Roberts (JKR) model, can play an essential role in adhesion mechanics of soft particles. An open problem in the adhesion mechanics of soft elastic particles is how to achieve an explicit expression for the surface energy outside the contact zone in terms of the two JKR-type variables ( a, δ), where a is the radius of the contact zone and δ is the relative approach of two bodies. The present work aims to develop an ellipsoidal cap model for the surface energy outside the contact zone of a soft elastic particle on a rigid substrate in terms of the two JKR-type variables ( a, δ). An explicit expression for the surface energy outside the contact zone is derived, and simple asymptotic equations are obtained to determine the two unknowns ( a, δ). The validity and accuracy of the derived expression and asymptotic equations are verified by good agreement with the Young–Dupre equation in the absence of an external applied force, and are also justified by good agreement of the predicted pull-off force with known results available in recent literature.

Hydraulic jumps in a shallow flow down a slightly inclined substrate

This work examines free-surface flows down an inclined substrate. The slope of the free surface and that of the substrate are both assumed small, whereas the Reynolds number $Re$ remains unrestricted. A set of asymptotic equations is derived, which includes the lubrication and shallow-water approximations as limiting cases (as $Re\rightarrow 0$ and $Re\rightarrow \infty$, respectively). The set is used to examine hydraulic jumps (bores) in a two-dimensional flow down an inclined substrate. An existence criterion for steadily propagating bores is obtained for the $({\it\eta},s)$ parameter space, where ${\it\eta}$ is the bore’s downstream-to-upstream depth ratio, and $s$ is a non-dimensional parameter characterising the substrate’s slope. The criterion reflects two different mechanisms restricting bores. If $s$ is sufficiently large, a ‘corner’ develops at the foot of the bore’s front – which, physically, causes overturning. If, in turn, ${\it\eta}$ is sufficiently small (i.e. the bore’s relative amplitude is sufficiently large), the non-existence of bores is caused by a stagnation point emerging in the flow.

Uniform Equations in the Inlet and Exit Zones of Heavily Loaded Point and Line Elastohydrodynamically Lubricated Contacts Involved in Various Steady Motions

Heavily loaded point elastohydrodynamically lubricated (EHL) contacts involved in steady purely transitional, skewed transitional, and transitional with spinning motions are considered. It is shown that in the central parts of the inlet and exit zones of such heavily loaded point EHL contacts the asymptotic equations governing the EHL problem along the lubricant flow streamlines for the above types of contact motions can be reduced to two sets of asymptotic equations: one in the inlet and one in the exit zones. The latter sets of equations are identical to the asymptotic equations describing lubrication process in the inlet and exit zones of the corresponding heavily loaded line EHL contact (Kudish, I. I., 2013, Elastohydrodynamic Lubrication for Line and Point Contacts: Asymptotic and Numerical Approaches, Chapman and Hall/CRC). For each specific motion of a point contact, a separate set of formulas for the lubrication film thickness is obtained. For different types of contact motions, these film thickness formulas differ significantly (Kudish, I. I., 2013, Elastohydrodynamic Lubrication for Line and Point Contacts: Asymptotic and Numerical Approaches, Chapman and Hall/CRC). For heavily loaded contacts, the discovered relationship between point and line EHL problems allows to apply to point contacts most of the results obtained for line contacts (Kudish, I. I., 2013, Elastohydrodynamic Lubrication for Line and Point Contacts: Asymptotic and Numerical Approaches, Chapman and Hall/CRC; Kudish, I. I., and Covitch, M. J., 2010, Modeling and Analytical Methods in Tribology, Chapman and Hall/CRC).

Entanglement of helicity and energy in kinetic Alfvén wave/whistler turbulence

The role of magnetic helicity is investigated in kinetic Alfvén wave and oblique whistler turbulence in presence of a relatively intense external magnetic fieldb0e∥. In this situation, turbulence is strongly anisotropic and the fluid equations describing both regimes are the reduced electron magnetohydrodynamics (REMHD) whose derivation, originally made from the gyrokinetic theory, is also obtained here from compressible Hall magnetohydrodynamics (MHD). We use the asymptotic equations derived by Galtier and Bhattacharjee (2003Phys. Plasmas10, 3065–3076) to study the REMHD dynamics in the weak turbulence regime. The analysis is focused on the magnetic helicity equation for which we obtain the exact solutions: they correspond to the entanglement relation,n + ñ= −6, wherenandñare the power law indices of the perpendicular (tob0) wave number magnetic energy and helicity spectra, respectively. Therefore, the spectra derived in the past from the energy equation only, namelyn= −2.5 andñ= −3.5, are not the unique solutions to this problem but rather characterize the direct energy cascade. The solutionñ= −3 is a limit imposed by the locality condition; it is also the constant helicity flux solution obtained heuristically. The results obtained offer a new paradigm to understand solar wind turbulence at sub-ion scales where it is often observed that −3 <n< −2.5.

Uniform Equations in the Inlet and Exit Zones of Heavily Loaded Point EHL Contacts Involved in Various Steady Motions

Heavily loaded point EHL contacts involved in steady purely transitional, skewed transitional, and transitional with spinning motions are considered. It is shown that in the central parts of the inlet and exit zones of such heavily loaded point EHL contacts the asymptotic equations governing the EHL problem along the lubricant flow streamlines for the above types of contact motions can be reduced to two sets of asymptotic equations: one in the inlet and one the exit zones. The latter sets of equations are identical to the asymptotic equations describing lubrication process in the inlet and exit zones of the corresponding heavily loaded line EHL contact [1]. For each specific motion of a point contact a separate set of formulas for the lubrication film thickness is obtained. For different types of contact motions these film thickness formulas differ significantly [1]. For heavily loaded contacts the discovered relationship between point and line EHL problems allow to apply to point contacts most of the results obtained for line contacts [1,2].

Linear trends in cloud top height from passive observations in the oxygen A-band

Measurements by the hyperspectral spectrometers GOME, SCIAMACHY and GOME-2 are used to determine the rate of linear change (and trends) in cloud top height (CTH) in the period between June 1996 and May 2012. The retrievals are obtained from Top-Of-Atmosphere (TOA) backscattered solar light in the oxygen A-band using the Semi-Analytical CloUd Retrieval Algorithm SACURA. The physical framework relies on the asymptotic equations of radiative transfer, valid for optically thick clouds. Using linear least-squares techniques, a global trend of −1.78 ± 2.14 m yr−1 in deseasonalized CTH has been found, in the latitude belt within ±60°, with diverging tendencies over land (+0.27 ± 3.2 m yr−1) and ocean (−2.51 ± 2.8 m yr−1). The El Niño–Southern Oscillation (ENSO), strongly coupled to CTH, forces clouds to lower altitudes. The global ENSO-corrected trend in CTH amounts to −0.49 ± 2.22 m yr−1. At a global scale, no explicit regional pattern of statistically significant trends (at 95% confidence level, estimated with bootstrap technique) have been found, which would be representative of typical natural synoptical features. One exception is North Africa, which exhibits the strongest upward trend in CTH sustained by an increasing trend in water vapour.

Trends in cloud top height from passive observations in the oxygen A-band

Measurements by the hyperspectral spectrometers GOME, SCIAMACHY, and GOME-2 are used to determine the rate of linear change (and trends) in cloud top height (CTH) in the period between June 1996 and May 2012. The retrievals are obtained from Top-Of-Atmosphere (TOA) backscattered solar light in the oxygen A-band using the Semi-Analytical CloUd Retrieval Algorithm SACURA. The physical framework relies on the asymptotic equations of radiative transfer, valid for optically thick clouds. Using linear least-squares techniques, a global trend of −1.78 ± 2.14 m yr−1 in deseasonalised CTH has been found, in the latitude belt within ±60°, with diverging tendencies over land (+0.27 ± 3.2 m yr−1) and ocean (−2.51 ± 2.8 m yr−1). The El Niño-Southern Oscillation (ENSO), strongly coupled to CTH, forces clouds to lower altitudes. The global ENSO-corrected trend in CTH amounts to −0.49 ± 2.22 m yr−1. At a~global scale, no explicit regional pattern of statistically significant trends (at 95 % confidence level, estimated with bootstrap technique) have been found, which would be representative of typical natural synoptical features. One exception is North Africa, which exhibits the strongest upward trend in CTH sustained by an increasing trend in water vapor.