Self‐contained two‐layer shallow‐water theory of strong internal bores

We show that interfacial gravity waves comprising strong hydraulic jumps (bores) can be described by a two-layer hydrostatic shallow-water (SW) approximation without invoking additional front conditions. The theory is based on a new SW momentum equation which is derived in locally conservative form containing a free parameter $\alpha.$ This parameter, which defines the relative contribution of each layer to the pressure at the interface, affects only hydraulic jumps but not continuous waves. The Rankine-Hugoniot jump conditions for the momentum and mass conservation equations are found to be mathematically equivalent to the classical front conditions, which were previously thought to be outside the scope of SW approximation. Dimensional arguments suggest that $\alpha$ depends on the density ratio. For nearly equal densities, both layers are expected to affect interfacial pressure with approximately equal weight coefficients, which corresponds to $\alpha\approx0.$ The front propagation velocity for $\alpha=0$ agrees well with experimental and numerical results in a wide range of bore strengths. A remarkably better agreement with high-accuracy numerical results is achieved by $\alpha=\sqrt{5}-2,$ which yields the largest height that a stable gravity current can have.Comment: 28 pages, 9 figures, extended version with appendix (to appear in Stud. Appl. Math.

Comment on "Linear stability of a rotating channel flow subjected to a static magnetic field " [Phys. Fluids 34, 054116 (2022)]

Recently, Sengupta and Ghosh [Phys. Fluids 34, 054116, (2022)] published a linear stability analysis of a pressure-driven channel flow which is subject to rotation around a spanwise axis and a uniform magnetic field applied in the same direction. Unfortunately, the formulation of the magnetohydrodynamic part of the problem contains an elementary error which makes the obtained results unphysical. The error is due to unfounded omission of the electric potential contribution in the induced electric current which, thus, does not satisfy the charge conservation.Comment: 2 page

Fractality of metal pad instability threshold in rectangular cells

We analyse linear stability of interfacial waves in an idealised model of an aluminium reduction cell consisting of two stably stratified liquid layers which carry a vertical electric current in a collinear external magnetic field. If the product of electric current and magnetic field exceeds a certain critical threshold depending on the cell design, the electromagnetic coupling of gravity wave modes can give rise to a self-amplifying rotating interfacial wave which is known as the metal pad instability. Using the eigenvalue perturbation method, we show that, in the inviscid limit, rectangular cells of horizontal aspect ratios $\alpha=\sqrt{m/n}$, where $m$ and $n$ are any two odd numbers, can be destabilised by an infinitesimally weak electromagnetic interaction while cells of other aspect ratios have finite instability thresholds. This fractal distribution of critical aspect ratios, which form an absolutely discontinuous dense set of points interspersed with aspect ratios with non-zero stability thresholds, is confirmed by accurate numerical solution of the linear stability problem. Although the fractality vanishes when viscous friction is taken into account, the instability threshold is smoothed out gradually and its principal structure, which is dominated by the major critical aspect ratios corresponding to moderate values of $m$ and $n$, is well-preserved up to relatively large dimensionless viscous friction coefficients $\gamma\sim 0.1$. With a small viscous friction, the most stable are cells with $\alpha^{2}\approx2.13$ which have the highest stability threshold corresponding to the electromagnetic interaction parameter $\beta\approx 4.7$.Comment: 21 pages, 6 figures (final version; to appear in J. Fluid Mech.

Evaluation of Latvia-China trade potential

International trade always has been a source of wealth for countries engaged in the process. Lately, European Union (EU) and Latvia is facing new challenges in for of trade restrictions and protectionism with the trade partners. One of the latest cases of Russian asymmetric sanctions in the form of food trade embargo left EU with difficult task of trade redirection and more intense local market competition. With the UK leaving EU case on the horizon all EU countries are facing the same problem – where to find new markets with high demand potential. The largest EU economies, Germany, France, Italy, Spain and Netherlands have already expanded their trade all over the world to diversify the risk of being too much dependant on several trade partners. This is also the case for Latvia – there is necessity to find new potential markets. The aim of the research is to evaluate Latvia - China trade potential in terms of Latvia’s export opportunities. Methods used in the research: Study of the literature on the export market evaluation and trade data analysis. Research involves usage of potential export market evaluation with Decision Support Model. Research is very topical in context of Latvia’s necessity of market diversification since Russia has food trade embargo (since August 7, 2014) and it is unclear future about trade regulations with the UK with the process of UK leaving European Union (BREXIT).peer-reviewe

Absolute versus convective helical magnetorotational instability in a Taylor-Couette flow

We analyze numerically the magnetorotational instability of a Taylor-Couette flow in a helical magnetic field (HMRI) using the inductionless approximation defined by a zero magnetic Prandtl number (Pm=0). The Chebyshev collocation method is used to calculate the eigenvalue spectrum for small amplitude perturbations. First, we carry out a detailed conventional linear stability analysis with respect to perturbations in the form of Fourier modes that corresponds to the convective instability which is not in general self-sustained. The helical magnetic field is found to extend the instability to a relatively narrow range beyond its purely hydrodynamic limit defined by the Rayleigh line. There is not only a lower critical threshold at which HMRI appears but also an upper one at which it disappears again. The latter distinguishes the HMRI from a magnetically-modified Taylor vortex flow. Second, we find an absolute instability threshold as well. In the hydrodynamically unstable regime before the Rayleigh line, the threshold of absolute instability is just slightly above the convective one although the critical wave length of the former is noticeably shorter than that of the latter. Beyond the Rayleigh line the lower threshold of absolute instability rises significantly above the corresponding convective one while the upper one descends significantly below its convective counterpart. As a result, the extension of the absolute HMRI beyond the Rayleigh line is considerably shorter than that of the convective instability. The absolute HMRI is supposed to be self-sustained and, thus, experimentally observable without any external excitation in a system of sufficiently large axial extension.Comment: 16 pages, 15 figures; minor revision, Phys. Rev. E (in press

Alternative transient eddy-current flowmetering methods for liquid metals

We present a comprehensive numerical analysis of alternative transient eddy-current flowmetering methods for liquid metals. This type of flowmeter operates by tracking eddy-current markers excited by the magnetic field pulses in the flow of a conducting liquid. Using a simple mathematical model, where the fluid flow is replaced by a translating cylinder, a number possible alternative measurement schemes are considered. The velocity of the medium can be measured by tracking zero crossing points and spatial or temporal extrema of the electromotive force (emf) induced by transient eddy currents in the surrounding space. Zero crossing points and spatial extrema of the emf travel synchronously with the medium whereas temporal extrema experience an initial time delay which depends on the conductivity and velocity of the medium. Performance of transient eddy-current flowmetering depends crucially on the symmetry of system. Eddy current asymmetry of a few per cent makes the detection point drift with a velocity corresponding to a magnetic Reynolds number $Rm\sim 0.1.$ With this level of asymmetry transient eddy-current flowmetering can be reliably applicable only to flows with $Rm\gtrsim 0.1.$ A more accurate symmetry adjustment or calibration of flowmeters may be necessary at lower velocities.Comment: 16 pages, 9 figures (to appear in Flow Meas. Instrum.

An Exploratory Analysis of Latvian Brands Competitiveness in China

In the past several years, China is entering a new stage of country`s economic development. Due to these changes of foreign brands also are switching their position towards Chinese consumers. The attitudes that consumers have towards international brands have been studied in the western countries for some time now. On 6th November 2016, the 5th Summit of China and Central and Eastern European Countries (CEECs 16+1 Cooperation) was successfully held in Riga, Latvia. On the behalf of Guidelines which were made to help the cooperation between Central and Eastern Europe and China, some Latvian companies adopted an approach that emphasizes relation building with local agencies. However, these companies don`t do enough marketing planning and their activities in this field are implicit, because they do not understand Chinese consumers and their shopping habits. In order to enter Chinese market and expand the business here, entrepreneurs must understand the mindset and preferences of Chinese consumers. Through some marketing survey in Beijing, Shanghai, Guangzhou, Chengdu, Nanjing, Hangzhou, the author collected the marketing research data of consumer preference by questionnaires among the six largest cities in China. Through some marketing theories, SWOT analysis, data statistics, the research gets a comprehensive analysis result. By taking a fresh look at the marketing researches amidst latent consumer conception and behavior further, this study generates gaining experiences of Latvian brands building in China market. Then the conclusion will show the suggestions of how Latvian international brands adjust their marketing changes to raise their brands competitiveness in Chinese consumption patterns and to address the impact of consumer behaviors and marketing activities successfully