A new class of periodic solutions in the Kovalevskaya case of a rigid body in rotation about a fixed point

The equation of motion of a rigid body in Kovaleveskaya case is reduced to a plane motion. By using the method of small parameters introduced by Poincaré, the existence of a periodic solution is established.     

Geochemistry of an island-arc plutonic suite: Wadi Dabr intrusive complex, Eastern Desert, Egypt

The Wadi Dabr intrusive complex, west of Mersa-Alam, Eastern Desert, Egypt ranges in composition from gabbro to diorite, quartz diorite and tonalite. The gabbroic rocks include pyroxene-horn blend e gabbro, hornblende gabbro, quartz-hornblende gabbro, metagabbro and amphibolite. Mineral chemistry data for the gabbroic rocks indicate that the composition of clinopyroxenes ranges from diopside to augite and the corresponding magma is equivalent to a volcanic-arc basalt. Plagioclase cores range from An₇₅ to An₃₄ for the gabbroic varieties, except for the metagabbro which has An _11–18. The brown amphiboles are primary phases and classified as calcic amphiboles, which range from tschermakitic hornblende to magnesiohornblende. Green hornblende and actinolite are secondary phases. Hornblende barometry and hornblende-plagioclase themometry for the gabbroic rocks estimate crystallisation conditions of 2–5 kb and 885–716°C.The intrusive rocks cover an extensive silica range (47.86–72.54 wt%) and do not exhibit simple straight-line variation on Harker diagrams for many elements (e.g. TiO₂, Al₂O₃, FeO^*, MgP, CaO, P₂O₅, Cr, Ni, V, Sr, Zr and Y). Most of these elements exhibit two geochemical trends suggesting two magma sources.The gabbroic rocks are relatively enriched in large ion lithophile elements (K, Rb, Sr and Ba) and depleted in high field strength elements (Nb, Zr, Ti and Y) which suggest subduction-related magma. Rare earth element (REE) data demonstrate that the gabbroic rocks have a slight enrichment of light REE [(La/Yb)_N=2.67−3.91] and depletion of heavy REE ((Tb/Yb)_N=1.42−1.47], which suggest the parent magma was of relatively primitive mantle source.The diorites and tonalites are clearly calc-alkaline and have negative anomalies of Nb, Zr, and Y which also suggest subduction-related magma. They are related to continental trondhjemites in terms of Rb---Sr, K---Na---Ca, and to volcanic-arc granites in terms of Rb---and Nb---Y.The Wadi Dabr intrusive complex is analogous to intrusions emplaced in immature ensimatic island-arcs and represents a mixture of mantle (gabbroic rocks) and crustal fusion products (diorites and tonalites) modified by fractional processes.     

Management of shock loads wastewater produced from water heaters industry

Water heater manufacturing represents one sector of household electrical appliance industry. It includes several batch processes which resulted in a highly polluted wastewater as shock loads. The objective of this study was to manage the shock loads wastewater with a simple and cost-effective approach prior to final discharge into municipality. To achieve this objective, two approaches were studied. The first approach was the chemical treatment of the accumulated shock loads wastewater using alum and an anionic polymer. Although this approach produced a very high-quality effluent, it was economically and technically infeasible. The second approach was a controlled release of the shock loads to the normal daily discharge in a way that guarantees the compliance of the end-off-pipe with the National Regulatory Standards. This solution required establishment of an equalization tank for normal daily flow and a holding tank for controlled release of the shock loads. Mathematical calculations were carried out to determine the most violating parameters in order to calculate the mixing ratio the of shock loads with the normal daily flow. Full engineering design of the proposed solution was carried out. This approach was implemented and proved to be simple, easy to operate, cost-effective and can be replicated in similar batch processing manufacturing plants.     

Chromospheric activity in $\epsilon$ Eridani: results from theoretical wave studies

This work discusses theoretical models of chromospheric heating for \(\epsilon\) Eridani by shock waves. Self-consistent, nonlinear and time-dependent ab-initio numerical computations for the excitation of the atmosphere (i.e., arrays of flux tubes) are pursued based on waves generated in stellar convective zones. Based on previous studies the magnetic filling factor is estimated according to the stellar rotational period, although general models are described as well. The Ca II H+K fluxes are computed assuming partial redistribution (PRD). Time-dependent ionization notably affects the resulting Ca II fluxes, as expected. The emergent Ca II H+K fluxes are based on two-component models, consisting of a dominant magnetic component (as given by longitudinal tube waves) and a subordinate acoustic component. The Ca II fluxes as obtained are smaller by about a factor of 2 than those given by observations. Possible reasons for this discrepancy include (1) inherent limitations of our theoretical approach as it is based on 1-D rather than 3-D modelling and/or (2) the existence of additional heating processes in \(\epsilon\) Eridani (a young star) not included here.     

On the galactic motion

A certain potential function is studied as a possible model for the galactic potential. Some solutions are obtained. Also the numerical study of some of the orbits and their stability is carried out.     

The heating of solar magnetic flux tubes II. Nonadiabatic longitudinal tube waves

The formation of shocks and shock heating by radiatively damped longitudinal waves in solar magnetic flux tubes of different filling factors is studied. We consider three flux tubes of filling factors: 1%, 20%, and exponentially spreading which represent normal, enhanced network regions and the interior of supergranulation cells respectively. Monochromatic waves with periods 60 s and energy fluxes of 4.0 · 10⁸ erg cm^?2 s^?1 are assumed to propagate in the tubes. We find that the H^?-continuum losses and the Mg II line emission are much reduced in the tube of small filling factor while the mean temperatures are roughly similar in both tubes. The exponential flux tube shows little or no shock heating and no radiation damping. Shocks form earlier in the tube of high filling factor, and have larger strength.     

Periodic solutions in the Kovalevskaya case of a rigid body in rotation about a fixed point

The equation of motion of a rigid body in the Kovalevskaya case is reduced to a plane motion. By using the method of small parameters introduced by Poincaré the existence of a periodic solution is established.     

The periodic solution of a rigid body in a central newtonian field

The equations of motion of a rigid body about a fixed point in a central Newtonian field is reduced to the equation of plane motion under the action of potential and gyroscopic forces, using the isothermal coordinates on the inertia ellipsoid.The construction of periodic solutions nearby equilibrium points, by using the Liapunov theorem of holomorphic integral are obtained and the necessary and sufficient conditions for the stability of the system are given.     

Neoproterozoic nascent island arc volcanism from the Nubian Shield of Egypt: Magma genesis and generation of continental crust in intra-oceanic arcs

The Neoproterozoic Wadi Ranga metavolcanic rocks, South Eastern Desert of Egypt, constitute a slightly metamorphosed bimodal sequence of low-K submarine tholeiitic mafic and felsic volcanic rocks. The mafic volcanic rocks are represented by massive and pillow flows and agglomerates, composed of porphyritic and aphyric basalts and basaltic andesites that are mostly amygdaloidal. The felsic volcanic rocks embrace porphyritic dacites and rhyolites and tuffs, which overlie the mafic volcanic rocks. The geochemical characteristics of Wadi Ranga volcanic rocks, especially a strong Nb depletion, indicate that they were formed from subduction-related melts. The clinopyroxene phenocrysts of basalts are more akin to those crystallizing from island-arc tholeiitic magmas. The tholeiitic nature of the Wadi Ranga volcanics as well as their LREE-depleted or nearly flat REE patterns and their low K₂O contents suggest that they were developed in an immature island arc setting. The subchondritic Nb/Ta ratios (with the lowest ratio reported for any arc rocks) and low Nb/Yb ratios indicate that the mantle source of the Wadi Ranga mafic volcanic rocks was more depleted than N-MORB-source mantle. Subduction signature was dominated by aqueous fluids derived from slab dehydration, whereas the role of subducted sediments in mantle-wedge metasomatization was subordinate, implying that the subduction system was sediment-starved and far from continental clastic input. The amount of slab-derived fluids was enough to produce hydrous magmas that follow the tholeiitic but not the calc-alkaline differentiation trend. With Mg# > 64, few samples of Wadi Ranga mafic volcanic rocks are similar to primitive arc magmas, whereas the other samples have clearly experienced considerable fractional crystallization.The low abundances of trace elements, together with low K₂O contents of the felsic metavolcanic rocks indicate that they were erupted in a primitive island arc setting. The felsic volcanic rocks are characterized by lower K/Rb ratios compared to the mafic volcanic rocks, higher trace element abundances (~ 2 to ~ 9 times basalt) on primitive arc basalt-normalized pattern and nearly flat chondrite-normalized REE patterns, which display a negative Eu anomaly. These features are largely consistent with fractional crystallization model for the origin of the felsic volcanic rocks. Moreover, SiO₂-REE variations for the Wadi Ranga volcanic rocks display steadily increasing LREE over the entire mafic to felsic range and enriched La abundances in the felsic lavas relative to the most mafic lavas, features which are consistent with production of the felsic volcanic rocks through fractional crystallization of basaltic melts. The relatively large volume of Wadi Ranga silicic volcanic rocks implies that significant volume of silicic magmas can be generated in immature island arcs by fractional crystallization and indicates the significant role of intra-oceanic arcs in the production of Neoproterozoic continental crust. We emphasize that the geochemical characteristics of these rocks such as their low LILE and nearly flat REE patterns can successfully discriminate them from other Egyptian Neoproterozoic felsic volcanic rocks, which have higher LILE, Zr and Nb and fractionated REE patterns.     

Neoproterozoic contaminated MORB of Wadi Ghadir ophiolite,NE Africa: Geochemical and Nd and Sr isotopic constraints

The ophiolitic metabasalts (pillowed and sheeted dikes) of Wadi Ghadir area, Eastern Desert, Egypt, were analyzed for their major, trace and rare earth elements, Nd and Sr isotopes and the chemistry of their plagioclase, amphibole and chlorite was also reported. Geochemically these rocks range from tholeiitic basalt to basaltic andesite. The generally low MgO, Cr and Ni and high Zr contents are consistent with derivation of these rocks from an evolved magma. The high TiO₂ contents (mostly between 1.76% and 2.23%) classify Wadi Ghadir ophiolitic metabasalts as MORB ophiolite. The chondrite-normalized REE patterns of most samples display small LREE-enrichment with (La/Yb)_n ranging from 1.44 to 2.56. The MORB-normalized spider diagram shows variable LILE abundances, which are either similar to or enriched relative to MORB, and most samples display small Nb depletion. The abundances of some LILE (Ba, Rb and K) as well as Na and Si were modified by post-magmatic seafloor hydrothermal alteration. Enrichment of the least mobile LILE (Th & U) indicates that Wadi Ghadir ophiolitic metabasalts are akin to C (contaminated)-MORB. These geochemical characteristics are similar to BABB modified by contamination.Wadi Ghadir metabasalts have low initial Sr ratios (0.7010–0.7034) which are similar to those of MORB, while their ε_Nd(t) values (+7.7–+4.5) are either more or less positive than the value of depleted mantle (DM). The more positive ε_Nd(t) values indicate DM source for these basalts, while the less positive ε_Nd(t) values reflect the involvement of slightly older component in Wadi Ghadir ophiolite. We suggest that the parent magma of metabasalts was contaminated by slightly older material, most probably oceanic-arc crustal rocks, which caused enrichment in LREE, and by analogy LILE, but did not significantly affect Nd isotopic systematics or modify Sr isotopes.Such contaminated MORB character also revealed by other ophiolitic metavolcanics in the Central Eastern Desert, contrasting the N-MORB character of the Gerf ophiolite in the South Eastern Desert. Moreover, the present work suggests the increase of the degree of contamination of the ophiolitic metabasalts from south to north in the Central Eastern Desert.