STUDY OF SELF-SUPERPOSABLE LIQUID IN OBLATE SPHEROIDAL SHAPE

The paper studiesthe self-superposable motion of a liquid of a fluid which is incompressible in nature in oblate spheroidal shape. An incompressible fluid is defined as the fluid whose volume or density does not change with pressure. Thus, the main aim of this paper is to solve the basic equations of fluid dynamics in oblate spheroidal coordinates considering self-superposable nature of the fluid. The paper includes the study of nature of vorticity and irrotationality and has not considered the boundary conditions in theanalysis. Lastly, the paper determines the pressure distribution and the solutions contain a set of constants.

Volume Integral Equation Method Solution for Spheroidal Inclusion Problem

In this paper, the volume integral equation method (VIEM) is introduced for the numerical analysis of an infinite isotropic solid containing a variety of single isotropic/anisotropic spheroidal inclusions. In order to introduce the VIEM as a versatile numerical method for the three-dimensional elastostatic inclusion problem, VIEM results are first presented for a range of single isotropic/orthotropic spherical, prolate and oblate spheroidal inclusions in an infinite isotropic matrix under uniform remote tensile loading. We next considered single isotropic/orthotropic spherical, prolate and oblate spheroidal inclusions in an infinite isotropic matrix under remote shear loading. The authors hope that the results using the VIEM cited in this paper will be established as reference values for verifying the results of similar research using other analytical and numerical methods.

The Theory of the Hydrogen Molecule Ion, Scalar Beams, and Scattering by Spheroids

<p>Schrodinger's equation for the hydrogen molecule ion and the Helmholtz equation are separable in prolate and oblate spheroidal coordinates respectively. They share the same form of the angular equation. The first task in deriving the ground state energy of the hydrogen molecule ion, and in obtaining finite solutions of the Helmholtz equation, is to obtain the physically allowed values of the separation of variables parameter. The separation parameter is not known analytically, and since it can only have certain values, it is an important parameter to quantify. Chapter 2 of this thesis investigates an exact method of obtaining the separation parameter. By showing that the angular equation is solvable in terms of confluent Heun functions, a new method to obtain the separation parameter was obtained. We showed that the physically allowed values of the separation of variables parameter are given by the zeros of the Wronskian of two linearly dependent solutions to the angular equation. Since the Heun functions are implemented in Maple, this new method allows the separation parameter to be calculated to unlimited precision. As Schrodinger's equation for the hydrogen molecule ion is related to Helmholtz's equation, this warranted investigation of scalar beams. Tightly focused optical and quantum particle beams are described by exact solutions of the Helmholtz equation. In Chapter 3 of this thesis we investigate the applicability of the separable spheroidal solutions of the scalar Helmholtz equation as physical beam solutions. By requiring a scalar beam solution to satisfy certain physical constraints, we showed that the oblate spheroidal wave functions can only represent nonparaxial scalar beams when the angular function is odd, in terms of the angular variable. This condition ensures the convergence of integrals of physical quantities over a cross-section of the beam and allows for the physically necessary discontinuity in phase at z = 0 on the ellipsoidal surfaces of otherwise constant phase. However, these solutions were shown to have a discontinuous longitudinal derivative. Finally, we investigated the scattering of scalar waves by oblate and prolate spheroids whose symmetry axis is coincident with the direction of the incident plane wave. We developed a phase shift formulation of scattering by oblate and prolate spheroids, in parallel with the partial wave theory of scattering by spherical obstacles. The crucial step was application of a finite Legendre transform to the Helmholtz equation in spheroidal coordinates. Analytical results were readily obtained for scattering of Schrodinger particle waves by impenetrable spheroids and for scattering of sound waves by acoustically soft spheroids. The advantage of this theory is that it enables all that can be done for scattering by spherical obstacles to be carried over to the scattering by spheroids, provided the radial eigenfunctions are known.</p>

The Theory of the Hydrogen Molecule Ion, Scalar Beams, and Scattering by Spheroids

<p>Schrodinger's equation for the hydrogen molecule ion and the Helmholtz equation are separable in prolate and oblate spheroidal coordinates respectively. They share the same form of the angular equation. The first task in deriving the ground state energy of the hydrogen molecule ion, and in obtaining finite solutions of the Helmholtz equation, is to obtain the physically allowed values of the separation of variables parameter. The separation parameter is not known analytically, and since it can only have certain values, it is an important parameter to quantify. Chapter 2 of this thesis investigates an exact method of obtaining the separation parameter. By showing that the angular equation is solvable in terms of confluent Heun functions, a new method to obtain the separation parameter was obtained. We showed that the physically allowed values of the separation of variables parameter are given by the zeros of the Wronskian of two linearly dependent solutions to the angular equation. Since the Heun functions are implemented in Maple, this new method allows the separation parameter to be calculated to unlimited precision. As Schrodinger's equation for the hydrogen molecule ion is related to Helmholtz's equation, this warranted investigation of scalar beams. Tightly focused optical and quantum particle beams are described by exact solutions of the Helmholtz equation. In Chapter 3 of this thesis we investigate the applicability of the separable spheroidal solutions of the scalar Helmholtz equation as physical beam solutions. By requiring a scalar beam solution to satisfy certain physical constraints, we showed that the oblate spheroidal wave functions can only represent nonparaxial scalar beams when the angular function is odd, in terms of the angular variable. This condition ensures the convergence of integrals of physical quantities over a cross-section of the beam and allows for the physically necessary discontinuity in phase at z = 0 on the ellipsoidal surfaces of otherwise constant phase. However, these solutions were shown to have a discontinuous longitudinal derivative. Finally, we investigated the scattering of scalar waves by oblate and prolate spheroids whose symmetry axis is coincident with the direction of the incident plane wave. We developed a phase shift formulation of scattering by oblate and prolate spheroids, in parallel with the partial wave theory of scattering by spherical obstacles. The crucial step was application of a finite Legendre transform to the Helmholtz equation in spheroidal coordinates. Analytical results were readily obtained for scattering of Schrodinger particle waves by impenetrable spheroids and for scattering of sound waves by acoustically soft spheroids. The advantage of this theory is that it enables all that can be done for scattering by spherical obstacles to be carried over to the scattering by spheroids, provided the radial eigenfunctions are known.</p>

Non-Equilibrium Thermodynamics-Based Convective Drying Model Applied to Oblate Spheroidal Porous Bodies: A Finite-Volume Analysis

Commonly based on the liquid diffusion theory, drying theoretical studies in porous materials has been directed to plate, cylinder, and sphere, and few works are applied to non-conventional geometries. In this sense, this work aims to study, theoretically, the drying of solids with oblate spheroidal geometry based on the thermodynamics of irreversible processes. Mathematical modeling is proposed to describe, simultaneously, the heat and mass transfer (liquid and vapor) during the drying process, considering the variability of the transport coefficients and the convective boundary conditions on the solid surface, with particular reference to convective drying of lentil grains at low temperature and moderate air relative humidity. All the governing equations were written in the oblate spheroidal coordinates system and solved numerically using the finite-volume technique and the iterative Gauss–Seidel method. Numerical results of moisture content, temperature, liquid, vapor, and heat fluxes during the drying process were obtained, analyzed, and compared with experimental data, with a suitable agreement. It was observed that the areas near the focal point of the lentil grain dry and heat up faster; consequently, these areas are more susceptible to the appearance of cracks that can compromise the quality of the product. In addition, it was found that the vapor flux was predominant during the drying process when compared to the liquid flux.

Pollen morphology of Malvaceae genera from Saudi Arabia and its taxonomic significance

Pollen morphology of 20 species belong to seven genera (Abutilon, Althaea, Hibiscus, Malva, Pavonia, Senra and Sida) of Malvaceae from Saudi Arabia were studied by using light microscope (LM) and scanning electron microscope (SEM). Quantitative and qualitative pollen morphological characters which vary among investigated taxa are found in the pollen polarity, symmetry, size, shape, polar axis, equatorial diameter, P/E ratio, average height and width of spine, aperature character and spine index. The pollen grains vary from spheroidal, prolate spheroidal, oblate spheroidal to suboblate. All taxa were characterized by relatively large to medium sized pollen grains, numerous pores scattered irregularly all over the grain, and echinate sculpturing. Sida ovata is the largest size pollen grain (138.95) µm. On the other hand, Malva parviflora showed the smallest pollen size (52.28 µm). The average height and width of spine varied greatly among studied taxa. The highest spines (20.65µm) found in Sida ovata, while the shortest (3.19 µm) was found in Abutilon pannosum. Results of the pollen shape, size, and exine sculpture characters offered useful data for evaluating the taxonomy of Malvaceae both on subgeneric and sectional levels. A key for the identification of the investigated taxa based on pollen grains characters is also provided

Pollen Morphology of the Genus Tragopogon (Asteraceae)

Tragopogon L. (Cichorioideae, Lactuceae, Scorzonerinae) is an Old World genus with 150 species. Pollen morphology has proved useful in the systematics of some genera and species of Asteraceae as well as in that of some of its genera and species. The pollen morphology of 24 taxa of the genus Tragopogon was investigated in detail by scanning electron microscopy (SEM). The pollen grain type ranged from suboblate, oblate-spheroidal to prolate-spheroidal in equatorial view and hexagonal, obtuse-hexagonal to hexagonal-angular in polar view. In this study separation of the species of the sections Majores, Profundisulcati, Sosnovsky, Chromopappus, Rubriflori according to Flora Iranica is presented from the other species of Tragopogon. T. jezdianus, T. porphyrocephalus, T. rezaiyensis are suggested to belong to Rubriflori section. The results indicate that the palynological characters of the genus Tragopogon are valuable for taxonomic applications and are useful for classification.

Proxapertites from Walat Formation, Sukabumi, West Java, Indonesia

Proxapertites have become one of the most significant indicators of ancient rock in Indonesia. Walat Formation is one of the oldest rocks exposed in Sukabumi, West Java, Indonesia. These Proxapertites have not been described in detail about their characteristics in previous studies, especially on Walat Formation. Therefore, knowing the characteristics of Proxapertites becomes interesting, especially in Walat Formation, which can be a reference for the characteristics of the late Eocene Proxapertites (37.8 - 33.9 million) in Indonesia. Acetolysis method was carried out for the preparation of pollen and spores; description and determination were carried out to see the characteristics of Proxapertites that present in Walat Formation. Result, there are three types of Proxapertites. Proxapertites operculatus have sizes 23 – 86 µ with average 40.5 (P) and 51.1 (E) µ, index PE 0,43 - 1, Peroblate – Subspheroidal – Oblate Spheroidal, Fine-Reticulate ornamentation, and Asymmetric Monosulcate aperture. Proxapertites cursus have sizes 23 – 86 µ with average 39.8 (P) and 49.8 (E) µ, index PE 0.51 - 1, Oblate – Subspheroidal – Oblate Spheroidal, Reticulate ornamentation, and Asymmetric Monosulcate aperture. Whereas Proxapertites psilatus have sizes 29 – 75 µ with average 42.3 (P) and 52.5 (E) µ, index PE 0.58 - 1, Oblate – Subspheroidal – Oblate Spheroidal, Psilate ornamentation, and Asymmetric Monosulcate aperture. These three Proxapertites can be distinguished by their type of ornamentation. Meanwhile, other aspects have similar characteristics and are affected by the appearance of individual pollen on the slide during preparation.