Abstract:
The jet flow problem concerning the discharge of a fluid (from an orifice in a container)into the atmosphere is studied herein in the framework of the Helmholtz-Kirchhoff model. Theproblem is reduced to the study of a system of nonlinear equations. Using Leray-Schauder's fixedpoint theorem we prove that the system of functional equations has at least one solution. Then wepresent a semi-inverse method which gives us the possibility to calculate numerically the unknownfree lines for symmetric jets whose walls consist of semi-infinite straight lines and arcs of circle andfor non-symmetric jets whose walls consist of semi-infinite straight lines.

Abstract:
This article deals with the flow of Newtonian fluids through axially-symmetric corrugated tubes. An analytical method to derive the relation between volumetric flow rate and pressure drop in laminar flow regimes is presented and applied to a number of simple tube geometries of converging-diverging nature. The method is general in terms of fluid and tube shape within the previous restrictions. Moreover, it can be used as a basis for numerical integration where analytical relations cannot be obtained due to mathematical difficulties.

Abstract:
It has been noted in the past, in the study of the wall-roughness impedance, that a metallic pipe with corrugated walls supports propagation of a high-frequency mode that is in resonance with a relativistic beam. This mode can be excited by a beam whose length is a fraction of the wavelength. In this paper, we study another option of excitation of the resonant mode in a metallic pipe with corrugated walls---via the mechanism of the free electron laser instability. This mechanism works if the bunch length is much longer than the wavelength of the radiation.

Abstract:
Starting from the rigorous excitation equation, the propagation of waves through a 2D waveguide with the periodically corrugated finite-length insert is examined in detail. The corrugation profile is chosen to obey the property that its amplitude is small as compared to the waveguide width, whereas the sharpness of the asperities is arbitrarily large. With the aid of the method of mode separation, which was developed earlier for inhomogeneous-in-bulk waveguide systems [Waves Random Media \textbf{10}, 395 (2000)], the corrugated segment of the waveguide is shown to serve as the effective scattering barrier whose width is coincident with the length of the insert and the average height is controlled by the sharpness of boundary asperities. Due to this barrier, the mode spectrum of the waveguide can be substantially rarefied and adjusted so as to reduce the number of extended modes to the value arbitrarily less than that in the absence of corrugation (up to zero), without changing considerably the waveguide average width.

Abstract:
The paper presents a numerical investigation of non-Newtonian modeling effects on unsteady periodic flows in a two-dimensional (2D) pipe with two idealized stenoses of 75% and 50% degrees, respectively. The governing Navier-Stokes equations have been modified using the Cartesian curvilinear coordinates to handle complex geometries. The investigation has been carried out to characterize four different non-Newtonian constitutive equations of blood, namely, the (i) Carreau, (ii) Cross, (iii) Modified Casson, and (iv) Quemada models. The Newtonian model has also been analyzed to study the physics of fluid and the results are compared with the non-Newtonian viscosity models. The numerical results are represented in terms of streamwise velocity, pressure distribution, and wall shear stress (WSS) as well as the vorticity, streamlines, and vector plots indicating recirculation zones at the poststenotic region. The results of this study demonstrate a lower risk of thrombogenesis at the downstream of stenoses and inadequate blood supply to different organs of human body in the Newtonian model compared to the non-Newtonian ones. 1. Introduction Stenosis is characterized by localized arterial narrowing that is initiated due to deposition of lipid, cholesterol, and some other substances on the endothelium and is of major concern to most of the Western world. Atherosclerotic lesions preferentially occur in arteries and arterioles in regions of high curvature or bifurcations and junctions causing major changes in flow structure and consequently large changes in fluid loading on vessel walls [1]. Such plaques or arterial constrictions usually disturb normal blood flow through the artery and there is considerable evidence that hydrodynamic factors can play a significant role in the development and progression of these lesions. It has been established that once a mild stenosis is developed inside the arterial lumen, the resulting flow disorder further influences the development of the disease and the arterial deformability to some extent, which eventually changes the regional blood rheology as well [2]. The rheological behavior of blood can be identified by non-Newtonian viscosity. Halder [3] demonstrated that the rheology of blood and the fluid dynamical properties of blood flow can play an important role in the basic understanding, diagnosis, and treatment of many cardiovascular and arterial diseases. Now, stenosis not only develops in one position of artery but also it may develop at more than one location of the cardiovascular system. However, in many medical

Abstract:
Mode coupling due to curvature in circumferentially corrugated circular waveguides is analyzed in this paper. Starting from Maxwell's equations in the annular orthogonal curvilinear coordinate-system, the integral formulae for coupling coefficients are derived. On the basis of them, the explicit expressions of coupling coefficients between positive or opposite modes are derived.

Abstract:
Benney's theory of evolution of disturbances in shear flows over smooth and flat boundary is extended to study for shear flows over combined spatially and temporally corrugated walls. Perturbation and multiple-scales analyses are employed for the case where both amplitude of the corrugations and the amplitude of wave motion are small. Analyses for instability of modulated mean shear flows with respect to spanwise-periodic disturbance rolls and for the subsequent vortex formation and vortex stability are presented, and the effects of the corrugated walls on the resulting flow and vortices are determined. It is found that particular corrugated walls can originate and control the longitudinal vortices, while some other types of corrugated walls can enhance instability of such vortices.

Abstract:
The multi-instanton solutions by 'tHooft and Jackiw, Nohl & Rebbi are generalized to curvilinear coordinates. Expressions can be notably simplified by the appropriate gauge transformation. This generates the compensating addition to the gauge potential of pseudoparticles. Singularities of the compensating connection are irrelevant for physics but affect gauge dependent quantities.

Abstract:
We define the \emph{curvilinear rank} of a degree $d$ form $P$ in $n+1$ variables as the minimum length of a curvilinear scheme, contained in the $d$-th Veronese embedding of $\mathbb{P}^n$, whose span contains the projective class of $P$. Then, we give a bound for rank of any homogenous polynomial, in dependance on its curvilinear rank.