Numerical Solution of the Time-Depending Flow of Immiscible Fluids with Fuzzy Boundary Conditions

Fluid flow modeling using fuzzy boundary conditions is one of the viable areas in biofluid mechanics, drug suspension in pharmacology, as well as in the cytology and electrohydrodynamic analysis of cerebrospinal fluid data. In this article, a fuzzy solution for the two immiscible fluid flow problems is developed, which is motivated by biomechanical flow engineering. Two immiscible fluids, namely micropolar and Newtonian fluid, are considered with fuzzy boundary conditions in the horizontal channel. The flow is considered unsteady and carried out by applying a constant pressure gradient in the X-direction of the channel. The coupled partial differential equations are modeled for fuzzy profiles of velocity and micro-rotation vectors then the numerical results are obtained by the modified cubic B - spline differential quadrature method. The evolution of membership grades for velocity and microrotation profiles has been depicted with the fuzzy boundaries at the channel wall. It is observed that Micropolar fluid has a higher velocity change than Newtonian fluid, and both profiles indicate a declining nature toward the interface.

Harmonic waves of a given azimuthal number in a micropolar cylindrical waveguide

Рассматривается система двух связанных векторных дифференциальных уравнений линейной теории микрополярной упругости, сформулированная в терминах перемещений и микровращений в случае гармонической зависимости перемещений и микровращений от времени. Вводятся потенциалы перемещений и микровращений. Выполнено расщепление связанных векторных дифференциальных уравнений микрополярной теории упругости для потенциалов на несвязанные винтовые уравнения, опираясь на пропорциональность (с разными масштабными факторами) вихревых составляющих перемещений и микровращений только одному вихревому винтовому полю. Найдено представление векторов перемещений и микровращений с помощью четырех винтовых векторов. Оно обеспечивает выполнимость связанных векторных дифференциальных уравнений линейной теории микрополярной упругости. Проблема нахождения вихревых составляющих перемещений и микровращений приведена к решению четырех несвязанных между собой векторных винтовых дифференциальных уравнений. Получено представление перемещений и микровращений с помощью двух несвязанных метагармонических векторов. Выполнено разделение пространственных переменных в уравнениях Гельмгольца в цилиндрической системе координат. Определены решения скалярного и векторного уравнений Гельмгольца в бесконечной цилиндрической области, содержащие ряд произвольных постоянных. В явном виде найдены представления векторов перемещений и микровращений в длинном линейном микрополярном цилиндре, содержащие восемь произвольных постоянных. Такого рода решения определяют формы гармонических волн перемещений и микровращений, распространяющихся вдоль оси длинного кругового цилиндра. Полученные представления для гармонических волн перемещений и микровращений имеют смысл только для волн, характеризующихся заданным азимутальным числом. The coupled system of vector differential equations of the linear theory of micropolar elasticity presented in terms of displacements and micro-rotations in the case of a harmonic dependence of physical fields on time is considered in the three different variants of which the two are due to W. Nowacki and H. Neuber. A new scheme of splitting the coupled vector differential equation of the linear theory of micropolar elasticity into uncoupled ones is proposed. The scheme is based on proportionality of the vortex parts of the displacements and micro-rotations to the single vector, which satisfies the screw equation. The problem of determination of the vortex parts of the displacements and micro-rotations fields is reduced to solution of four uncoupled screw differential equations. A new representation of displacement and micro-rotation vectors is obtained by using two uncoupled metaharmonic vectors. The separation of spatial variables in the Helmholtz metaharmonic equations in a cylindrical coordinate net is described. Solutions of the scalar and vector Helmholtz equations in an infinite cylindrical domain containing a series of arbitrary constants are obtained. Representation of displacement and micro-rotation vectors in a long micropolar cylinder containing eight arbitrary constants are explicitly found. The corresponding solutions are proved to determine the modes of harmonic waves of displacements and micro-rotations propagating along the axis of a long circular cylinder. The obtained modes of the harmonic displacements and micro-rotations waves are valid only for those characterized by a given azimuthal number.

Detecção e avaliação de efeitos geodinâmicos em uma porção das estações GNSS brasileiras

Neste trabalho apresentamos uma metodologia para detecção de possíveis efeitos geodinâmicos, utilizando dados GNSS, modelos de velocidade SIRGAS, Geológico e Geofísico. As observações GNSS foram processadas para obtenção da variação da coordenada vertical e horizontal. Os modelos de velocidade VEMOS2009 e 2015 serviram como base comparativa para verificar os resultados obtidos com o processamento GNSS. A partir do conjunto de vetores de velocidades das estações da Rede Brasileira de Monitoramento Contínuo (RBMC) obtidos do NNR-NUVEL-1A, estimaram-se os elementos definidores do movimento da placa SOAM. Em seguida, foram realizadas algumas análises e comparações com os vetores de rotação da placa SOAM obtidas pelo modelo APKIM2008. As análises conduzidas tiveram por base observações contínuas, desde 2007 até 2016, junto a uma porção de estações GNSS pertencentes à RBMC (estações SIRGAS-CON). Os resultados considerando os modelos de velocidade de SIRGAS mostraram que após o terremoto no Chile em algumas regiões do Brasil houveram efeitos geodinâmico. Em comparação aos modelos geológico e geofísico observou-se que o campo de velocidade definido no processamento GNSS conseguiu retratar a realidade, como caso de estudo foi considerado a estação de Imbituba. Detection and assessment of geodynamic effects on a portion of Brazilian GNSS stations A B S T R A C TThis work presents a methodology for detecting possible geodynamic effects using GNSS data, SIRGAS, Geological, and Geophysical velocity models. GNSS observations were processed to obtain the variation for the vertical (up) and horizontal coordinates. The VEMOS2009 and 2015 velocity models served as a comparative basis to verify the results obtained with GNSS processing. From the set of velocity vectors of stations belonging to the Brazilian Network for Continuous Monitoring of the GNSS (RBMC) obtained from NNR-NUVEL-1A, estimated the defining elements of the movement of the SOAM plate. Then, it performed some analyses and comparisons with the SOAM plate rotation vectors obtained by the APKIM2008 model. The studies conducted were based on continuous observations, from 2007 to 2016, on a portion of GNSS stations belonging to RBMC (SIRGAS-CON stations). The results considering the SIRGAS velocity models showed that after the earthquake in Chile, in some regions of Brazil, there were geodynamic effects. Comparison to the geological and geophysical models, it was observed that the velocity field defined in GNSS processing was able to portray reality, as well as case study, which was considered the Imbituba station.Keywords: geodynamic, geophysics, geologic, lithospheric plate.

A New SLM-UFMC Model for Universal Filtered Multi-Carrier to Reduce Cubic Metric and Peak to Average Power Ratio in 5G Technology

The new generation of wireless communication systems has adopted different waveforms. The universal filtered multicarrier is one of the adopted candidates that has symmetry with various numerology designs. However, the high peak to average power ratio is one of the major limitations faced by universal filter multicarrier (UFMC) designers. Moreover, recent studies utilize cubic metric along with the peak to average power ratio (PAPR) to show the power back-off effect of the signal in which the PAPR metric identifies the maximum peak and the cubic metric (CM) identifies the Out of Band emission and In-Band distortion. Most of the current solutions, such as amplitude clipping, tone reservation, and active constellation extension, decrease the PAPR but cause degradation to the bit error rate. Selected mapping is one of the promising techniques that is recently used to solve the PAPR and CM problems without causing bit error rate (BER) degradation. In this paper, the selected mapping (SLM) is integrated with UFMC to reduce the PAPR and CM without affecting the BER of 5G networks. The SLM-UFMC solution model is simulated by MATLAB and the results show that the SLM-UFMC model presents better PAPR and CM performance without BER degradation. The PAPR has been decreased to 1.5 dB with respect to eight-phase rotation vectors and the CM decreased to 1.25 dB compared to the conventional UFMC.

A condition that implies full homotopical complexity of orbits for surface homeomorphisms

We consider closed orientable surfaces $S$ of genus $g>1$ and homeomorphisms $f:S\rightarrow S$ isotopic to the identity. A set of hypotheses is presented, called a fully essential system of curves $\mathscr{C}$ and it is shown that under these hypotheses, the natural lift of $f$ to the universal cover of $S$ (the Poincaré disk $\mathbb{D}$), denoted by $\widetilde{f},$ has complicated and rich dynamics. In this context, we generalize results that hold for homeomorphisms of the torus isotopic to the identity when their rotation sets contain zero in the interior. In particular, for $C^{1+\unicode[STIX]{x1D716}}$ diffeomorphisms, we show the existence of rotational horseshoes having non-trivial displacements in every homotopical direction. As a consequence, we found that the homological rotation set of such an $f$ is a compact convex subset of $\mathbb{R}^{2g}$ with maximal dimension and all points in its interior are realized by compact $f$-invariant sets and by periodic orbits in the rational case. Also, $f$ has uniformly bounded displacement with respect to rotation vectors in the boundary of the rotation set. This implies, in case where $f$ is area preserving, that the rotation vector of Lebesgue measure belongs to the interior of the rotation set.

Planet formation and stability in polar circumbinary discs

Context. Dynamical studies suggest that most circumbinary discs (CBDs) should be coplanar (i.e. the rotation vectors of the binary and the disc should be aligned). However, some theoretical works show that under certain conditions a CBD can become polar, which means that its rotation vector is orthogonal with respect to the binary orbital plane. Interestingly, very recent observations show that polar CBDs exist in nature (e.g. HD 98800). Aims. We test the predictions of CBD alignment around eccentric binaries based on linear theory. In particular, we compare prograde and retrograde CBD configurations. Then, assuming planets form in these systems, we thoroughly characterise the orbital behaviour and stability of misaligned (P-type) particles. This is done for massless and massive particles. Methods. The evolution of the CBD alignment for various configurations was modelled through three-dimensional hydrodynamical simulations. For the orbital characterisation and the analysis stability, we relied on long-term N-body integrations and structure and chaos indicators, such as Δe and MEGNO. Results. We confirm previous analytical predictions on CBD alignment, but find an unexpected symmetry breaking between prograde and retrograde configurations. More specifically, we observe polar alignment for a retrograde misaligned CBD that was expected to become coplanar with respect to the binary disc plane. Therefore, the likelihood of becoming polar for a highly misaligned CBD is higher than previously thought. Regarding the stability of circumbinary P-type planets (also know as Tatooines), polar orbits are stable over a wide range of binary parameters. In particular, for binary eccentricities below 0.4 the orbits are stable for any value of the binary mass ratio. In the absence of gas, planets with masses below 10−5 M⊙ have negligible effects on the binary orbit. Finally, we suggest that mildly eccentric equal-mass binaries should be searched for polar Tatooines.

Low-Complexity Blind Selected Mapping Scheme for Peak-to-Average Power Ratio Reduction in Orthogonal Frequency-Division Multiplexing Systems

Orthogonal frequency-division multiplexing (OFDM) is an attractive multicarrier technique for the simplicity of equalization and high data throughput. However, the transmitted OFDM signal has a very high peak-to-average power ratio (PAPR), which severely degrades the performance of practical OFDM systems and reduces the efficiency of high-power amplifiers (HPA). The selected mapping (SLM) scheme is an effective PAPR reduction method of OFDM signals. However, this approach usually requires side information (SI) transmission, which increases the difficulty of the hardware implementation with high complexity and reduces the data transmission rate. In this paper, based on designing phase rotation vectors in the time domain, a novel blind SLM method with low complexity is proposed to reduce the PAPR of OFDM signals. At the transmitter, the proposed method properly designs the phase rotation vectors in the time domain, which can be considered as an equivalent wireless channel without SI transmission. At the receiver, the effect of phase rotation vectors can be removed by the conventional channel estimation method, and the data demodulation processing can be easily performed by the frequency domain equalization. Simulation results show that the proposed scheme can achieve low complexity in PAPR reduction and has great robustness in bit error rate (BER) performance compared to the other low-complexity SLM PAPR schemes.

Homotopical complexity of a billiard flow on the 3D flat torus with two cylindrical obstacles

We study the homotopical rotation vectors and the homotopical rotation sets for the billiard flow on the unit flat torus with two disjoint and orthogonal toroidal (cylindrical) scatterers removed from it. The natural habitat for these objects is the infinite cone erected upon the Cantor set $\text{Ends}(G)$ of all ‘ends’ of the hyperbolic group $G=\unicode[STIX]{x1D70B}_{1}(\mathbf{Q})$. An element of $\text{Ends}(G)$ describes the direction in (the Cayley graph of) the group $G$ in which the considered trajectory escapes to infinity, whereas the height function $s$ ($s\geq 0$) of the cone gives us the average speed at which this escape takes place. The main results of this paper claim that the orbits can only escape to infinity at a speed not exceeding $\sqrt{3}$ and, in any direction $e\in \text{Ends}(\unicode[STIX]{x1D70B}_{1}({\mathcal{Q}}))$, the escape is feasible with any prescribed speed $s$, $0\leq s\leq 1/(\sqrt{6}+2\sqrt{3})$. This means that the radial upper and lower bounds for the rotation set $R$ are actually pretty close to each other. Furthermore, we prove the convexity of the set $\mathit{AR}$ of constructible rotation vectors, and that the set of rotation vectors of periodic orbits is dense in $\mathit{AR}$. We also provide effective lower and upper bounds for the topological entropy of the studied billiard flow.

Spacelike surface geometry

In this paper, by considering [Formula: see text] and [Formula: see text] parameter curves on spacelike surface [Formula: see text], [Formula: see text] and [Formula: see text], respectively, and any spacelike curve [Formula: see text] that passes through the intersection point of these parameter curves, we have found the Darboux instantaneous rotation vectors of Darboux trihedrons of these three curves, as follows: [Formula: see text] [Formula: see text] [Formula: see text] and we have obtained the relationship between these vectors as [Formula: see text] where [Formula: see text] and [Formula: see text] are the spacelike angles between tangent vectors of [Formula: see text] and [Formula: see text] curves, and of [Formula: see text] and [Formula: see text] curves, respectively. [Formula: see text] is the unit normal vector of the surface. Besides, we have given Euler, Liouville, Bonnet formulas and Gauss curvature of the spacelike surface with new statement.

Ground states and zero-temperature measures at the boundary of rotation sets

We consider a continuous dynamical system $f:X\rightarrow X$ on a compact metric space $X$ equipped with an $m$-dimensional continuous potential $\unicode[STIX]{x1D6F7}=(\unicode[STIX]{x1D719}_{1},\ldots ,\unicode[STIX]{x1D719}_{m}):X\rightarrow \mathbb{R}^{m}$. We study the set of ground states $GS(\unicode[STIX]{x1D6FC})$ of the potential $\unicode[STIX]{x1D6FC}\cdot \unicode[STIX]{x1D6F7}$ as a function of the direction vector $\unicode[STIX]{x1D6FC}\in S^{m-1}$. We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of $\unicode[STIX]{x1D6F7}$. In particular, for each $\unicode[STIX]{x1D6FC}$ the set of rotation vectors of $GS(\unicode[STIX]{x1D6FC})$ forms a non-empty, compact and connected subset of a face $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ of the rotation set associated with $\unicode[STIX]{x1D6FC}$. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$. We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any $m\in \mathbb{N}$ examples with an exposed boundary point (that is, $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of $GS(\unicode[STIX]{x1D6FC})$ is a non-trivial line segment.