Half-Monopole in the Weinberg-Salam Model

We present new axially symmetric half-monopole configuration of the SU(2)$\times$U(1) Weinberg-Salam model of electromagnetic and weak interactions. The half-monopole configuration possesses net magnetic charge $2\pi/e$ which is half the magnetic charge of a Cho-Maison monopole. The electromagnetic gauge potential is singular along the negative $z$-axis. However the total energy is finite and increases only logarithmically with increasing Higgs field self-coupling constant $\lambda^{1/2}$ at $\sin^2\theta_W=0.2312$. In the U(1) magnetic field, the half-monopole is just a one dimensional finite length line magnetic charge extending from the origin $r=0$ and lying along the negative $z$-axis. In the SU(2) 't Hooft magnetic field, it is a point magnetic charge located at $r=0$. The half-monopole possesses magnetic dipole moment that decreases exponentially fast with increasing Higgs field self-coupling constant $\lambda^{1/2}$ at $\sin^2\theta_W=0.2312$.Comment: 14 pages, 3 Figure

MAP, MAC, and Vortex-rings Configurations in the Weinberg-Salam Model

We report on the presence of new axially symmetric monopoles, antimonopoles and vortex-rings solutions of the SU(2)$\times$U(1) Weinberg-Salam model of electromagnetic and weak interactions. When the $\phi$-winding number $n=1$, and 2, the configurations are monopole-antimonopole pair (MAP) and monopole-antimonopole chain (MAC) with poles of alternating sign magnetic charge arranged along the $z$-axis. Vortex-rings start to appear from the MAP and MAC configurations when the winding number $n=3$. The MAP configurations possess zero net magnetic charge whereas the MAC configurations possess net magnetic charge of $4\pi n/e$. In the MAP configurations, the monopole-antimonopole pair is bounded by the ${\cal Z}^0$ field flux string and there is an electromagnetic current loop encircling it. The monopole and antimonopole possess magnetic charges $\pm\frac{4\pi n}{e}\sin^2\theta_W$ respectively. In the MAC configurations there is no string connecting the monopole and the adjacent antimonopole and they possess magnetic charges $\pm\frac{4\pi n}{e}$ respectively. The MAC configurations possess infinite total energy and zero magnetic dipole moment whereas the MAP configurations which are actually sphalerons possess finite total energy and magnetic dipole moment. The configurations were investigated for varying values of Higgs self-coupling constant $0\leq \lambda\leq 40$ at Weinberg angle $\theta_W=\frac{\pi}{4}$.Comment: 31 pages, 10 figures, 2 table

Electrically Charged One and a Half Monopole Solution

Recently, we have discussed the coexistence of a finite energy one-half monopole and a 't Hooft-Polyakov monopole of opposite magnetic charges. In this paper, we would like to introduce electric charge into this new monopoles configuration, thus creating a one and a half dyon. This new dyon possesses finite energy, magnetic dipole moment and angular momentum and is able to precess in the presence of an external magnetic field. Similar to the other dyon solutions, when the Higgs self-coupling constant, $\lambda$, is nonvanishing, this new dyon solution possesses critical electric charge, total energy, magnetic dipole moment, and dipole separation as the electric charge parameter, $\eta$, approaches one. The electric charge and total energy increase with $\eta$ to maximum critical values as $\eta\rightarrow1$ for all nonvanishing $\lambda$. However, the magnetic dipole moment decreases with $\eta$ when $\lambda\geq0.1$ and the dipole separation decreases with $\eta$ when $\lambda\geq1$ to minimum critical values as $\eta\rightarrow1$.Comment: 24 pages, 7 figures. arXiv admin note: text overlap with arXiv:1208.4893, arXiv:1112.149

A Geometrical Perspective for the Bargaining Problem

A new treatment to determine the Pareto-optimal outcome for a non-zero-sum game is presented. An equilibrium point for any game is defined here as a set of strategy choices for the players, such that no change in the choice of any single player will increase the overall payoff of all the players. Determining equilibrium for multi-player games is a complex problem. An intuitive conceptual tool for reducing the complexity, via the idea of spatially representing strategy options in the bargaining problem is proposed. Based on this geometry, an equilibrium condition is established such that the product of their gains over what each receives is maximal. The geometrical analysis of a cooperative bargaining game provides an example for solving multi-player and non-zero-sum games efficiently

Computational studies of shear-dependent non-newtonian droplet formation at microfluidics T-junction with experimental justification

When Reynolds number (Re) is typically small, the dominant forces governing droplet formation in a microfluidic system includes surface tension, viscous, adhesion, and inertial forces, and the rheology of fluid becomes significantly important when non-Newtonian fluids are involved. The aim of this thesis is to systematically investigate the non-Newtonian shear-thinning effect of sodium carboxymethylcellulose (CMC) on the physical process of droplet formation. A two-phase conservative level-set formulation is adopted to capture the droplets breakup dynamics and relevant hydrodynamics. Detailed two-dimensional (2D) computational microfluidics flow simulations were carried out to examine systematically the influence of different controlling parameters such as degree of shear-thinning (ηo/η∞), relaxation time (λCY), flow rates (Qc, Qd), viscosities (ηc, ηd), surface wettability (θ), and interfacial tensions (σ) on CMC microdroplets formation in a Newtonian continuum. Experimental tests and numerical model justification were performed in conjunction with grid refinement to support the computational analysis and ensure its accuracy and numerical stability. The breakup process of CMC microdroplets in the cross-flowing immiscible liquids in microfluidic device with T-shaped geometry was predicted well. Data for the rheological and physical properties obeying the Carreau-Yasuda stress model were experimentally obtained to support the computational work. In present study, it is worth noting that the dynamics of shear-thinning breakup process is very sensitive to the rheological quantities (ηo/η∞, λCY) under a range of typical shear rate that associated with microchannel. The systematic variation in these rheological quantities has demonstrated different velocity profile and droplet properties. This variation significantly affects the size of droplet and breakup regime, which has never been studied previously. In contrast to previous findings based on Newtonian solutions, the dependence of flow regimes, breakup time and generation frequency on CMC polymer concentration is distinctly different in dilute and semi-dilute concentration regimes, which only exists in polymer solutions. In dilute regimes, the breakup dynamics is similar to pure Newtonian solutions, which is droplet breakup sharply at the corner of T-Junction, as the polymers are at sufficiently low concentration. Conversely, in semi-dilute regime, the presence of highly polymer molecules leads to an elongated fluid thread, a delay in breakup time, and lower production rate of droplet. Interestingly, the existence of thin polymeric filament can be observed prior to breakup for shear-thinning solution. This feature is rarely observed in Newtonian solution, but a similar phenomenon that was previously attributed to elastic effects in the fluid. Besides, the instabilities in the filament leading to the formation and breakup of satellite droplets. In the view of essential role of interfacial tension, adhesion, viscous and inertial forces, the size of shear-thinning CMC droplets is always found to be smaller than the size of Newtonian droplets as it is believed that the shear-thinning thread encounter less resistance, resulting in rapid breakup phenomenon. Present investigations enhance the understanding of the polymer structural features that govern the droplet behaviour in different flow condition. This may contributes a conceptual framework to rheological application in pharmaceutical field, especially drug delivery system, which focuses on the stability and diffusion of drug particles in dispersion into the outer fluid

Particles of One-Half Topological Charge

We would like to show the existence of finite energy SU(2) Yang-Mills-Higgs particles of one-half topological charge. The magnetic fields of these solutions at spatial infinity correspond to the magnetic field of a positive one-half magnetic monopole located at the origin and a semi-infinite Dirac string which carries a magnetic flux of $\frac{2\pi}{g}$ going into the center of the sphere at infinity. Hence the net magnetic charge of the configuration is zero. The solutions possess gauge potentials that are singular along one-half of the z-axis, elsewhere they are regular. There are two distinct configurations of these particles with different total energies and magnetic dipole moments. Their total energies are found to increase with the strength of the Higgs field self-coupling constant $\lambda$.Comment: 27 pages, 9 figure

Microdroplets Advancement in Newtonian and Non- Newtonian Microfluidic Multiphase System

With recent advancement in droplet microfluidics for both microdroplet encapsulation and fission, it is of paramount importance to understand the flow physics for both Newtonian and non-Newtonian fluids in microdroplet encapsulation and fission as the development of the field is approaching to its maturity. The chapter aims to review and discuss the fluid flow behavior of the multiphase system, mathematical models as well as the fundamental phenomena driving force of microdroplet encapsulation and fission multiphase system. Together, the recent advances in technologies that enable fabrication and application of droplets encapsulation and fission from both Newtonian and non-Newtonian microfluidic multiphase system will be reviewed as well