Solution Branches Research Articles

Mixed convective-quadratic radiative MoS 2–SiO 2/H 2 O hybrid nanofluid flow over an exponentially shrinking permeable Riga surface with slip velocity and convective boundary conditions: Entropy and stability analysis

In this work, the flow of a hybrid Mo S 2 − Si O 2 /water nanofluid across a porous exponentially diminishing Riga surface with the effects of quadratic heat radiation is taken into account. We also considered the effects of mixed convection, slip velocity, and convective thermal boundary conditions. Before numerically solving the governing partial differential equations, the MATLAB function bvp4c was used to get a comprehensive description of all relevant flow characteristics. The ramifications of physical characteristics concerning fluid flow and thermal profile are investigated using graphs and tables. The primary objective of this study is to investigate the Nusselt number, skin friction coefficient, and entropy production. Using the second law of thermodynamics, the irreversibility factor is computed. For a given value of the mass suction parameter, the findings also indicate the occurrence of dual-nature solutions in the shrinking sheet area, with a stable upper solution branch and unstable bottom solution branch. In addition, the first solution produces a positive minimum eigenvalue, whereas the second solution produces a negative eigenvalue, showing the stability of the first solution. For heat transfer enhancement, the existence of quadratic thermal radiative parameter impacts is more advantageous. With a larger value for the modified Hartman number, the velocity field has expanded. Also, recent research demonstrates that raising the thermal Biot number enhances the thermal gradient. Increases in nanoparticle volume fraction and thermal radiation parameters lead to a commensurate rise in entropy production. Because of these findings, we can better regulate the temperature of various environments by controlling the flow of heat.

Homoclinic bifurcation and switching of edge state in plane Couette flow.

We identify the presence of three homoclinic bifurcations that are associated with edge states in a system that is governed by the full Navier-Stokes equation. In plane Couette flow with a streamwise period slightly longer than the minimal unit, we describe a rich bifurcation scenario that is related to new time-periodic solutions and the Nagata steady solution [M. Nagata, J. Fluid Mech. 217, 519-527 (1990)]. In this computational domain, the vigorous time-periodic solution (PO3) with comparable fluctuation amplitude to turbulence and the lower branch of the Nagata steady solution are considered as edge states at different ranges of Reynolds number. These edge states can help in understanding the mechanism of subcritical transition to turbulence in wall-bounded shear flows. At the Reynolds numbers at which the homoclinic bifurcations occur, we find the creation (or destruction) of the time-periodic solutions. At a higher Reynolds number, we observe the edge state switching from the lower-branch Nagata steady solution to PO3 at the creation of this vigorous cycle due to the homoclinic bifurcation. Consequently, the formation of the boundary separating the basins of attraction of the laminar attractor and the time-periodic/chaotic attractor also switches to the respective stable manifolds of the edge states, providing a change in the behavior of a typical amplitude of perturbation toward triggering the transition to turbulence.

A Posteriori Validation of Generalized Polynomial Chaos Expansions

Generalized polynomial chaos (gPC) expansions are a powerful tool for studying differential equations with random coefficients, allowing, in particular, one to efficiently approximate random invariant sets associated with such equations. In this work, we use ideas from validated numerics in order to obtain rigorous a posteriori error estimates together with existence results about gPC expansions of random invariant sets. This approach also provides a new framework for conducting validated continuation, i.e., for rigorously computing isolated branches of solutions in parameter-dependent systems, which generalizes in a straightforward way to multiparameter continuation. We illustrate the proposed methodology by rigorously computing random invariant periodic orbits in the Lorenz system, as well as branches and 2 dimensional manifolds of steady states of the Swift–Hohenberg equation.

Periodic Oscillations in a \(2{N}\) -Body Problem

Hip-hop solutions of the -body problem are solutions that satisfy, at every instance of time, that the bodies with the same mass are at the vertices of two regular -gons, and each one of these -gons is at a plane that are equidistant from a fixed plane forming an antiprism. In this paper, we first prove that for every and every there exists a family of periodic hip-hop solutions. For every solution in these families the oriented distance to the plane , which we call , is an odd function that is also even with respect to for some . For this reason we call solutions in these families double symmetric solutions. By exploring more carefully our initial set of periodic solutions, we numerically show that some of the branches established in our existence theorem have bifurcations that produce branches of solutions with the property that the oriented distance function is not even with respect to any ; we call these solutions single symmetric solutions. We prove that no single symmetric solution is a choreography. We also display explicit double symmetric solutions that are choreographies.

Analysis of High-Temperature Superconducting Current Leads: Multiple Solutions, Thermal Runaway, and Protection

The multiple steady states of Ag/Bi2212-composite high-Tc superconducting leads modeling current delivery to a superconducting magnet have been numerically calculated. The model is based on longitudinal conduction combined with convective heat dissipation from a helium gas stream along the conductor. Because of the nonlinearities introduced by the voltage–current relationship and the temperature-dependent material properties, up to three solutions have been identified within the range of parameters considered. Linear stability analysis reveals that two of them are stable, i.e., the superconducting and the normal branches, while the remaining one is unstable. The limit points separating the stable from the unstable steady states form the blow-up threshold, beyond which any further increase in the operating current results in a thermal runway. Interesting findings are that for low filling ratios no bounded solution exists when the length of the lead exceeds the lower limit point, while very high maximum temperatures may be encountered along the normal solution branch. The effect of various parameters such as the conduction–convection parameter, the applied current, and the reduction in coolant flow (LOFA) on the bifurcation structure and their stabilization effect on the blow-up threshold are also evaluated. Apart from the steady and unsteady operating modes, the multiplicity analysis is also used to identify the range of the design and operating variables where safe operation, with a sufficient margin from the onset of instabilities, may be established, thus facilitating the protection of the leads and the device connected to it.

Characterizing harmonic and subharmonic solutions of the bi-stable piezoelectric harvester with a modified Harmonic Balance approach

In this study, broadband frequency-response of a bi-stable piezo-magneto-elastic harvester is extended by analytically characterizing many subharmonic-n solutions. The energy principle-based model considers the continuous analytical mode-shapes of a compound beam, rotational inertia of the tip mass, and nonlinear geometry effect. A modified higher-order Harmonic Balance (HB) approach proposed here is effective not only in characterizing the frequency-responses of harmonic-1 and subharmonic-n solutions, but also in estimating the chaotic response zones of any multi-stable harvester. Balancing between the computational cost and solution convergence, intra-well (IW) and cross-well (CW) period-n solutions of general-type upto period-4 and odd-type upto period-7 are characterized. Effectiveness and convergence of the modified HB approach are demonstrated through good qualitative and quantitative agreements observed between the numerical sweep and HB results. Dynamical behaviors of several period-n solutions from different branches are revealed through the self-evident plots of displacement and voltage time histories, frequency spectra, and phase-portraits. Novel observations regarding the role of various sub- and super-harmonics in producing only CW orbits and mixed-orbits; and subsequently, asymmetric solution branches appearing due to the nonlinear geometry effect are reported. Besides the customary voltage and power output analyses, the assessment of maximum and average strains induced in the piezoelectric material is presented. Such an assessment has proved quite useful in identifying the alternate low-strain inducing and next best power-producing solutions to harmonic-1 responses. The characterized subharmonic-n solutions produce 20μW–90 mW of RMS power and span the entire range of 1–30 Hz considered in this investigation.

Dual Solutions and Stability Analysis of Cu-H2O-Casson Nanofluid Convection past a Heated Stretching/Shrinking Slippery Sheet in a Porous Medium

In this study, we examined the impact of Cu-H2O nanoparticles on two-dimensional Casson nanofluid flows past permeable stretching/shrinking sheet embedded in a Darcy-Forchheimer porous medium in the presence of slipperiness of surface, suction/injection, viscous dissipation, and convective heating. Using some realistic assumptions and appropriate similarity transformations, the governing nonlinear partial differential equations were formulated and transformed into a system of nonlinear ordinary differential equations and then numerically solved by using the shooting technique. Numerical results are displayed for dimensionless fluid velocity and temperature profiles, skin friction, and the local Nusselt number. The impacts of different governing physical parameters on these quantities are presented and discussed using graphs, tables, and a chart. For the specific range of shrinking sheet, the result shows that dual solutions exist, and temporal stability analysis is performed by introducing small disturbances to determine the stable solutions. It is detected that the upper branch solution is hydrodynamically stable and substantially realistic; however, the lower branch solution is unstable and physically unachievable. The fluid flow stability is obtained by enhancing the suction, surface slipperiness, and viscous dissipation parameters. However, augmenting the values of the Casson factor, Cu-H2O nanoparticle volume fraction, porous medium, porous medium inertia, and convective heating parameters increases the blow-up stability of the fluid flow. The rate of heat transfer enhances with the increment in the Casson factor, porous medium, porous medium inertia, suction, velocity ratio, nanoparticle volume fraction, and convective heating parameters, whereas it reduces as the slipperiness of the surface and viscous dissipation parameters rise. Increment of Cu-H2O nanoparticle volume fraction into the Casson fluid boosts the heat transfer enhancement rate higher for the shrinking sheet surface.

Lack of robustness and accuracy of many numerical schemes for phase-field simulations

In this paper, we study the stability, accuracy and convergence behavior of various numerical schemes for phase-field modeling through a simple ODE model. Both theoretical analysis and numerical experiments are carried out on this ODE model to demonstrate the limitation of most numerical schemes that have been used in practice. One main conclusion is that the first-order fully implicit scheme is the only robust algorithm for phase-field simulations while all other schemes (that have been analyzed) may have convergence issue if the time step size is not exceedingly small. More specifically, by rigorous analysis in most cases, we have the following conclusions: (i) The first-order fully implicit scheme converges to the correct steady state solution for all time step sizes. In the case of multiple solutions, one of the solution branches always converges to the correct steady state solution. (ii) The first-order convex splitting scheme, which is equivalent to the first-order fully implicit scheme with a different time scaling, always converges to the correct steady state solution but may seriously lack numerical accuracy for transient solutions. (iii) For the second-order fully implicit and convex splitting schemes, for any time step size [Formula: see text], there exists an initial condition [Formula: see text], with [Formula: see text], such that the numerical solution converges to the wrong steady state solution. (iv) For [Formula: see text], all second-order schemes studied in this paper converge to the correct steady state solution although severe numerical oscillations occur for most of them if the time step size is not sufficiently small. (v) An unconditionally energy-stable scheme (such as the modified Crank–Nicolson scheme) is not necessarily better than a conditionally energy-stable scheme (such as the Crank–Nicolson scheme). Most, if not all, of the above conclusions are expected to be true for more general Allen–Cahn and other phase-field models.

Superradiance and black resonator strings encounter helical black strings

We construct a cohomogeneity-1 helical black string in six-dimensional Einstein gravity. The helical solution branches from the onset of the gravitational superradiant instability of the equal-spinning Myers-Perry black string. The isometry group of the helical black string is ℝT × U(1)Z × SU(2), where the first two are helical isometries generated by linear combinations of time translation, shifts along the string, and rotation, each of which is individually broken by the superradiant instability. The helical black string is stationary, non-axisymmetric, and has nonzero horizon velocity despite the absence of momentum in the string direction. The entropy of the helical black string is higher than that of the Myers-Perry black string, but lower than cohomogeneity-2 “black resonator strings” (recently found) when the solutions overlap in the microcanonical ensemble. The entropy of the helical black string approaches zero when the horizon velocity along the string reaches its maximum given by the speed of light. Nevertheless, we find no evidence for the existence of regular horizonless solutions in this limit.

Beyond Schwarzschild–de Sitter spacetimes. II. An exact non-Schwarzschild metric in pure R2 gravity and new anomalous properties of R2 spacetimes

In our recent publication (Phys. Rev. D 106, 104004 (2022)), we advanced a program that Buchdahl originated but prematurely abandoned circa 1962 (Nuovo Cimento 23, 141 (1962)). Therein we obtained an exhaustive class of metrics that constitute the branch of non-trivial solutions to the pure $R^{2}$ field equation in vacuo. The Buchdahl-inspired metrics in general possess non-constant scalar curvature, thereby defeating the generalized Lichnerowicz theorem previously advocated for quadratic gravity. We found that the said theorem makes an overly strong assumption on the asymptotic falloff in the spatial derivatives of the Ricci scalar, rendering it violable against the Buchdahl-inspired metrics. In this paper, we shall further extend our work mentioned above by showing that, within the class of Buchdahl-inspired metrics, the asymptotically flat member takes on an exact closed analytical expression. The new metric is characterized by a horizon radius $r_{\text{s}}$ and the Buchdahl parameter $k$, the latter of which arises via the higher-derivative nature of $R^{2}$ gravity. For $k=0$, the new metric recovers the classic Schwarzschild metric. Equipped with the exact expression of the new metric, we shall analytically construct the Kruskal-Szekeres diagram for pure $R^{2}$ spacetime. We find that the Buchdahl parameter $k$ fundamentally modifies the properties of $R^{2}$ spacetime structures in a variety of ways.

Scrutinization of Unsteady Magnetohydrodynamics Williamson Nanofluid Flow and Heat Transfer Past a Permeable Shrinking Sheet

This study contributes to the thermal characteristics and flow behaviour of magnetohydrodynamics (MHD) unsteady flow of Williamson nanofluid over a permeable stretching/shrinking sheet in the presence of velocity slip at the boundary. The PDEs with the corresponding constraints are transformed into a set of non-linear ODEs using similarity transformations and are then solved numerically using fourth order Runge-Kutta integration scheme along with shooting technique. The results show that for the flow past a shrinking sheet, dual solutions exist for a certain range of physical parameters. The analysis shows that velocity profile is an increasing function of magnetic parameter M (0.0 ≤ M ≤ 0.3), velocity slip parameter δ (0.0 ≤ δ ≤ 0.25) and suction parameter s (2.0 ≤ s ≤ 2.2) while decreasing function of Williamson parameter β (0.0 ≤ β ≤ 0.8) and unsteadiness parameter A (−3.5 ≤ A ≤ 0.0) for the first solution branch. The temperature profile augments due to the increase of M and δ for the first solution branch. Computational outcomes of the skin friction coefficient, the Nusselt number, the Sherwood number are tabulated with suitable interpretations. The temporal stability analysis reveals that among the two solutions, first solution is stable and physically realizable. Our method of finding dual solutions and analyzing stability analysis is of practical importance to those interested in fluid dynamics as it provides one with a way to determine whether a steady state solution is physically meaningful or not.

Nonlinear optimal perturbations and formation mechanism of localized wave packet in channel flow

The subcritical transition in a two-dimensional channel flow is characterized by the emergence of localized wave packet (LWP). In this paper, the nonlinear nonmodal optimization method is used, and when the target time is long enough the minimal seed, the minimal-energy perturbation triggering the transition is found to be the localized wave packet tilted upstream with energy scaling as Re−3.8 for Re>2500. The formation of LWP from a minimal seed includes three stages. First, the disturbance kinetic energy of the minimal seed increases to a local maximum through a transient growth governed by the Orr mechanism, and second, large scale vortex structures are generated due to the nonlinear forcing mainly contributed by the wave components and causes a streamwise symmetry breaking, i.e., a long but weak upstream tail and a strong downstream front are formed. As a result, the perturbation structure approaches the unstable lower branch solution and is attracted and captured at the last stage by the stable upper branch solution of LWP in the phase space.

Stability analysis of dual viscous flows saturating a vertical porous pipe

The linear stability analysis of a mixed convection viscous flow in a vertical porous pipe is here investigated. The contribution of viscous heating is assumed to be non negligible. A fully developed flow regime is assumed for the basic state. The local balance equations for this state display dual stationary solutions. The dual branches of stationary solutions are determined numerically. Since the pipe is characterised by an isothermal lateral surface, the viscous heating is the sole cause of the buoyancy force. In order to investigate the stability of the basic dual solutions, small amplitude disturbances with the form of normal modes are superposed to the basic state. The solution of the eigenvalue problem obtained allows one to determine the growth rate associated to both the basic solution branches. The sign of the growth rate determines whether the particular basic solution is stable or unstable.

Multiple Rings and Asymmetric Structures in the Disk of SR 21

Crescent-like asymmetric dust structures discovered in protoplanetary disks indicate dust aggregations. Thus, researching these structures helps us understand the planet formation process. Here we analyze ALMA data of the protoplanetary disk around the T-Tauri star SR 21, which has asymmetric structures that were detected in previous submillimeter observations. Imaged at ALMA Band 6 (1.3 mm) with a spatial resolution of about 0.″04, the disk is found to consist of two rings and three asymmetric structures, with two of the asymmetric structures being in the same ring. Compared to the Band 6 image, the Band 3 (2.7 mm) image also shows the three asymmetric structures, but with some clumps. The elongated asymmetric structures in the outer ring could be due to the interactions of a growing planet. Based on fitting the Band 3 and Band 6 dust continuum data, two branches of solutions for the maximum dust size in the disk are suggested: one is larger than 1 mm, and the other is smaller than 300 μm. High-resolution continuum observations at longer wavelengths as well as polarization observations can help break the degeneracy. We also suggest that the prominent spiral previously identified in VLT/SPHERE observations south of the star at 0.″25 may be the scattered-light counterpart of the inner arc, and the structure is a dust-trapping vortex in nature. The discovered features in SR 21 make it a good target for studying the evolution of asymmetric structures and planet formation.

Understanding oscillons: Standing waves in a ball

Oscillons are localized long-lived pulsating states in the three-dimensional ${\ensuremath{\phi}}^{4}$ theory. We gain insight into the spatiotemporal structure and bifurcation of the oscillons by studying time-periodic solutions in a ball of a finite radius. A sequence of weakly localized Bessel waves---nonlinear standing waves with the Bessel-like $r$-dependence---is shown to extend from eigenfunctions of the linearized operator. The lowest-frequency Bessel wave serves as a starting point of a branch of periodic solutions with exponentially localized cores and small-amplitude tails decaying slowly toward the surface of the ball. A numerical continuation of this branch gives rise to the energy-frequency diagram featuring a series of resonant spikes. We show that the standing waves associated with the resonances are born in the period-multiplication bifurcations of the Bessel waves with higher frequencies. The energy-frequency diagram for a sufficiently large ball displays sizeable intervals of stability against spherically symmetric perturbations.

Static and Eigenvalue Analysis of Electrostatically Coupled and Tunable Shallow Micro-Arches for Sensing-Based Applications.

This paper investigated the mechanical performance of an electrostatically tunable microbeams-based resonators. The resonator was designed based on two initially-curved microbeams that are electrostatically coupled, offering the potential for improved performance compared to single-beam based resonators. Analytical models and simulation tools were developed to optimize the resonator design dimensions and to predict its performance, including its fundamental frequency and motional characteristics. The results show that the electrostatically-coupled resonator exhibits multiple nonlinear phenomena including mode veering and snap-through motion. A coexistence of two stable branches of solutions for a straight beam case was even obtained due to the direct effect of the coupling electrostatic force with the other curved beam. Indeed, the results are promising for the better performance of coupled resonators compared to single-beam resonators and offer a platform for future MEMS applications including mode-localized based micro-sensors.

Homogeneous and isotropic cosmology in general teleparallel gravity

We derive the most general homogeneous and isotropic teleparallel geometries, defined by a metric and a flat, affine connection. We find that there are five branches of connection solutions, which are connected via several limits, and can further be restricted to the torsion-free and metric-compatible cases. We apply our results to several classes of general teleparallel gravity theories and derive their cosmological dynamics for all five branches. Our results show that for large subclasses of these theories the dynamics reduce to that of closely related metric or symmetric teleparallel gravity theories, while for other subclasses up to two new scalar degrees of freedom participate in the cosmological dynamics.

Hiemenz stagnation point flow with computational modelling of variety of boundary conditions

This work explains the flow of a G-H2O nanofluid under the special case of MHD stagnation point flow, and the detailed investigation of the Navier’s stokes equations extracted analytically. The main methodology is given work of PDEs is converted into ODEs using the appropriate similarity transformations. The momentum equations solved analytically to derive the solution domain, the impact of thermal radiation is seen in energy equation, four different scenarios are used to solve the energy equation. Aim of the present work is to study the theoretical analysis and it can be discussed for dual nature behavior by providing different physical parameters, these parameters control the domain, momentum and heat transpiration. In the heat transfer analysis solutions are derived in terms of incomplete gamma function and confluent hypergeometric form. The current work is examined using graphene nanoparticles, and the value of Pr is fixed at 6.2. The present problem is the benchmark solution for the results and it is significance in industrial and technological applications in fluid-based systems involving shrinkable/stretchable materials. At the end we get Velocity decreases with increases of VC for upper branch of solution and increases with increases of VC for lower branch of solution in the case shrinking sheet.

Modelling and nonlinear analysis of the wave-induced vibrations of a single hybrid riser conveying two-phase flow

This paper analyses the nonlinear dynamic response of a single hybrid riser pipe to wave excitations. Detailed nonlinear equations of motion are derived for the riser pipe as it conveys two-phase flow. Hence its dynamics are described by two coupled nonlinear equations, relating to both longitudinal and transverse displacements. Wave excitations forcing the system are imposed as oscillating inline flow. In addition, considering that the riser is a structure that is vibrating in a moving fluid, the inline force is modelled with the extended Morison equation. These equations are solved by the method of multiple scale perturbation. Theoretical results show that longitudinal and transverse responses are coupled at some specified natural frequencies. Similarly, the riser pipe is also prone to primary and super-harmonic resonance excitations at some wave frequencies. This presents a complex nonlinear multi-frequency excitation problem identical to the numerical studies conducted on a typical single hybrid riser pipe based on the frequency response curves. The uncoupled primary resonance behaviour of the transverse displacements for all the void fractions studied exhibited a hardening nonlinear behaviour of non-unique responses and jump phenomenon. However, a distinctive feature of the harmonics was observed where the curves separate into two to form an isolated branch of periodic solutions. These are known as Isola. Also, the presence of internal resonance results in the transfer of energy between the planar axis resulting in undesirable axial resonance peaks which are destabilized by the presence of the nonlinear anti-resonance effect.

MHD Marangoni convection heat transfer of Ag-Cu hybrid nanofluid under a stretching/shrinking sheet with the effect of suction

This paper examines the heat transfer characteristics of magnetohydrodynamics (MHD), suction, and Marangoni convection under the stretching/shrinking Ag–Cu hybrid nanofluid surface flow. First, the governing partial differential equations (PDEs) were transformed into ordinary differential equations (ODEs), and the numerical result was obtained using the boundary value problem solver (bvp4c) in MATLAB. The development of the Nusselt number, the velocity profile and the temperature profile was plotted, discussed and inspected. Next, this paper undergoes stability analysis and heat transfer rate comparison between water, nanofluid and hybrid nanofluid. The dual solutions were observed, and the upper branch solution is determined to be stable. Compared to water, the heat transfer rates of Ag–Cu hybrid nanofluid and Cu nanofluid were accelerated by 2.84% and 2.75%, respectively.