Solution Branches Research Articles

Influence of the Inclination Angle on Mixed Convection and Heat Transfer in a “T” Shaped Double Enclosure

The effect of the tilt angle on mixed convection and related heat transfer in a “T” shaped double enclosure with four heated obstacles on the bottom surface is numerically investigated. The considered obstacles are constantly kept at a relatively high (fixed) temperature, while the cavity’s upper wall is cooled. The finite volume approach is used to solve the mass, momentum, and energy equations with the SIMPLEC algorithm being exploited to deal with the pressure-velocity coupling. Emphasis is put on the influence of the tilt angle on the solution symmetry, flow structure, and heat exchange through the walls. The following parameters and related ranges are considered: Rayleigh number 104 ≤ Ra ≤ 5.105, tilt angle 0° ≤ φ ≤ 90°, Reynolds number 100 ≤ Re ≤ 1000, Prandtl number Pr = 0.72, block height B = 0.5, opening width C = 0.15, and distance between blocks D = 0.5. The results reveal different branches of solutions on varying Re and φ . They also show that the symmetry of the solution regarding the P2 axis is retained for all cases with no tilt and for values of Re between 100 and 1000.

КРАЙОВI ЗАДАЧI ДЛЯ РIВНЯННЯ ЛЯПУНОВА

The boundary value problems for the Lyapunov equation in the resonant (irregular) case in Banach and Hilbert spaces, when the solution of the equation does not exist for all right-hand sides and its uniqueness may be violated, have been investigated. The conditions for bifurcation and branching of solutions in linear and nonlinear cases, including with a moving right end of the segment on which the corresponding boundary value problem is considered, are found.

Stability analysis for heat transfer flow in micropolar hybrid nanofluids.

Objective: hybrid nanofluids have superior thermal efficiency and physical durability in contrast to regular nanofluids. The stagnation point flow of MHD micropolar hybrid nanofluids over a deformable sheet with viscous dissipation is investigated. Methodology: the controlling partial differential equations are converted to nonlinear ordinary differential equations using the transmuted similarity, and are subsequently solved using the bvp4c solver in MATLAB. The hybrid nanofluids consist of aluminum and copper nanoparticles, dispersed in a base fluid of water. Results: multiple solutions are obtained in the given problem for the case of shrinking as well as for the stretching sheet due to the variation in several influential parameters. Non-unique solutions, generally, exist for the case of shrinking sheets. In addition, the first branch solution is physically stable and acceptable according to the stability analysis. The friction factor is higher for the branch of the first solution and lower in the second branch due to the higher magnetic parameters, while the opposite behavior is seen in the case of the local heat transfer rate. Originality: the novelty of this model is that it finds multiple solutions in the presence of Cu and Al2O3 nanoparticles and also performs the stability analysis. In general, non-unique solutions exist for the phenomenon of shrinking sheets.

Global bifurcation of positive solutions for a class of superlinear elliptic systems

We are concerned with the global bifurcation of positive solutions for semilinear elliptic systems of the form { − Δ u = λ f ( u , v ) in Ω , − Δ v = λ g ( u , v ) in Ω , u = v = 0 on ∂ Ω , where λ ∈ R is the bifurcation parameter, Ω ⊂ R N , N ≥ 2 is a bounded domain with smooth boundary ∂ Ω . We establish the existence of an unbounded branch of positive solutions, emanating from the origin, which is bounded in positive λ -direction. The nonlinearities f , g ∈ C 1 ( R × R , ( 0 , ∞ ) ) are nondecreasing for each variable and have superlinear growth at infinity. The proof of our main result is based upon bifurcation theory. In addition, as an application for our main result, when f and g subject to the upper growth bound, by a technique of taking superior limit for components, then we may show that the branch must bifurcate from infinity at λ = 0 .

Effects of the magnetic field penetration into the metal on the Brillouin flow in a crossed-field gap

In this paper, the relativistic Brillouin flow in a crossed-field gap is investigated. For this, the case of a planar magnetron is considered. In contrast to previous studies, it is assumed that the electron discharge occurs in a timescale that is long compared to the magnetic diffusion time in the metal. It is found that the Brillouin flow properties and the overall scenario for the loss of magnetic insulation are different from the short pulse case. In particular, it is shown that two branches of equilibrium Brillouin flow solutions can coexist inside the gap region: one linearly stable and the other linearly unstable. As the parameters are varied, the two branches coalesce and cease to exist in a bifurcation that leads to a complete loss of magnetic insulation. Nevertheless, the mere existence of the unstable solution inside the gap is shown to affect the electron dynamics causing cathode–anode currents. An expression for the onset of the unstable solution is obtained and compared to the relativistic Hull cutoff condition for the short pulse case. It is found that the loss of magnetic insulation occurs for lower accelerating potentials in the present case. This effect is noticeable even for weakly relativistic cases.

Computation of Stagnation Point Convection Flow of Carbon Nanotube Nanofluids From a Stretching Sheet With Melting: Dual Solutions

Abstract A theoretical study in stagnation point flow is presented where melting heat transfer effects of carbon nanotube (CNT) from a stretching surface is appeared. Both carbon nanotubes like single-wall CNT (SWCNT) and multiwall CNT (MWCNT) are homogeneously dispersed in the base fluid. As the ordinary (or base) fluids, water and kerosene oil are employed. A set of nonlinear ordinary differential equations with appropriate boundary conditions is formed by transforming the governing equations via similarity transformations. The transformed nonlinear ordinary differential equations are then solved numerically using the bvp4c solver in matlab, an efficient numerical finite difference method. The impact of nanoparticle volume fraction, velocity, melting, stretching parameter, and CNT type on transport characteristics are explored and visualized graphically and in tabular forms. Verification of the matlab computations with available data in certain limiting cases is included showing excellent agreement. Existence of dual (upper and lower branch) solution is shown for a certain range of stretching sheet parameter. The obtained dual solutions are examined for velocity and temperature in detail. A stability analysis demonstrates that the first solution is a stable solution, and the second solution is an unstable solution. Local skin friction and local Nusselt number are also computed in order to determine critical values that can permit dual solutions. It is observed that when a dimensionless melting parameter is greater than 1, SWCNT nanofluids attain greater velocities than MWCNT nanofluids for water as well as kerosene oil base fluids. Moreover, the flow is accelerated for SWCNT compared with MWCNT for both water and kerosene oil. With increasing stretching parameter, the heat transfer rate (Nusselt number) increases, whereas skin friction coefficients decrease. Higher skin friction and Nusselt number are obtained for SWCNTs compared to MWCNTs due to their greater density and thermal conductivity. The study is relevant to phase change manufacturing fluid dynamics of nanomaterials.

Детерминизм генезиса крупномасштабных структур в астрофизике

The criteria for the formation of non-stationary pseudo-periodic structures in a system of gravitating particles, described by the Vlasov--Poisson system of equations. Conditions of branching of solutions of a nonlinear integral equation for a generalized gravitational potential, leading to the emergence of coherent complex states of relative equilibrium in non-stationary systems of massive particles, is studied.

Three-dimensional rotating viscous flow past a permeable stretching/shrinking sheet with convective boundary condition

The study of three-dimensional rotating boundary layer flow approaching a stretching or shrinking surface under convective boundary conditions is presented in this paper. This study expands the scope of previous researchers' work to encompass broader scenarios, including situations involving mass transfer (suction) on the wall, the Biot number and cases featuring a shrinking surface. It is found that the increase of suction and rotating parameters leads to a noticeable rise in both the local skin friction coefficients and the local Nusselt number, and the solutions to the governing ordinary differential equations exhibit a dual-branch nature, comprising both upper and lower branch solutions, within a specific range of the governing parameters.

Skyrmion and meron ordering in quasi-two-dimensional chiral magnets

We revisit the well-known theoretical phase diagram for a two-dimensional chiral magnet with an easy-axis uniaxial anisotropy and an applied magnetic field. A particular emphasis is given to merons---quanta of the phase transition between the cycloid and the hexagonal skyrmion lattice, which possess fractional topological charges $\ifmmode\pm\else\textpm\fi{}1/2$. A localized state of two coupled merons with opposite topological charges exhibits the nonreciprocal current-driven dynamics: it experiences no skyrmion Hall effect, but due to the asymmetry of the attracting intermeron potential gives rise to asymmetric current-velocity characteristics. We also address the problem of skyrmion ordering into hexagonal and square skyrmion crystals. In an entire field range, the square lattice turns out a saddle point state between differently oriented hexagonal lattices. Merons separating elongated skyrmions were also shown to form a local energy minimum by forming a hexagonal lattice shadowing the ordinary lattice of skyrmions. In addition, we focus on the anisotropy-driven transformations of modulated phases and their isolated counterparts, which exist as detached branches of solutions on both sides from the critical anisotropy value. Moreover, a family of isolated skyrmions with both rotational senses can be found near this anisotropy value. Our results are useful from both the fundamental point of view, since they complement previous findings, and from the point of view of applications, since they provide guidelines for developing meron-based spintronic devices.

On non‐local elliptic equations with sublinear nonlinearities involving an eigenvalue problem

A non‐local elliptic equation of Kirchhoff‐type for with Dirichlet boundary conditions is investigated for the cases where . It is well known that with the non‐local effect removed and , a branch of positive solutions bifurcates from infinity at and no positive solution exists whenever for some (see K. J. Brown, Calc. Var. 22, 483‐494, 2005),where is the principal eigenvalue of the linear problem . As a consequence of the non‐local effect, our analysis has found no bifurcation from infinity, and at least one positive solution is always permitted for . Moreover, regions with three positive solutions are found for small value of . Comparisons are also made of the results here with those of the elliptic problem in the absence of the non‐local term under the same prescribed conditions using numerical simulations.

Critical points and stability analysis in MHD radiative non‐Newtonian nanoliquid transport phenomena with artificial neural network prediction

In the present framework, flow and thermal transport behavior of non‐Newtonian viscoelastic fluid induced by stretching/shrinking of the horizontal sheet under the influence of Lorentz force, volumetric heat source/sink, and radiation (assuming optically thick medium) has been investigated. Multiple solutions (Branches) have been predicted numerically using Lie symmetry transformation and Runge Kutta Dormand Prince (RKDP) algorithm‐based Shooting method for the different controlling parameters, stretching/shrinking , and suction with substantial impact with UCM parameter, . Some of the results have also been compared with MATLAB built‐in solvers to validate our in‐house code. The deviations in critical (turning) points have been noticed for different values of and . The temporal stability analysis is performed for these parameters , and the upper (lower) branch is only physically feasible (infeasible). The combined effect on Nusselt number and skin‐friction coefficient is also depicted in the form of contour plots. The outcomes from artificial neural network (ANN) using Levenberg‐Marquardt algorithm give a preferred estimate for current numerical simulation. In ANN prediction, the optimal number of neurons in the hidden layer is used, having minimum value of RMSE and RMRE with maximum for different cases of non Newtonian stretching/shrinking sheet. Finally, the results conclude that the applied ANN model can accurately predict the output results.

A direct construction of a full family of Whitham solitary waves

Starting with the periodic waves earlier constructed for the gravity Whitham equation, we parameterise the solution curves through relative wave height, and use a limiting argument to obtain a full family of solitary waves. The resulting branch starts from the zero solution, traverses unique points in the wave speed–wave height space, and reaches a singular highest wave at φ ( 0 ) = μ 2 \varphi (0) = \frac {\mu }{2} . The construction is based on uniform estimates improved from earlier work on periodic waves for the same equation, together with limiting arguments and a Galilean transform to exclude vanishing waves and waves levelling off at negative surface depth. In fact, the periodic waves can be proved to converge locally uniformly to a wave with negative tails, which is then transformed to the desired branch of solutions. The paper also contains some proof concerning uniqueness and continuity for signed solutions (improved touching lemma).

Fundamental competition of smooth and non-smooth bifurcations and their ghosts in vibro-impact pairs

A combined analysis of smooth and non-smooth bifurcations captures the interplay of different qualitative transitions in a canonical model of an impact pair, a forced capsule in which a ball moves freely between impacts on either end of the capsule. The analysis, generic for the impact pair context, is also relevant for applications. It is applied to a model of an inclined vibro-impact energy harvester device, where the energy is generated via impacts of the ball with a dielectric polymer on the capsule ends. While sequences of bifurcations have been studied extensively in single- degree-of-freedom impacting models, there are limited results for two-degree-of-freedom impacting systems such as the impact pair. Using an analytical characterization of impacting solutions and their stability based on the maps between impacts, we obtain sequences of period doubling and fold bifurcations together with grazing bifurcations, a particular focus here. Grazing occurs when a sequence of impacts on either end of the capsule are augmented by a zero-velocity impact, a transition that is fundamentally different from the smooth bifurcations that are instead characterized by eigenvalues of the local behavior. The combined analyses allow identification of bifurcations also on unstable or unphysical solutions branches, which we term ghost bifurcations. While these ghost bifurcations are not observed experimentally or via simple numerical integration of the model, nevertheless they can influence the birth or death of complex behaviors and additional grazing transitions, as confirmed by comparisons with the numerical results. The competition between the different bifurcations and their ghosts influences the parameter ranges for favorable energy output; thus, the analyses of bifurcation sequences yield important design information.

Multiplicity for a strongly singular quasilinear problem via bifurcation theory

A [Formula: see text]-Laplacian elliptic problem in the presence of both strongly singular and [Formula: see text]-superlinear nonlinearities is considered. We employ bifurcation theory, approximation techniques and sub-supersolution method to establish the existence of an unbounded branch of positive solutions, which is bounded in positive [Formula: see text]-direction and bifurcates from infinity at [Formula: see text]. As consequence of the bifurcation result, we determine intervals of existence, nonexistence and in particular cases, global multiplicity.

Stagnation point flow of a water-based graphene-oxide over a stretching/shrinking sheet under an induced magnetic field with homogeneous-heterogeneous chemical reaction

Water has received a great deal of attention as lubrication for production due to its efficiency, affordable, and friendly to the environment. GO (graphene oxide) ingredients have the capability to upsurge the efficiency of water lubrication owing to theire extraordinary machine-driven/mechanical properties, water dispensability, and variety of practical uses. With such a motivation, it is anticipated the effect of the induced magnetic field (IMF) on the fluid flow close to a standstill point created by a stretching/shrinking sheet with a homogeneous-heterogeneous chemical reaction and dispensed with water-based GO nanoparticles. The leading governing partial differential equations (PDEs) of the problem are converted into ordinary differential equations (ODEs) via employing similarity variables. After that, the resulting equations are worked out numerically with help of the bvp4c (boundary-value problem of fourth-order) solver. Figures are employed as representations to demonstrate the stimulation of pertinent parameters such as the reciprocal magnetic parameter, the magnetic parameter, the Prandtl number, the Schmidt number, the ratio of the diffusion coefficients, radiation, nanoparticle volume fraction, heat source/sink, and heterogeneous reaction parameter on the profile of induced magnetic, velocity, temperature, concentration, rate of heat transfer, and friction factor. It is noted that both the gradients are enhanced by the nanoparticle fraction volume. Moreover, dual solutions can be established in the limited values of stretching/shrinking parameter. Moreover, the shear stress upsurges by almost 4.18% and 0.52% for the upper and lower solution branches with higher values of the solid nanoparticles volume fraction, respectively, whilst it shrinkages up to 7.56% in the upper branch outcome owing to the higher influence of the magnetic parameter but increases by almost 14.56% for the lower branch solution.

Features of hybridized AA7072 and AA7075 alloys nanomaterials with melting heat transfer past a movable cylinder with Thompson and Troian slip effect

A number of scientists have indicated interest in the promising area of research that involves improving heat transmission by utilizing nanoparticles. The dynamic properties of nanofluids reflect several applications in the cooling process, thermal engineering, heat exchangers, magnetic cell separation, energy production, hyperthermia, etc. In light of the appealing implications of nanofluids, the current research aims to explore the melting process of MHD hybrid AA7072 and AA7075 alloy nanoparticles from a movable cylinder. It also incorporates the irregular Thompson and Troian slip effect to simulate the intricate features of the nanofluid flow. The primary nonlinear partial differential equations (NPDEs) are converted into nonlinear ordinary differential equations (NODEs) with the use of the similarity technique. These converted equations are unraveled numerically via applying the built-in program bvp4c in MATLAB. It is visually examined that physical parameters affect the temperature and skin friction. The outcomes suggest that double solutions are found in a certain range of movable cylinder parameter. In addition, the skin friction is decelerated due to the critical shear rate and velocity slip factors while it is enhanced with the influence of suction and nanoparticles volume fraction. Moreover, the temperature is uplifted due to nanoparticles volume fraction and is declined due to Stefan number. In case of quanitative descriptions, the skin friction boosted up by almost 3.64% and 1.18% for the particular upper and lower branch outcomes due to the superior influences of nanoparticle volume fraction. Also, the skin friction decelerated for the branch of upper and lower solutions at about 5.26% and 22.55%, respectively, due to the larger velocity slip parameter.

Multiscalar Gauss-Bonnet gravity: Scalarized black holes beyond spontaneous scalarization

Recently, a new nonlinear mechanism for black hole scalarization, different from the standard spontaneous scalarization, was demonstrated to exist for scalar Gauss-Bonnet theories in which no tachyonic instabilities can occur. Thus the Schwarzschild black hole is linearly stable but instead nonlinear instability can kick in. In the present paper we expand on this idea in the case of multiscalar Gauss-Bonnet gravity with exponential coupling functions of third and fourth leading order in the scalar field. The main motivation comes from the fact that these theories admit hairy compact objects with zero scalar charge, thus zero scalar-dipole radiation, which automatically evades the binary pulsar constraints on the theory parameters. We demonstrate numerically the existence of scalarized black holes for both coupling functions and for all possible maximally symmetric scalar field target spaces. The thermodynamics and the stability of the obtained solution branches is also discussed.

Contribution of suction phenomenon and thermal slip effects for radiated hybrid nanoparticles (Al2O3 −Cu/H2O) with stability framework

This thermal case pronounced the stability framework for stagnation point flow of magnetized alumina and copper nanoparticles with due exponentially shrinking permeable surface. The thermal stability and enhancement of water base liquid had been taken into account with uniform impulsion of hybrid nanomaterials. The induced flow results via exponentially shrinking permeable surface. The similarity transformation simplifies the mathematical model where governing formulated system for hybrid nanofluid is altered into the nondimensional form. A numerical solver called bvp4c is employed in MATLAB software to aid in the problem-solving process, and dual branches have been found. The significance of pertaining parameters associated to the flow model is inspected in view of thermal properties. The findings show that there are two branches for suction strength [Formula: see text] and magnetic strength [Formula: see text]. The bifurcation values [Formula: see text] and [Formula: see text] reduce for the occurrence of dual branches as the solid volume percentages of copper increase. Furthermore, for the upper branch solutions, the skin friction and heat transfer rate rise as [Formula: see text] increases. The temporal stability analysis determines the stability of the dual branches, and it is discovered that only one of them is stable and physically applicable. The presence of suction parameter effectively controls the thermal transportation phenomenon.

Self-sustainment of coherent structures in counter-rotating Taylor–Couette flow

We investigate the local self-sustained process underlying spiral turbulence in counter-rotating Taylor–Couette flow using a periodic annular domain, shaped as a parallelogram, two of whose sides are aligned with the cylindrical helix described by the spiral pattern. The primary focus of the study is placed on the emergence of drifting–rotating waves (DRW) that capture, in a relatively small domain, the main features of coherent structures typically observed in developed turbulence. The transitional dynamics of the subcritical region, far below the first instability of the laminar circular Couette flow, is determined by the upper and lower branches of DRW solutions originated at saddle-node bifurcations. The mechanism whereby these solutions self-sustain, and the chaotic dynamics they induce, are conspicuously reminiscent of other subcritical shear flows. Remarkably, the flow properties of DRW persist even as the Reynolds number is increased beyond the linear stability threshold of the base flow. Simulations in a narrow parallelogram domain stretched in the azimuthal direction to revolve around the apparatus a full turn confirm that self-sustained vortices eventually concentrate into a localised pattern. The resulting statistical steady state satisfactorily reproduces qualitatively, and to a certain degree also quantitatively, the topology and properties of spiral turbulence as calculated in a large periodic domain of sufficient aspect ratio that is representative of the real system.

Viscous froth model applied to the dynamic simulation of bubbles flowing in a channel: three-bubble case

A two-dimensional foam system comprised of three bubbles is studied via simulations with the viscous froth model. Bubbles are arranged in a so-called staircase configuration and move along a channel due to imposed driving back pressure. This flowing three-bubble system has been studied previously on the basis that it interpolates between a simpler staircase structure (a simple lens, which breaks up via so-called topological transformations if driven at high pressure) and an infinite staircase (which sustains arbitrarily large driving pressure without breaking). Depending on bubble size relative to channel size, different solution branches for the three-bubble system were found: certain branches terminate (as for the simple lens) in topological transformations and others reach (as for an infinite staircase) a geometrically invariant migrating state. The methodology used previously was, however, a purely steady state one, and hence did not interrogate stability of the various branches, nor the role of imposing different driving pressures upon topological transformation type. To address this, unsteady state three-bubble simulations are realized here. Stable solution branches without topological transformation exist for comparatively low driving pressures. For sufficiently high imposed back pressures, however, topological transformations occur, albeit with imposed pressure now influencing the transformation type.