Steady Solutions Research Articles

Existence and uniqueness of solutions to the Peierls–Nabarro model in anisotropic media

We study the existence and uniqueness of solutions to the vector field Peierls–Nabarro (PN) model for curved dislocations in a transversely isotropic medium. Under suitable assumptions for the misfit potential on the slip plane, we reduce the 3D PN model to a nonlocal scalar Ginzburg–Landau equation. For a particular range of elastic coefficients, the nonlocal scalar equation with explicit nonlocal positive kernel is derived. We prove that any stable steady solution has a one-dimensional profile. As a result, we obtain that solutions to the scalar equation, as well as the original 3D system, are characterized as a one-parameter family of straight dislocations. This paper generalizes results found previously for the full isotropic case to an anisotropic setting.

An extended 2D Gao beam model

This work derives and simulates a two-dimensional extension of the nonlinear Gao beam, by adding the cross-sectional shear variable, similarly to the extension of the usual Bernoulli–Euler beam into the Timoshenko beam. The model allows for oscillatory motion about a buckled state, as well as adds vertical shear of the cross sections, thus reflecting better nonlinear thick beams. The static model is derived from the principle of virtual elastic energy, and is in the form of a highly nonlinear coupled system for the beams transverse vibrations and the motion of the cross sections. The model allows for general distributive transversal and longitudinal loads and a compressive horizontal load acting on its edges. The model is simulated numerically, using the dynamic version for better insight into the steady solutions. The terms that distinguish our numerical solutions from the solutions of the Gao beam, described in the literature, are highlighted. The numerical scheme and its characteristic finite element matrices allow us to obtain simulation results that demonstrate the type of vibrations of our extended and modified beam, and also the differences between these solutions and those of the Gao beam model.

An extended Hasegawa–Mima equation for nonlinear drift wave turbulence in general magnetic configurations

We propose an extended Hasegawa–Mima equation describing the evolution of nonlinear drift wave turbulence in general magnetic configurations. Such HMGM equation can be derived within the kinetic framework of guiding center motion or from a two-fluid model of an ion-electron plasma by application of a drift wave turbulence ordering that does not involve conditions on spatial derivatives of magnetic field and plasma density. The HMGM equation is therefore appropriate to describe the evolution of drift wave turbulence in strongly inhomogeneous magnetized plasmas, such as magnetospheric and stellarator plasmas, involving complex magnetic field geometries and non-uniform plasma density distributions. We find conservation laws (mass, energy, and generalized enstrophy) of the HMGM equation, study its algebraic (Hamiltonian) structure, and prove a nonlinear stability criterion for steady solutions through the energy-Casimir method. We then apply these results to describe drift waves and infer the existence of stable toroidal zonal flows with radial shear in dipole magnetic fields.

THE EFFECT OF VISCOUS DISSIPATION ON THE ONSET OF INSTABILITY OF MAGNETO HYDRODYNAMIC FLOW IN A POROUS MEDIUM

The study of flow of fluid with its motion was study in the nineteenth century by Maurice Couette which was as a result of two parallel plates in which the movement are relative, such that the surfaces is moving laterally. The plates with varying radii, could be flat, parallel or two concentric which is generally referred to as plane Couette. Characteristics of fluid at rest or in motion may undergo changes and become unstable. The critical value or range of the parameters of flow which will give rise to instabilities is one of the problems of stability analysis. The stability of fluid motion may be tested by perturbing the fluid with a small sinusoidal disturbances on the parameter. In plane Couette where moving plates drive the flow and plane Poiseuille flow where a pressure gradient drives the flow, in the situation of weak flow and strong distortional elasticity, an asymptotic analysis will yields closed - form steady solutions with identical wall conditions which will focus on simulations will expose the effects due to wall anchoring conflicts and illustrate the induced morphology of the orientation distribution, stored viscoelastic stresses and non - Newtonian flow.

Effects of Pressure Gradient on Fluid Flow and Energy Distribution in a Bending Square Channel

A precise knowledge on fluid flow and energy distribution in a bending channel is important for the thermal management of various engineering problems. The literature currently lacks a comprehensive understanding on how fluid characteristics and heat transfer in a bending channel are affected by pressure gradients, as well as when and how fluids transit into a turbulent state under continuous pressure gradient. The present study aims to explore the bifurcation structure of the steady solutions, linear stability and velocity distribution of the solutions and the transitional behavior from stead-state to other flow states (i.e., periodic or chaotic) for an extensive domain of the Dean number 0 < Dn ≤ 5000. The geometry of the problem is that the outer and bottom walls are heated while the inner and top are at room temperature. A proper grid analysis and validation are performed as well. The study successfully analyzed four branches of steady solutions where stability exists only in the first branch up to Dn = 2593.3709 and 2- to 7-vortex solutions are observed in the secondary flow. The flow transitions in the unsteady solutions are analyzed exquisitely by performing time-advancement of the solutions followed by inquisition of the phase space of time-dependent solutions. Results show that, if Dn > 2593.3709, the steady-state flow becomes periodic followed by a chaotic solution and 2- to 5-vortex solutions are observed for the unsteady flow. It is noticed that, if Dn > 0, the value of the Nusselt number (Nu) was initially higher on the cooling wall compared to the heated wall and at Dn = 102.67, Nu is found to be equal on both sidewalls and then it started increasing rapidly on the heated walls. The present study figured out how centrifugal force impacts fluid flow for larger pressure gradient, which assists fluid mixing and consequently enhances heat transfer in the fluid.

Examining the Quasi-Steady Airflow Assumption in Irregular Vocal Fold Vibration

The quasi-steady flow assumption (QSFA) is commonly used in the field of biomechanics of phonation. It approximates time-varying glottal flow with steady flow solutions based on frozen glottal shapes, ignoring unsteady flow behaviors and vocal fold motion. This study examined the limitations of QSFA in human phonation using numerical methods by considering factors of phonation frequency, air inertance in the vocal tract, and irregular glottal shapes. Two sets of irregular glottal shapes were examined through dynamic, pseudo-static, and quasi-steady simulations. The differences between dynamic and quasi-steady/pseudo-static simulations were measured for glottal flow rate, glottal wall pressure, and sound spectrum to evaluate the validity of QSFA. The results show that errors in glottal flow rate and wall pressure predicted by QSFA were small at 100 Hz but significant at 500 Hz due to growing flow unsteadiness. Air inertia in the vocal tract worsened predictions when interacting with unsteady glottal flow. Flow unsteadiness also influenced the harmonic energy ratio, which is perceptually important. The effects of glottal shape and glottal wall motion on the validity of QSFA were found to be insignificant.

A robust vector tracking loop structure based on potential bias analysis

This paper proposes a robust vector tracking loop structure based on potential bias analysis. The influence of four kinds of biases on the existing two implementations of Vector Tracking Loops (VTLs) is illustrated by theoretical analysis and numerical simulations, and the following findings are obtained. Firstly, the initial user state bias leads to steady navigation solution bias in the relative VTL, while new measurements can eliminate it in the absolute VTL. Secondly, the initial code phase bias is transferred to the following navigation solutions in the relative VTL, while new measurements can eliminate it in the absolute VTL. Thirdly, the user state bias induced by erroneous navigation solution of VTLs can be eliminated by both of the two VTLs. Fourthly, the multipath/NLOS likely affects the two VTLs, and the induced tracking bias in the duration of the multipath/NLOS would decrease the performance of VTLs. Based on the above analysis, a robust VTL structure is proposed, where the absolute VTL is selected for its robustness to the two kinds of initialization biases; meanwhile, the instant bias detection and correction method is used to improve the performance of VTLs in the duration of the multipath/NLOS. Numerical simulations and experimental results verify the effectiveness of the proposed robust VTL structure.

Wall mode dynamics and transition to chaos in magnetoconvection with a vertical magnetic field

Quasistatic magnetoconvection of a fluid with low Prandtl number ( ${\textit {Pr}}=0.025$ ) with a vertical magnetic field is considered in a unit-aspect-ratio box with no-slip boundaries. At high relative magnetic field strengths, given by the Hartmann number ${\textit {Ha}}$ , the onset of convection is known to result from a sidewall instability giving rise to the wall-mode regime. Here, we carry out three-dimensional direct numerical simulations of unprecedented length to map out the parameter space at ${\textit {Ha}} = 200, 500, 1000$ , varying the Rayleigh number ( ${\textit {Ra}}$ ) over the range $6\times 10^5 \lesssim {\textit {Ra}} \lesssim 5\times 10^8$ . We track the development of stable equilibria produced by this primary instability, identifying bifurcations leading to limit cycles and eventually to chaotic dynamics. At ${\textit {Ha}}=200$ , the steady wall-mode solution undergoes a symmetry-breaking bifurcation producing a state that features a coexistence between wall modes and a large-scale roll in the centre of the domain, which persists to higher ${\textit {Ra}}$ . However, under a stronger magnetic field at ${\textit {Ha}}=1000$ , the steady wall-mode solution undergoes a Hopf bifurcation producing a limit cycle which further develops to solutions that shadow an orbit homoclinic to a saddle point. Upon a further increase in ${\textit {Ra}}$ , the system undergoes a subsequent symmetry break producing a coexistence between wall modes and a large-scale roll, although the large-scale roll exists only for a small range of ${\textit {Ra}}$ , and chaotic dynamics primarily arise from a mixture of chaotic wall-mode dynamics and arrays of cellular structures.

Numerical analyses of the flow past a short rotating cylinder

This work studies the three-dimensional flow dynamics around a rotating circular cylinder of finite length, whose axis is positioned perpendicular to the streamwise direction. Direct numerical simulations and global stability analyses are performed within a parameter range of Reynolds number $Re=DU_\infty /\nu <500$ (based on cylinder diameter $D$ , uniform incoming flow velocity $U_\infty$ ), length-to-diameter ratio ${\small \text{AR}}=L/D\leq 2$ and dimensionless rotation rate $\alpha =D\varOmega /2U_\infty \leq 2$ (where $\varOmega$ is rotation rate). By solving Navier–Stokes equations, we investigated the wake patterns and explored the phase diagrams of the lift and drag coefficients. For a cylinder with ${\small \text{AR}}=1$ , we found that when the rotation effect is weak ( $0\leq \alpha \lesssim 0.3$ ), the wake pattern is similar to the unsteady wake past the non-rotating finite-length cylinder, but with a new linear unstable mode competing to dominate the saturation state of the wake. The flow becomes stable for $0.3\lesssim \alpha \lesssim 0.9$ when $Re<360$ . When the rotation effect is strong ( $\alpha \gtrsim 0.9$ ), new low-frequency wake patterns with stronger oscillations emerge. Generally, the rotation effect first slightly decreases and then sharply increases the $Re$ threshold of the flow instability when $\alpha$ is relatively small, but significantly decreases the threshold at high $\alpha$ ( $0.9<\alpha \leq 2$ ). Furthermore, the stability analyses based on the time-averaged flows and on the steady solutions demonstrate the existence of multiple unstable modes undergoing Hopf bifurcation, greatly influenced by the rotation effect. The shapes of these global eigenmodes are presented and compared, as well as their structural sensitivity, visualising the flow region important for the disturbance development with rotation. This research contributes to our understanding of the complex bluff-body wake dynamics past this critical configuration.

On the exponential decay in time of solutions to a generalized Navier–Stokes–Fourier system

We consider a non-Newtonian incompressible heat conducting fluid with a prescribed nonuniform temperature on the boundary and with the no-slip boundary condition for the velocity. We assume no external body forces. For the power-law like models with the power law index bigger than or equal to 11/5 in three dimensions, we identify a class of solutions fulfilling the entropy equality and converging to the equilibria exponentially in a proper metric. In fact, we show the existence of a Lyapunov functional for the problem. Consequently, the steady solution is nonlinearly stable and attracts all proper weak solutions.

Note on the start-up of Couette flow for viscoelastic fluids

This paper is concerned with the numerical modeling of viscoelastic fluids in non-steady shear motions. Time-dependent solutions for three-constant differential models are obtained at the start-up of the planar Couette flows. The influences of (i) the Reynolds number, (ii) the value of κ− material parameter (the ratio between the retardation time and relaxation time), and (iii) the initial condition for the normal stress on the velocity and stresses distributions in the gap are investigated using the numerical solutions obtained with Mathematica software. The focus of the study is the analysis of the Jaumann model (characterized by the corotational derivative) in transitory simple shear rheological tests, as a function of initial conditions for stresses. The steady solutions, corroborated with the non-monotonicity of the steady flow curve, confirm the kink presence in the steady velocity distributions and the formation of shear bandings at Re ≥ 1. The analyses of the strain- and stress-controlled simulations performed at different initial and boundary conditions offer possible explanations of some spurious data recorded in shear measurements of complex viscoelastic fluids. The findings have important consequences for performing transient shear experiments; specifically, it is demonstrated that reproducibility and correlations between the tests require the control of initial normal stresses in the sample.

A computational modeling on two-dimensional laminar flow and thermal characteristics through a strongly bent square channel

In order to have a precise knowledge on how pressure gradients and buoyancy force affect fluid flow and energy distribution in a bending channel, it is important to perform a comprehensive study on flow characteristics and heat transfer mechanisms that trigger out the transition of fluids into a turbulent state, subject to a sustained pressure gradient. The present paper explores a computational modeling on two-dimensional fluid flow and thermal characteristics in a bent square channel of strong curvature. The Newton–Raphson (N-R) iteration method is applied to obtain a bifurcation structure depending on the pressure-driven force, the Dean number (De), covering 0 < De ≤ 5000. As a consequence, four branches of asymmetric steady solutions are identified for each of the cases of the Grashof number, Gn (=1000, 1500, and 2000), where only the first branch is found to exhibit asymmetric two-vortex solutions while the remaining branches encompass two- to four-vortex solutions. The similarity and disparity in the branching structure are also demonstrated. Then, adopting the Adam–Bashforth (A-B) method together with Crank–Nicholson (C-N) formula, the unsteady solutions (US) have been explored, validated by power spectrum density (PSD) and phase space Within the realm of US, two- and three-vortex solutions are found and these solutions exhibit transitions from steady to chaotic behavior profoundly. Effects of the Grashof number with convective heat transfer (CHT) are also compared. By analyzing the Nusselt number (Nu), it is observed that in case of highly chaotic flow, CHT experiences substantial enhancement. This intensified CHT arises from increased turbulence and mixing, facilitating more efficient thermal energy exchange under such chaotic flow conditions.

Numerical study of solution structure and nonlinear behavior of Dean flow with vortex structure in a bending square duct

Fluid flow and heat transfer in bending channels are topics of much research interest because of increasing demand in various fields, particularly in the medical and industrial arenas. This paper presents a numerical study of fluid flow and heat transfer in a bending channel with a square cross section. Numerical calculations using spectral methods were performed for a curvature of 0.001 and a Dean number (Dn) in the range of 0<Dn≤5000. A temperature difference was maintained between the horizontal walls for a Grashof number of Gr = 1000, with the bottom wall heated and the outer and inner walls thermally insulated. Applying Newton–Raphson iteration and path continuation, two branching structures of steady solutions with two to eight vortices were obtained. The first branch comprises multi-vortex up to eight and it occurs at Dn≥3500 while the second branch comprises to multi-vortex up to a maximum of four. The first branch forms symmetric solution while the second branch for its complexity forms asymmetric solutions. For unsteady solutions, time-evolution calculations were performed to investigate the nonlinear behavior, and it was found that with increasing Dn, the unsteady flow experienced various flow instabilities. The time evolution was plotted in phase space to clarify the unsteady flow characteristics. Distinctive contours of the secondary flow patterns, streamwise velocity distributions, and isotherms were also obtained, and the unsteady flow was found to comprise two to six vortices. Finally, convective heat transfer was explored by obtaining temperature contours, and the secondary flow was found to magnify the convective heat transfer significantly. Because of the increase of several secondary vortices in the chaotic solutions, heat transfer occurred markedly in the flow.

Sonic–supersonic–shock–subsonic–sonic solution to Euler–Poisson system with varying background charges

In this paper, steady transonic shock solutions to the Euler–Poisson system are investigated with respect to varying background charge. More precisely, the density of the background charge is prescribed as a piecewise constant function whose discontinuity is determined only by the shock front. The Euler–Poisson system with varying background charge can describe the motion of the electrons in the special semiconductor devices. We establish the well-posedness of sonic–supersonic–shock–subsonic–sonic solutions, with sonic physical boundary conditions at the upstream and downstream respectively.

Morawetz’s contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed type

This article is a survey of Cathleen Morawetz’s contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed elliptic-hyperbolic type. The main focus is on Morawetz’s fundamental work on the nonexistence of continuous transonic flows past profiles, Morawetz’s program regarding the construction of global steady weak transonic flow solutions past profiles via compensated compactness, and a potential theory for regular and Mach reflection of a shock at a wedge. The profound impact of Morawetz’s work on recent developments and breakthroughs in these research directions and related areas in pure and applied mathematics are also discussed.

Modulation theory for solitary waves generated by viscous flow over a step

The effect of dissipation on the flow of a fluid over a step or jump is examined. This flow has been described theoretically and numerically by using the framework of the forced Korteweg–de Vries–Burgers equation. This model includes the influence of the viscosity of the fluid beyond the Korteweg–de Vries approximation. Modulation theory solutions for the undular bores generated upstream and downstream of the forcing are found. Predictions show that the effect of viscosity plays a key role in determining the properties of waves such as the upstream and downstream solitary wave amplitudes and the widths of the bores. The numerical simulations have shown that the unsteady flow consists of upstream and downstream undular bores, which are connected by a locally steady solution over a step. These numerical solutions are compared with modulation theory results with the excellent agreement obtained. Data from the undular bores are all tabulated. Also, the numerical solutions of the Euler equations are compared with the results of the model, which gave good comparisons. Moreover, the comparisons with the experimental measurements of Lee et al. (1989) together with the Navier–Stokes equations show evidence of the significance of this model.

Hidden convexity in the heat, linear transport, and Euler’s rigid body equations: A computational approach

A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such ‘gauge invariance’ is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by noncausal dual problems.

A Numerical Investigation of Hydro-Magnetic Mixed Convection Flow of Viscous Dissipative Fluid in a Channel Filled with Porous Material

The purpose of this paper is to inspect the unsteady MHD mixed convective Couette flow of viscous dissipative fluid in a vertical channel filled with porous material. The steady state analytical solutions of temperature, velocity, frictional force and heat transfer rate are obtained using the Homotopy perturbation method (HPM). The implicit finite difference technique is used to solve the transient leading equations numerically. Plots and tables with the dynamic flow parameters and dimensionless variables are displayed. During this computation, it was found that raising the Darcy parameter (Da) and the mixed convection (Gre) parameters both push the fluid flow upwards, but increasing the magnetic field's values had the opposite effect (M). HIGHLIGHTS Effect of viscous Dissipation on unsteady MHD Mixed Convection Flow with Porous Material is investigated Increasing Darcy number improves the flow pattern Rate of heat transfer could be enhance by considering high values of Brinkman number GRAPHICAL ABSTRACT

The effects of heterogeneity on solute transport in porous media: anomalous dispersion

Mixing of solute in sub-surface flows, for example in the leakage of contaminants into groundwater, is quantified by a dispersion coefficient that depends on the dispersivity of the medium and the transport velocity. Many previous models of solute transport data have assumed Fickian behaviour with constant dispersivity, but found that the inferred dispersivity increases with the distance from the point of injection. This approach assumes that the dispersion on either side of the advective front is symmetric and that the concentration there is close to half of the injected value. However, field data show consistent asymmetry about the advective front. Here, motivated by experimental data and a fractal interpretation of porous media, we consider a simplified heterogeneous medium described by a dispersivity with a power-law dependence on the downstream distance from the source and explore the nature of the asymmetry obtained in the solute transport. In a heterogeneous medium of this type, we show that asymmetry in solute transport gradually increases with the increase in heterogeneity or fractal dimension, and the concentration at the advective front becomes increasingly different from 50% of that at the inlet. In particular, for a sufficiently heterogeneous medium, the concentration profiles at late-time approach a non-trivial steady solution and, as a result, the concentration at a downstream location will never reach that at the inlet. By fitting the Fickian solution to our results, we are able to connect the parameters from our model to those found from experimental data, providing a more physically grounded approach for interpreting them.

Horizontal length of finite-amplitude thermal convection cells with temperature-dependent viscosity

Temperature-dependent viscosity convection is investigated for various horizontal wavelengths of the convective cells. Finite-amplitude steady solutions are obtained by the Newton method in a two-dimensional layer for various values of the Rayleigh number and strength of temperature-dependence of viscosity, and their stability is examined through numerical time integrations. The viscosity η of the model varies with temperature T as η∝exp−γT, where the parameter γ denotes the strength of the temperature-dependency of η. Although approximately square convection cells are stable when γ is small, the stable convective structure elongates horizontally as γ increases in the middle range of γ less than about 10. When γ exceeds that range, the stable convection approaches a square cell.Scaling relations for the Nusselt number that include the effect of the horizontal wavelength are developed. The results obtained by the numerical steady solutions are well explained by the proposed novel scaling relations. When the solutions with the maximum Nusselt number are traced using the scaling relations for various γ, we find that the convective cells elongate gradually as γ increases until γ<8.6, and then the convection becomes narrower. The most elongated convection is expected to appear at the threshold with a horizontal length λ of 6.6, which may not depend on the Rayleigh number. Our results suggest that rocky exoplanets (such as super-Earths), which will be studied in detail in the future, may have surface plates with various horizontal scales.