scholarly journals Structure and Branching of Unstable Modes in a Swirling Flow

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 99
Author(s):  
Vadim Akhmetov
Keyword(s):  
Vortex Breakdown ◽  
Swirling Flow ◽  
Maximum Growth ◽  
Critical Parameters ◽  
Return Flow ◽  
Neutral Stability ◽  
Wave Disturbances ◽  
Branching Points ◽  
Hydrodynamic Stability Theory ◽  
The Stability

Swirling has a significant effect on the main characteristics of flow and can lead to its fundamental restructuring. On the flow axis, a stagnation point with zero velocity is possible, behind which a return flow zone is formed. The apparent instability leads to the formation of secondary vortex motions and can also be the cause of vortex breakdown. In the paper, a swirling flow with a velocity profile of the Batchelor vortex type has been studied on the basis of the linear hydrodynamic stability theory. An effective numerical method for solving the spectral problem has been developed. This method includes the asymptotic solutions at artificial and irregular singular points. The stability of flows was considered for the values of the Reynolds number in the range 10≤Re≤5×106. The calculations were carried out for the value of the azimuthal wavenumber parameter n=−1. As a result of the analysis of the solutions, the existence of up to eight simultaneously occurring unstable modes has been shown. The paper presents a classification of the detected modes. The critical parameters are calculated for each mode. For fixed values of the Reynolds numbers 60≤Re≤5000, the curves of neutral stability are plotted. Branching points of unstable modes are found. The maximum growth rates for each mode are determined. A new viscous instability mode is found. The performed calculations reveal the instability of the Batchelor vortex at large values of the swirl parameter for long-wave disturbances.

On the stability of extending films: a model for the film casting process

1974 ◽  
Vol 66 (3) ◽  
pp. 613-622 ◽  
Author(s):  
Y. L. Yeow
Keyword(s):  
Simple Model ◽  
Eigenvalue Problems ◽  
Film Flow ◽  
Casting Process ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Film Casting ◽  
Hydrodynamic Stability Theory ◽  
The Stability ◽  
Stability Results

Isothermal Newtonian film flow is put forward as a simple model of the film casting process. Methods of linear hydrodynamic stability theory are applied to study the stability of the film flow. The relevant eigenvalue problems are formulated and solved numerically. Results are presented in the form of neutral-stability curves in the appropriate parameter space. For the case of two-dimensional disturbances stability results obtained here are compared with those of Pearson & Matovich (1969) and Gelder (1971) for the stability of isothermal Newtonian threadline flow.

scholarly journals On the Stability of Convection in a Non-Newtonian Vertical Fluid Layer in the Presence of Gold Nanoparticles: Drug Agent for Thermotherapy

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1302
Author(s):  
Khaled S. Mekheimer ◽  
Bangalore M. Shankar ◽  
Shaimaa F. Ramadan ◽  
Hosahalli E. Mallik ◽  
Mohamed S. Mohamed
Keyword(s):  
Gold Nanoparticles ◽  
Newtonian Fluid ◽  
Fluid Layer ◽  
Critical Parameters ◽  
Second Grade ◽  
Brownian Diffusion ◽  
Complex Eigenvalue ◽  
Neutral Stability ◽  
Eighth Order ◽  
The Stability

We consider the effect of gold nanoparticles on the stability properties of convection in a vertical fluid layer saturated by a Jeffreys fluid. The vertical boundaries are rigid and hold at uniform but different temperatures. Brownian diffusion and thermophoresis effects are considered. Due to numerous applications in the biomedical industry, such a study is essential. The linear stability is investigated through the normal mode disturbances. The resulting stability problem is an eighth-order ordinary differential complex eigenvalue problem that is solved numerically using the Chebyshev collection method. Its solution provides the neutral stability curves, defining the threshold of linear instability, and the critical parameters at the onset of instability are determined for various values of control parameters. The results for Newtonian fluid and second-grade fluid are delineated as particular cases from the present study. It is shown that the Newtonian fluid has a more stabilizing effect than the second-grade and the Jeffreys fluids in the presence of gold nanoparticles and, Jeffreys fluid is the least stable.

On the effects of boundary-layer growth on flow stability

1974 ◽  
Vol 66 (3) ◽  
pp. 465-480 ◽  
Author(s):  
M. Gaster
Keyword(s):  
Asymptotic Series ◽  
Reynolds Numbers ◽  
Travelling Wave ◽  
Layer Growth ◽  
Neutral Stability ◽  
Wave Disturbances ◽  
Flow Approximation ◽  
Asymptotic Series Solution ◽  
The Stability ◽  
Valid Solution

The stability of small travelling-wave disturbances in the flow over a flat plate is discussed. An iterative method is used to generate an asymptotic series solution in inverse powers of the Reynolds number Rx = Ux/v to the power one half. The neutral-stability boundaries given by the first two terms of this series are obtained and compared with experimental data. It is shown that the parallel flow approximation leads to a valid solution at very large Reynolds numbers.

scholarly journals Elastohydrodynamical instabilities of active filaments, arrays and carpets analyzed using slender body theory

2020 ◽  
Author(s):  
Ashok S. Sangani ◽  
Arvind Gopinath
Keyword(s):  
Critical Force ◽  
Slender Body ◽  
Critical Parameters ◽  
Neutral Stability ◽  
Slender Body Theory ◽  
Follower Forces ◽  
Freely Suspended ◽  
Filament Spacing ◽  
The Stability ◽  
Body Theory

ABSTRACTThe rhythmic motions and wave-like planar oscillations in filamentous soft structures are ubiquitous in biology. Inspired by these, recent work has focused on the creation of synthetic colloid-based active mimics that can be used to move, transport cargo, and generate fluid flows. Underlying the functionality of these mimics is the coupling between elasticity, geometry, dissipation due to the fluid, and active force or moment generated by the system. Here, we use slender body theory to analyze the linear stability of a subset of these - active elastic filaments, filament arrays and filament carpets - animated by follower forces. Follower forces can be external or internal forces that always act along the filament contour. The application of slender body theory enables the accurate inclusion of hydrodynamic effects, screening due to boundaries, and interactions between filaments. We first study the stability of fixed and freely suspended sphere-filament assemblies, calculate neutral stability curves separating stable oscillatory states from stable straight states, and quantify the frequency of emergent oscillations. When shadowing effects due to the physical presence of the spherical boundary are taken into account, the results from the slender body theory differ from that obtained using local resistivity theory. Next, we examine the onset of instabilities in a small cluster of filaments attached to a wall and examine how the critical force for onset of instability and the frequency of sustained oscillations depend on the number of filaments and the spacing between the filaments. Our results emphasize the role of hydrodynamic interactions in driving the system towards perfectly in-phase or perfectly out of phase responses depending on the nature of the instability. Specifically, the first bifurcation corresponds to filaments oscillating in-phase with each other. We then extend our analysis to filamentous (line) array and (square) carpets of filaments and investigate the variation of the critical parameters for the onset of oscillations and the frequency of oscillations on the inter-filament spacing. The square carpet also produces a uniform flow at infinity and we determine the ratio of the mean-squared flow at infinity to the energy input by active forces. We conclude by analyzing the bending and buckling instabilities of a straight passive filament attached to a wall and placed in a viscous stagnant flow - a problem related to the growth of biofilms, and also to mechanosensing in passive cilia and microvilli. Taken together, our results provide the foundation for more detailed non-linear analyses of spatiotemporal patterns in active filament systems.

The dynamics of a swirling flow in a pipe and transition to axisymmetric vortex breakdown

1997 ◽  
Vol 340 ◽  
pp. 177-223 ◽  
Author(s):  
S. WANG ◽  
Z. RUSAK
Keyword(s):  
Steady State ◽  
Vortex Breakdown ◽  
Swirling Flow ◽  
Theoretical Understanding ◽  
Axisymmetric Vortex ◽  
Constant Area ◽  
Breakdown Phenomenon ◽  
Length Constant ◽  
High Reynolds ◽  
The Stability

This paper provides a new study of the axisymmetric vortex breakdown phenomenon. Our approach is based on a thorough investigation of the axisymmetric unsteady Euler equations which describe the dynamics of a swirling flow in a finite-length constant-area pipe. We study the stability characteristics as well as the time-asymptotic behaviour of the flow as it relates to the steady-state solutions. The results are established through a rigorous mathematical analysis and provide a solid theoretical understanding of the dynamics of an axisymmetric swirling flow. The stability and steady-state analyses suggest a consistent explanation of the mechanism leading to the axisymmetric vortex breakdown phenomenon in high-Reynolds-number swirling flows in a pipe. It is an evolution from an initial columnar swirling flow to another relatively stable equilibrium state which represents a flow around a separation zone. This evolution is the result of the loss of stability of the base columnar state when the swirl ratio of the incoming flow is near or above the critical level.

Influence of viscosity temperature dependence on the spectral characteristics of the thermoviscous liquids flow stability equation

2019 ◽  
Vol 14 (1) ◽  
pp. 52-58 ◽  
Author(s):  
A.D. Nizamova ◽  
V.N. Kireev ◽  
S.F. Urmancheev
Keyword(s):  
Reynolds Number ◽  
Spectral Characteristics ◽  
Wave Number ◽  
Critical Parameters ◽  
Fluid Viscosity ◽  
Flat Channel ◽  
Stability Equation ◽  
The Stability ◽  
Thermoviscous Fluid ◽  
Sommerfeld Equation

The flow of a viscous model fluid in a flat channel with a non-uniform temperature field is considered. The problem of the stability of a thermoviscous fluid is solved on the basis of the derived generalized Orr-Sommerfeld equation by the spectral decomposition method in Chebyshev polynomials. The effect of taking into account the linear and exponential dependences of the fluid viscosity on temperature on the spectral characteristics of the hydrodynamic stability equation for an incompressible fluid in a flat channel with given different wall temperatures is investigated. Analytically obtained profiles of the flow rate of a thermovisible fluid. The spectral pictures of the eigenvalues of the generalized Orr-Sommerfeld equation are constructed. It is shown that the structure of the spectra largely depends on the properties of the liquid, which are determined by the viscosity functional dependence index. It has been established that for small values of the thermoviscosity parameter the spectrum compares the spectrum for isothermal fluid flow, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has a nontrivial solution. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of eigenvalues found with fixed Reynolds number and wavenumber parameters. It is shown that with a fixed Reynolds number and a wave number with an increase in the thermoviscosity parameter, the flow becomes unstable. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermally viscous fluid. The eigenfunctions constructed in the subsequent works show the behavior of transverse-velocity perturbations, their possible growth or decay over time.

Systems with Condensates and Pairing

2018 ◽  
Author(s):  
Klaus Morawetz
Keyword(s):  
Critical Parameters ◽  
Einstein Condensation ◽  
Cooper Pairs ◽  
Pair Breaking ◽  
Systematic Theory ◽  
Bose Einstein Condensation ◽  
T Matrix ◽  
The Stability ◽  
Bose Einstein

The Bose–Einstein condensation and appearance of superfluidity and superconductivity are introduced from basic phenomena. A systematic theory based on the asymmetric expansion of chapter 11 is shown to correct the T-matrix from unphysical multiple-scattering events. The resulting generalised Soven scheme provides the Beliaev equations for Boson’s and the Nambu–Gorkov equations for fermions without the usage of anomalous and non-conserving propagators. This systematic theory allows calculating the fluctuations above and below the critical parameters. Gap equations and Bogoliubov–DeGennes equations are derived from this theory. Interacting Bose systems with finite temperatures are discussed with successively better approximations ranging from Bogoliubov and Popov up to corrected T-matrices. For superconductivity, the asymmetric theory leading to the corrected T-matrix allows for establishing the stability of the condensate and decides correctly about the pair-breaking mechanisms in contrast to conventional approaches. The relation between the correlated density from nonlocal kinetic theory and the density of Cooper pairs is shown.

scholarly journals Vortex breakdown of the swirling flow in a Lean Direct Injection burner

2020 ◽  
Vol 32 (12) ◽  
pp. 125118
Author(s):  
Yazhou Shen ◽  
Mohamad Ghulam ◽  
Kai Zhang ◽  
Ephraim Gutmark ◽  
Christophe Duwig
Keyword(s):  
Vortex Breakdown ◽  
Swirling Flow ◽  
Direct Injection ◽  
Lean Direct Injection ◽  
Injection Burner

On stability of parallel flow of an incompressible fluid of variable density and viscosity

1962 ◽  
Vol 58 (4) ◽  
pp. 646-661 ◽  
Author(s):  
P. G. Drazin
Keyword(s):  
Incompressible Fluid ◽  
Water Waves ◽  
Salt Water ◽  
Reynolds Numbers ◽  
Variable Density ◽  
Stability Equation ◽  
Wave Disturbances ◽  
Long Wave ◽  
Vertical Variations ◽  
The Stability

ABSTRACTSome aspects of generation of water waves by wind and of turbulence in a heterogeneous fluid may be described by the theory of hydrodynamic stability. The technical difficulties of these problems of instability have led to obscurities in the literature, some of which are elucidated in this paper. The stability equation for a basic steady parallel horizontal flow under the influence of gravity is derived carefully, the undisturbed fluid having vertical variations of density and viscosity. Methods of solution of the equation for large Reynolds numbers and for long-wave disturbances are described. These methods are applied to simple models of wind blowing over water and of fresh water flowing over salt water.

An Operationally Optimized Seven Link Biped Robot Moving on Slopes

2013 ◽  
Author(s):  
Peiman Naseradinmousavi
Keyword(s):  
Simulated Annealing Algorithm ◽  
Nonlinear Modeling ◽  
Biped Robot ◽  
Gait Pattern ◽  
Critical Parameters ◽  
Modeling Process ◽  
Operational Optimization ◽  
Ankle Joints ◽  
Annealing Algorithm ◽  
The Stability

In this paper, we discuss operational optimization of a seven link biped robot using the well-known “Simulated Annealing” algorithm. Some critical parameters affecting the robot gait pattern are selected to be optimized reducing the total energy used. Nonlinear modeling process we published elsewhere is shown here for completeness. The trajectories of both the hip and ankle joints are used to plan the robot gait on slopes and undoubtedly those parameters would be the target ones for the optimization process. The results we obtained reveal considerable amounts of the energy saved for both the ascending and descending surfaces while keeping the robot stable. The stability criterion we utilized for both the modeling and then optimization is “Zero Moment Point”. A comparative study of human evolutionary gait and the operationally optimized robot is also presented.

Sign in / Sign up

Export Citation Format

Share Document