Linear stability of the dissipative, two-fluid, cylindrical Couette problem. Part 1. The stably-stratified hydrodynamic problem

1971 ◽  
Vol 45 (1) ◽  
pp. 91-110 ◽  
Author(s):  
G. P. Schneyer ◽  
S. A. Berger
Keyword(s):  
Eigenvalue Problems ◽  
Numerical Solutions ◽  
Outer Cylinder ◽  
Neutral Stability ◽  
Two Fluids ◽  
Multiple Eigenvalues ◽  
Fluid Properties ◽  
The Stability ◽  
Nuclear Rocket ◽  
Two Fluid

The stability of a two-fluid vortex is studied as a step towards understanding the separation and containment problems in a gaseous-core nuclear rocket. In particular, the linear hydrodynamic stability of two incompressible, immiscible, viscous fluids occupying separate annular regions of a cylindrical Couette apparatus is considered. Neglecting surface tension and gravity, a conservative assumption, the governing equations for arbitrary jumps in fluid properties are derived and numerical solutions to the resultant eigenvalue problems obtained. Results are presented for the effect on neutral stability of density and viscosity jumps, varying gap widths, and differing fluid-fluid interfacial positions. The solutions are limited, however, to the case of stably stratified fluids and a stationary outer cylinder.Two separate modes (multiple eigenvalues) have been discovered for all cases in which two fluids, differing in any property, are present. A rationale is presented for this phenomenon as well as for most of the other observed results.While most results are believed to be manifestations of the Taylor cylindrical Couette instability phenomenon, evidence is presented for the existence of additional ‘hidden’ eigenvalues attributable to the classical Kelvin–Helmholtz and/or the recently reported Yih viscosity-stratification instability phenomena.

scholarly journals Azimuthal magnetorotational instability with super-rotation

2018 ◽  
Vol 84 (1) ◽  
Author(s):  
G. Rüdiger ◽  
M. Schultz ◽  
M. Gellert ◽  
F. Stefani
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Numerical Solutions ◽  
Hartmann Number ◽  
Outer Cylinder ◽  
Magnetic Reynolds Number ◽  
Neutral Stability ◽  
Magnetorotational Instability ◽  
Magnetic Prandtl Number ◽  
The Stability

It is demonstrated that the azimuthal magnetorotational instability (AMRI) also works with radially increasing rotation rates contrary to the standard magnetorotational instability for axial fields which requires negative shear. The stability against non-axisymmetric perturbations of a conducting Taylor–Couette flow with positive shear under the influence of a toroidal magnetic field is considered if the background field between the cylinders is current free. For small magnetic Prandtl number $Pm\rightarrow 0$ the curves of neutral stability converge in the (Hartmann number,Reynolds number) plane approximating the stability curve obtained in the inductionless limit $Pm=0$. The numerical solutions for $Pm=0$ indicate the existence of a lower limit of the shear rate. For large $Pm$ the curves scale with the magnetic Reynolds number of the outer cylinder but the flow is always stable for magnetic Prandtl number unity as is typical for double-diffusive instabilities. We are particularly interested to know the minimum Hartmann number for neutral stability. For models with resting or almost resting inner cylinder and with perfectly conducting cylinder material the minimum Hartmann number occurs for a radius ratio of $r_{\text{in}}=0.9$. The corresponding critical Reynolds numbers are smaller than $10^{4}$.

On the Oscillatory Instability of Closed-Loop Thermosyphons

1985 ◽  
Vol 107 (4) ◽  
pp. 826-832 ◽  
Author(s):  
K. Chen
Keyword(s):  
Turbulent Flows ◽  
Closed Loop ◽  
Numerical Solutions ◽  
Angular Frequency ◽  
Oscillatory Instability ◽  
Neutral Stability ◽  
Horizontal Segment ◽  
Aspect Ratios ◽  
The Stability ◽  
Laminar And Turbulent Flows

The stability of natural convection flows in single-phase closed-loop thermosyphons is investigated. The thermosyphons considered in the present analysis are fluid-filled tubes bent into rectangular shapes. The fluid is heated over the lower horizontal segment and cooled over the upper horizontal segment. Analytical and numerical solutions are presented for a range of loop aspect ratios and radii for both laminar and turbulent flows. It is found that the steady-state results for thermosyphons with different aspect ratios and radii can be expressed in terms of a single dimensionless parameter. When this parameter is less than a critical value, the flow is always stable. Above this critical point, oscillatory instability exists for a narrow range of a friction parameter. The calculated neutral stability conditions show that the flow is least stable when the aspect ratio of the loop approaches unity. The frequency of the convection-induced oscillation is slightly higher than the angular frequency of a fluid particle traveling along the loop.

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 An Arnoldi-Frontal Approach for the Stability Analysis of Flows in a Collapsible Channel

2016 ◽  
Vol 08 (06) ◽  
pp. 1650073
Author(s):  
Yujue Hao ◽  
Zongxi Cai ◽  
Steven Roper ◽  
Xiaoyu Luo
Keyword(s):  
Finite Element ◽  
Stability Analysis ◽  
Krylov Subspace ◽  
Eigenvalue Problems ◽  
Complex Eigenvalue ◽  
Neutral Stability ◽  
Arnoldi Method ◽  
New Approach ◽  
Generalized Complex ◽  
The Stability

In this paper, we present a new approach based on a combination of the Arnoldi and frontal methods for solving large sparse asymmetric and generalized complex eigenvalue problems. The new eigensolver seeks the most unstable eigensolution in the Krylov subspace and makes use of the efficiency of the frontal solver developed for the finite element methods. The approach is used for a stability analysis of flows in a collapsible channel and is found to significantly improve the computational efficiency compared to the traditionally used QZ solver or a standard Arnoldi method. With the new approach, we are able to validate the previous results obtained either on a much coarser mesh or estimated from unsteady simulations. New neutral stability solutions of the system have been obtained which are beyond the limits of previously used methods.

The anomalous motion of superfluid helium in a rotating cavity

2000 ◽  
Vol 406 ◽  
pp. 199-219 ◽  
Author(s):  
KAREN L. HENDERSON ◽  
CARLO F. BARENGHI
Keyword(s):  
Superfluid Helium ◽  
Outer Cylinder ◽  
Navier Stokes ◽  
Normal Fluid ◽  
Two Fluids ◽  
Rotating Cavity ◽  
Vortex Lines ◽  
Stokes Fluid ◽  
Two Fluid ◽  
Ekman Cells

We numerically solve the nonlinear two-fluid Hall–Vinen–Bekharevich–Khalatnikov (HVBK) equations for superfluid helium confined inside a short Couette annulus. The outer cylinder and the ends of the annulus are held fixed whilst the inner cylinder is rotated. This simple flow configuration allows us to study how the vortex lines respond to a shear in the presence of boundaries. It also allows us to investigate further the boundary conditions associated with the HVBK model. The main result of our investigation is the anomalous motion of helium II when compared to a classical fluid. The superfluid Ekman cells always rotate in the opposite sense to a classical Navier–Stokes fluid due to the mutual friction between the two fluids, whilst the sense of rotation of the normal fluid Ekman cells depends on the parameter range considered. We also find that the tension of the vortex lines forces the superfluid to rotate about the inner cylinder almost like a rigid column.

The effect of transition layer inhomogeneity on the stability of compressible MHD fluids

2008 ◽  
Vol 74 (6) ◽  
pp. 827-837 ◽  
Author(s):  
L. RAJAEE ◽  
B. SHOKRI
Keyword(s):  
Magnetic Field ◽  
Transition Layer ◽  
Numerical Solutions ◽  
Compressible Fluids ◽  
Tangential Discontinuity ◽  
Helmholtz Instability ◽  
Two Fluids ◽  
Magnetic Field Change ◽  
The Stability ◽  
The Magnetic Field

AbstractThe Kelvin–Helmholtz instability on the interface of two magnetized compressible fluids with tangential discontinuity is studied in two situations. For a sharp interface, the stability conditions of the surface with tangential discontinuity is investigated. It is shown that in this case magnetohydrodynamic modes such as Alfvén and the magnetosonic waves can propagate. When a transition layer exists between two fluids and the density and magnetic field change across this layer, numerical solutions show that the increase of the Mach number and compressibility has a destabilizing effect while the magnetic field and density increase has a stabilizing effect.

scholarly journals Instability of a binary liquid film flowing down a slippery inclined heated plate

2021 ◽  
Author(s):  
Eglal Ellaban
Keyword(s):  
Liquid Film ◽  
Numerical Solutions ◽  
Soret Effect ◽  
Critical Conditions ◽  
Heated Plate ◽  
Neutral Stability ◽  
Inclined Plate ◽  
Binary Liquid ◽  
Chebyshev Spectral Collocation ◽  
The Stability

In this thesis we studied the stability of a binary liquid film flowing down a heated porous inclined plate. It is assumed that the heating induces concentration differences in the liquid mixture (Soret effect), which together with the differences in temperature affects the surface tension. A mathematical model is constructed by coupling the Navier- tokes equations governing the flow with equations for the concentration and temperature. The effect of substrate permeability is incorporated by applying a specific slip condition at the bottom of the liquid layer. We carry out a linear stability analysis in order to obtain the critical conditions for the onset of instability. We used a Chebyshev spectral collocation method to obtain numerical solutions to the resulting Orr-Sommerfeld type equations. We also obtained an asymptotic solution which yielded an expression for the state of neutral stability of long perturbations as a function of the parameters controlling the problem. We present our findings by illustrating and interpreting our results for the critical Reynolds number for instability.

scholarly journals On the inviscid stability of parallel bubbly flows

1997 ◽  
Vol 339 ◽  
pp. 261-274 ◽  
Author(s):  
LUCA D'AGOSTINO ◽  
FABRIZIO D'AURIA ◽  
CHRISTOPHER E. BRENNEN
Keyword(s):  
Void Fraction ◽  
Bubble Dynamics ◽  
Eigenvalue Problems ◽  
Numerical Solutions ◽  
Ambient Pressure ◽  
Equations Of Motion ◽  
Bubbly Flows ◽  
Rayleigh Equation ◽  
Multiple Shooting ◽  
The Stability

This paper investigates the effects of bubble dynamics on the stability of parallel bubbly flows of low void fraction. The equations of motion for the bubbly mixture are linearized for small perturbations and the parallel flow assumption is used to obtain a modified Rayleigh equation governing the inviscid stability problem. This is then used for the stability analysis of two-dimensional shear layers, jets and wakes. Inertial effects associated with the bubble response and energy dissipation due to the viscosity of the liquid, the heat transfer between the two phases, and the liquid compressibility are included. Numerical solutions of the eigenvalue problems for the modified Rayleigh equation are obtained by means of a multiple shooting method. Depending on the characteristic velocities of the various flows, the void fraction, and the ambient pressure, the presence of air bubbles can induce significant departures from the classical stability results for a single-phase fluid.

scholarly journals Instability of a binary liquid film flowing down a slippery inclined heated plate

2021 ◽  
Author(s):  
Eglal Ellaban
Keyword(s):  
Liquid Film ◽  
Numerical Solutions ◽  
Soret Effect ◽  
Critical Conditions ◽  
Heated Plate ◽  
Neutral Stability ◽  
Inclined Plate ◽  
Binary Liquid ◽  
Chebyshev Spectral Collocation ◽  
The Stability

In this thesis we studied the stability of a binary liquid film flowing down a heated porous inclined plate. It is assumed that the heating induces concentration differences in the liquid mixture (Soret effect), which together with the differences in temperature affects the surface tension. A mathematical model is constructed by coupling the Navier- tokes equations governing the flow with equations for the concentration and temperature. The effect of substrate permeability is incorporated by applying a specific slip condition at the bottom of the liquid layer. We carry out a linear stability analysis in order to obtain the critical conditions for the onset of instability. We used a Chebyshev spectral collocation method to obtain numerical solutions to the resulting Orr-Sommerfeld type equations. We also obtained an asymptotic solution which yielded an expression for the state of neutral stability of long perturbations as a function of the parameters controlling the problem. We present our findings by illustrating and interpreting our results for the critical Reynolds number for instability.

Linear stability of a two-fluid interface for electrohydrodynamic mixing in a channel

2007 ◽  
Vol 583 ◽  
pp. 347-377 ◽  
Author(s):  
F. LI ◽  
O. OZEN ◽  
N. AUBRY ◽  
D. T. PAPAGEORGIOU ◽  
P. G. PETROPOULOS
Keyword(s):  
Stability Analysis ◽  
Electric Field ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Neutral Stability ◽  
Surface Charges ◽  
Two Fluids ◽  
Long Wave ◽  
Normal Electric Field ◽  
Two Fluid

We study the electrohydrodynamic stability of the interface between two superposed viscous fluids in a channel subjected to a normal electric field. The two fluids can have different densities, viscosities, permittivities and conductivities. The interface allows surface charges, and there exists an electrical tangential shear stress at the interface owing to the finite conductivities of the two fluids. The long-wave linear stability analysis is performed within the generic Orr–Sommerfeld framework for both perfect and leaky dielectrics. In the framework of the long-wave linear stability analysis, the wave speed is expressed in terms of the ratio of viscosities, densities, permittivities and conductivities of the two fluids. For perfect dielectrics, the electric field always has a destabilizing effect, whereas for leaky dielectrics, the electric field can have either a destabilizing or a stabilizing effect depending on the ratios of permittivities and conductivities of the two fluids. In addition, the linear stability analysis for all wavenumbers is carried out numerically using the Chebyshev spectral method, and the various types of neutral stability curves (NSC) obtained are discussed.

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