Viscous potential flow analysis of Kelvin–Helmholtz instability in a channel

2001 ◽  
Vol 445 ◽  
pp. 263-283 ◽  
Author(s):  
T. FUNADA ◽  
D. D. JOSEPH
Keyword(s):  
Surface Tension ◽  
Critical Velocity ◽  
Potential Flow ◽  
Critical Value ◽  
Neutral Curve ◽  
Shear Stresses ◽  
Inviscid Fluid ◽  
Neutral Stability ◽  
Viscous Potential Flow ◽  
The Stability

We study the stability of stratified gas–liquid flow in a horizontal rectangular channel using viscous potential flow. The analysis leads to an explicit dispersion relation in which the effects of surface tension and viscosity on the normal stress are not neglected but the effect of shear stresses is. Formulas for the growth rates, wave speeds and neutral stability curve are given in general and applied to experiments in air–water flows. The effects of surface tension are always important and determine the stability limits for the cases in which the volume fraction of gas is not too small. The stability criterion for viscous potential flow is expressed by a critical value of the relative velocity. The maximum critical value is when the viscosity ratio is equal to the density ratio; surprisingly the neutral curve for this viscous fluid is the same as the neutral curve for inviscid fluids. The maximum critical value of the velocity of all viscous fluids is given by that for inviscid fluid. For air at 20°C and liquids with density ρ = 1 g cm−3 the liquid viscosity for the critical conditions is 15 cP: the critical velocity for liquids with viscosities larger than 15 cP is only slightly smaller but the critical velocity for liquids with viscosities smaller than 15 cP, like water, can be much lower. The viscosity of the liquid has a strong effect on the growth rate. The viscous potential flow theory fits the experimental data for air and water well when the gas fraction is greater than about 70%.

Adhesion of lipid tubules in an assembly

1998 ◽  
Vol 9 (5) ◽  
pp. 485-506 ◽  
Author(s):  
RICCARDO ROSSO ◽  
EPIFIANO G. VIRGA
Keyword(s):  
Surface Tension ◽  
External Field ◽  
Equilibrium Problem ◽  
Biological Systems ◽  
Critical Value ◽  
Stability Diagram ◽  
Energy Functional ◽  
Highly Nonlinear ◽  
The Stability ◽  
Lipid Tubules

We study a unilateral equilibrium problem for the energy functional of a lipid tubule subject to an external field. These tubules, which constitute many biological systems, may form assemblies when they are brought in contact, and so made to adhere to one another along at interstices. The contact energy is taken to be proportional to the area of contact through a constant, which is called the adhesion potential. This competes against the external field in determining the stability of patterns with flat interstices. Though the equilibrium problem is highly nonlinear, we determine explicitly the stability diagram for the adhesion between tubules. We conclude that the higher the field, the lower the adhesion potential needed to make at interstices energetically favourable, though its critical value depends also on the surface tension of the interface between the tubules and the isotropic fluid around them.

The stability of spatially periodic flows

1981 ◽  
Vol 108 ◽  
pp. 461-474 ◽  
Author(s):  
D. N. Beaumont
Keyword(s):  
Velocity Profile ◽  
Phase Speed ◽  
Neutral Curve ◽  
Long Waves ◽  
Inviscid Fluid ◽  
Periodic Flows ◽  
A Value ◽  
Parallel Flows ◽  
Spatially Periodic ◽  
The Stability

The stability characteristics for spatially periodic parallel flows of an incompressible fluid (both inviscid and viscous) are studied. A general formula for the determination of the stability characteristics of periodic flows to long waves is obtained, and applied to approximate numerically the stability curves for the sinusoidal velocity profile. The neutral curve for the sinusoidal velocity profile is obtained analytically. The stability of two broken-line velocity profiles in an inviscid fluid is studied and the results are used to describe the overall pattern for the sinusoidal velocity profile in the case of long waves. In an inviscid fluid it is found that all periodic flows (other than the trivial flow in which the basic velocity is constant) are unstable to long waves with a value of the phase speed determined by simple integrals of the basic flow. In a viscous fluid it is found that the sinusoidal velocity profile is very unstable with the inviscid solution being a good approximation to the solution of the viscous problem when the value of the Reynolds number is greater than about 20.

Instability of a gravity-modulated fluid layer with surface tension variation

2001 ◽  
Vol 434 ◽  
pp. 243-271 ◽  
Author(s):  
J. RAYMOND LEE SKARDA
Keyword(s):  
Surface Tension ◽  
Fluid Layer ◽  
Modulation Frequency ◽  
Galerkin Approximation ◽  
Neutral Stability ◽  
Modulation Amplitude ◽  
Neutral Stability Curve ◽  
Stability Curve ◽  
The Stability ◽  
Rayleigh Benard

Gravity modulation of an unbounded fluid layer with surface tension variations along its free surface is investigated. The stability of such systems is often characterized in terms of the wavenumber, α and the Marangoni number, Ma. In (α, Ma) parameter space, modulation has a destabilizing effect on the unmodulated neutral stability curve for large Prandtl number, Pr, and small modulation frequency, Ω, while a stabilizing effect is observed for small Pr and large Ω. As Ω → ∞ the modulated neutral stability curves approach the unmodulated neutral stability curve. At certain values of Pr and Ω, multiple minima are observed and the neutral stability curves become highly distorted. Closed regions of subharmonic instability are also observed. In (1/Ω, g1Ra)-space, where g1 is the relative modulation amplitude, and Ra is the Rayleigh number, alternating regions of synchronous and subharmonic instability separated by thin stable regions are observed. However, fundamental differences between the stability boundaries occur when comparing the modulated Marangoni–Bénard and Rayleigh–Bénard problems. Modulation amplitudes at which instability tongues occur are strongly influenced by Pr, while the fundamental instability region is weakly affected by Pr. For large modulation frequency and small amplitude, empirical relations are derived to determine modulation effects. A one-term Galerkin approximation was also used to reduce the modulated Marangoni–Bénard problem to a Mathieu equation, allowing qualitative stability behaviour to be deduced from existing tables or charts, such as Strutt diagrams. In addition, this reduces the parameter dependence of the problem from seven transport parameters to three Mathieu parameters, analogous to parameter reductions of previous modulated Rayleigh–Bénard studies. Simple stability criteria, valid for small parameter values (amplitude and damping coefficients), were obtained from the one-term equations using classical method of averaging results.

Hydrodynamic stability of the flow in the laminar boundary layer between parallel streams

1954 ◽  
Vol 50 (1) ◽  
pp. 105-124 ◽  
Author(s):  
R. C. Lock
Keyword(s):  
Boundary Layer ◽  
Surface Tension ◽  
Water Waves ◽  
Horizontal Wind ◽  
Neutral Stability ◽  
Wind Speeds ◽  
Sinusoidal Oscillations ◽  
Parallel Streams ◽  
The Stability ◽  
Steady Laminar

ABSTRACTA method is given for determining the stability of small sinusoidal oscillations in the steady laminar flow of a horizontal wind over the surface of a liquid at rest (with particular reference to the flow of air over water), taking into account viscosity, gravity and surface tension. It is shown that there are two fundamental types of oscillation of the system, which may be called ‘water’ waves and ‘air’ waves, and curves showing the conditions for neutral stability of these two types of wave are given for a range of wind speeds from 100 to 300 cm./sec.

scholarly journals On the long-wave instability of natural-convection boundary layers

1997 ◽  
Vol 335 ◽  
pp. 57-73 ◽  
Author(s):  
P. G. DANIELS ◽  
JOHN C. PATTERSON
Keyword(s):  
Boundary Layer ◽  
Travelling Waves ◽  
Vertical Wall ◽  
Phase Speed ◽  
Neutral Curve ◽  
Neutral Stability ◽  
Layer Width ◽  
Dimensional Boundary ◽  
Prandtl Numbers ◽  
The Stability

This paper considers the stability of the one-dimensional boundary layer generated by sudden heating of an infinite vertical wall. A quasi-steady approximation is used to analyse the asymptotic form of the lower branch of the neutral curve, corresponding to disturbances of wavelength much greater than the boundary-layer width. This leads to predictions of the critical wavenumber for neutral stability and the maximum phase speed of the travelling waves. Results are obtained for a range of Prandtl numbers and are compared with solutions of the full stability equations and with numerical simulations and experimental observations of cavity flows driven by sudden heating of the sidewalls.

scholarly journals Viscous Potential Flow Analysis of Electroaerodynamic Instability of a Liquid Sheet Sprayed with an Air Stream

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Mukesh Kumar Awasthi ◽  
Vineet K. Srivastava ◽  
M. Tamsir
Keyword(s):  
Electric Field ◽  
Potential Flow ◽  
Thin Sheet ◽  
Flow Analysis ◽  
Liquid Sheet ◽  
Liquid Stream ◽  
Viscous Potential Flow ◽  
Two Fluids ◽  
Air Stream ◽  
The Stability

The instability of a thin sheet of viscous and dielectric liquid moving in the same direction as an air stream in the presence of a uniform horizontal electric field has been carried out using viscous potential flow theory. It is observed that aerodynamic-enhanced instability occurs if the Weber number is much less than a critical value related to the ratio of the air and liquid stream velocities, viscosity ratio of two fluids, the electric field, and the dielectric constant values. Liquid viscosity has stabilizing effect in the stability analysis, while air viscosity has destabilizing effect.

VISCOUS CONTRIBUTIONS TO THE PRESSURE FOR THE POTENTIAL FLOW ANALYSIS OF MAGNETOHYDRODYNAMIC KELVIN–HELMHOLTZ INSTABILITY

2012 ◽  
Vol 04 (01) ◽  
pp. 1250001 ◽  
Author(s):  
MUKESH KUMAR AWASTHI ◽  
G. S. AGRAWAL
Keyword(s):  
Normal Stress ◽  
Potential Flow ◽  
Mechanical Energy ◽  
Flow Analysis ◽  
Shear Stresses ◽  
Helmholtz Instability ◽  
Viscous Potential Flow ◽  
Two Fluids ◽  
Tangential Stresses ◽  
Viscous Pressure

The present paper deals with the study of viscous contributions to the pressure for the viscous potential flow analysis of Kelvin–Helmholtz instability with tangential magnetic field at the interface of two viscous fluids. Viscosity enters through normal stress balance in the viscous potential flow theory and tangential stresses for two fluids are not continuous at the interface. Here, we have considered viscous pressure in the normal stress balance along with the irrotational pressure and it is assumed that the addition of this viscous pressure will resolve the discontinuity between the tangential stresses and the tangential velocities at the interface. The viscous pressure is derived by mechanical energy equation and this pressure correction applied to compute the growth rate of magnetohydrodynamic Kelvin–Helmholtz instability. A dispersion relation is obtained and stability criterion is given in the terms of critical value of relative velocity. It has been observed that the inclusion of irrotational shear stresses have stabilizing effect on the stability of the system.

The stability of a strut under thrust, when buckling is resisted by a force proportional to the displacement

1926 ◽  
Vol 23 (2) ◽  
pp. 120-129 ◽  
Author(s):  
S. Goldstein
Keyword(s):  
Surface Tension ◽  
Solid Surface ◽  
Wave Length ◽  
Critical Value ◽  
Breaking Stress ◽  
Surface Films ◽  
The Earth ◽  
Solid Film ◽  
Rough Calculation ◽  
The Stability

For a strut under thrust when buckling is resisted by a force proportional to the displacement, it is shown that for both a clamped and a pinned strut there is stability for any length when the thrust is less than a certain critical value; and the relation is found between the length and the first value of the thrust above this that will give buckling. A notion of the wave-length and number of nodes after buckling is obtained.The results are applied to the theory of the formation of mountains by horizontal compression, and it is shown that the crust of the earth would be able to transmit, without buckling, stresses right up to the breaking stress across regions of continental extent, unless the depth down to the level of no strain were less than 16 metres.An attempt is made to employ the results also to consider the stability of solid surface films under compression, and it is seen that other factors, making for stability, must enter besides rigidity and gravity. This is supplied by surface tension, and it then appears that collapse due to a weakness in the film must precede buckling.A rough calculation is given of the frequency of vibration of an atom in a solid film.

STUDY ON ELECTROHYDRODYNAMIC CAPILLARY INSTABILITY OF VISCOELASTIC FLUIDS WITH RADIAL ELECTRIC FIELD

2014 ◽  
Vol 06 (04) ◽  
pp. 1450037
Author(s):  
MUKESH KUMAR AWASTHI
Keyword(s):  
Dispersion Relation ◽  
Electric Field ◽  
Radial Electric Field ◽  
Critical Value ◽  
Linear Analysis ◽  
Wave Number ◽  
Dual Effect ◽  
Capillary Instability ◽  
Viscous Potential Flow ◽  
The Stability

We study the linear analysis of electrohydrodynamic capillary instability of the interface between a viscous fluid and viscoelastic fluid of Maxwell type, when the phases are enclosed between two horizontal cylindrical surfaces coaxial with the interface, and when fluids are subjected to the radial electric field. Here, we use an irrotational theory known as viscous potential flow (VPF) theory in which viscosity enters through normal stress balance but shearing stresses are assumed to be zero. A quadratic dispersion relation that accounts for the growth of axisymmetric waves is obtained and stability criterion is given in terms of a critical value of wave number as well as electric field. It is observed that the radial electric field has dual effect on the stability of the system.

Instability of a Radially Moving Cylindrical Surface: A Viscous Potential Flow Approach

2020 ◽  
Vol 142 (9) ◽  
Author(s):  
Mukesh Kumar Awasthi ◽  
Shivam Agarwal
Keyword(s):  
Stability Criterion ◽  
Potential Flow ◽  
Inertial Confinement Fusion ◽  
Fusion Reactor ◽  
Inertial Confinement ◽  
Order Ordinary Differential Equation ◽  
Viscous Potential Flow ◽  
Viscosity Effect ◽  
Confinement Fusion ◽  
The Stability

Abstract The effect of viscosity on the unstable interface of the cylindrical jet is analyzed through viscous potential flow approach. The jet is moving radially and jet interface is experiencing Rayleigh–Taylor type instability. Previous studies have completely ignored the viscosity effect while considering the instability of a radially moving cylindrical jet. The fluids inside and outside jet are incompressible as well as viscous. The theoretical analysis provides us a second-order ordinary differential equation to establish the instability/stability criterion. The radial velocity and acceleration both have significant impact on the stability of the jet. We found that as viscosity enters to the analysis, perturbations grow rapidly. In addition, the acquired stability criterion is applied to the cylindrical jets in HYLIFE-II which is basically an inertial confinement fusion reactor.

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