Effect of tube elasticity on the stability of Poiseuille flow

1977 ◽  
Vol 81 (4) ◽  
pp. 625-640 ◽  
Author(s):  
Vijay K. Garg
Keyword(s):  
Poiseuille Flow ◽  
Bessel Functions ◽  
Critical Reynolds Number ◽  
Elastic Tube ◽  
Dimensionless Frequency ◽  
Tube Material ◽  
Rigid Tube ◽  
Neutral Curves ◽  
The Stability ◽  
Axisymmetric Disturbances

The effect of tube elasticity on the stability of Poiseuille flow to infinitesimal axisymmetric disturbances is investigated. The disturbance equations for the fluid are solved numerically while those for the arbitrarily thick tube are solved analytically in terms of Bessel functions of complex argument. It is shown that an elastic tube can cause instability of Poiseuille flow, unlike a rigid tube, in which the flow is always stable. Neutral curves are presented for various values of the tube parameters. It is found that the critical Reynolds number varies almost as the square root of the Young's modulus of the tube material while the critical dimensionless frequency is almost invariant, being about 1·1 for the cases studied.

Stability of plane Poiseuille–Couette flow in a fluid layer overlying a porous layer

2017 ◽  
Vol 826 ◽  
pp. 376-395 ◽  
Author(s):  
Ting-Yueh Chang ◽  
Falin Chen ◽  
Min-Hsing Chang
Keyword(s):  
Couette Flow ◽  
Porous Layer ◽  
Poiseuille Flow ◽  
Fluid Layer ◽  
Maximum Velocity ◽  
Stability Characteristic ◽  
Layer System ◽  
Neutral Curves ◽  
Layer Mode ◽  
The Stability

This paper performs a linear stability analysis to investigate the stability of plane Poiseuille–Couette flow in a fluid layer overlying a porous medium saturated with the same fluid. The effect of superimposed Couette flow on the associated Poiseuille flow in such a two-layer system is explored carefully. The result shows that the presence of Couette flow may destabilize the Poiseuille flow at small depth ratio $\hat{d}$, defined by the ratio of the depth of the fluid layer to the depth of the porous layer, and induce a tri-modal structure to the neutral curves. At moderate $\hat{d}$, the Couette component generally produces a stabilization effect on the flow. When the velocity of the upper moving wall is large enough, a bi-modal behaviour of the neutral curves appears and a shift of instability mode occurs from the long-wave fluid-layer mode to the porous-layer mode with higher wavenumber. These stability characteristics are remarkably different from those of the plane Poiseuille–Couette flow in a single fluid layer in that the flow becomes absolutely stable when the wall velocity is over 70 % of the maximum velocity of the Poiseuille component of flow. The stability of pure Couette flow in such a two-layer system is also studied. It is found that the flow is still absolutely stable with respect to infinitesimal disturbances, which is the same as the stability characteristic of a single-layer plane Couette flow.

The stability of Poiseuille flow in a pipe of circular cross-section

1972 ◽  
Vol 54 (1) ◽  
pp. 93-112 ◽  
Author(s):  
Harold Salwen ◽  
Chester E. Grosch
Keyword(s):  
Cross Section ◽  
Poiseuille Flow ◽  
Circular Cross Section ◽  
Perturbation Velocity ◽  
The Matrix ◽  
Complete Set ◽  
Expansion Coefficients ◽  
Matrix Differential ◽  
The Stability ◽  
Axisymmetric Disturbances

The stability of Poiseuille flow in a pipe of circular cross-section to azimuthally varying as well as axisymmetric disturbances has been studied. The perturbation velocity and pressure were expanded in a complete set of orthonormal functions which satisfy the boundary conditions. Truncating the expansion yielded a matrix differential equation for the time dependence of the expansion coefficients. The stability characteristics were determined from the eigenvalues of the matrix, which were calculated numerically. Calculations were carried out for the azimuthal wavenumbersn= 0,…, 5, axial wavenumbers α between 0·1 and 10·0 and αR[les ] 50000,Rbeing the Reynolds number. Our results show that pipe flow is stable to infinitesimal disturbances for all values of α,Randnin these ranges.

Stability of fluid flow in a flexible tube to non-axisymmetric disturbances

2000 ◽  
Vol 407 ◽  
pp. 291-314 ◽  
Author(s):  
V. SHANKAR ◽  
V. KUMARAN
Keyword(s):  
Reynolds Number ◽  
Fluid Flow ◽  
Phase Velocity ◽  
Poiseuille Flow ◽  
Inviscid Limit ◽  
Mean Flow ◽  
Flexible Tube ◽  
Moderate Reynolds Number ◽  
The Stability ◽  
Axisymmetric Disturbances

The stability of fluid flow in a flexible tube to non-axisymmetric perturbations is analysed in this paper. In the first part of the paper, the equivalents of classical theorems of hydrodynamic stability are derived for inviscid flow in a flexible tube subjected to arbitrary non-axisymmetric disturbances. Perturbations of the form vi = v˜i exp [ik(x − ct) + inθ] are imposed on a steady axisymmetric mean flow U(r) in a flexible tube, and the stability of mean flow velocity profiles and bounds for the phase velocity of the unstable modes are determined for arbitrary values of azimuthal wavenumber n. Here r, θ and x are respectively the radial, azimuthal and axial coordinates, and k and c are the axial wavenumber and phase velocity of disturbances. The flexible wall is represented by a standard constitutive relation which contains inertial, elastic and dissipative terms. The general results indicate that the fluid flow in a flexible tube is stable in the inviscid limit if the quantity Ud[Gscr ]/dr [ges ] 0, and could be unstable for Ud[Gscr ]/dr < 0, where [Gscr ] ≡ rU′/(n2 + k2r2). For the case of Hagen–Poiseuille flow, the general result implies that the flow is stable to axisymmetric disturbances (n = 0), but could be unstable to non-axisymmetric disturbances with any non-zero azimuthal wavenumber (n ≠ 0). This is in marked contrast to plane parallel flows where two-dimensional disturbances are always more unstable than three-dimensional ones (Squire theorem). Some new bounds are derived which place restrictions on the real and imaginary parts of the phase velocity for arbitrary non-axisymmetric disturbances.In the second part of this paper, the stability of the Hagen–Poiseuille flow in a flexible tube to non-axisymmetric disturbances is analysed in the high Reynolds number regime. An asymptotic analysis reveals that the Hagen–Poiseuille flow in a flexible tube is unstable to non-axisymmetric disturbances even in the inviscid limit, and this agrees with the general results derived in this paper. The asymptotic results are extended numerically to the moderate Reynolds number regime. The numerical results reveal that the critical Reynolds number obtained for inviscid instability to non-axisymmetric disturbances is much lower than the critical Reynolds numbers obtained in the previous studies for viscous instability to axisymmetric disturbances when the dimensionless parameter Σ = ρGR2/η2 is large. Here G is the shear modulus of the elastic medium, ρ is the density of the fluid, R is the radius of the tube and η is the viscosity of the fluid. The viscosity of the wall medium is found to have a stabilizing effect on this instability.

On the stability of flow in an elliptic pipe which is nearly circular

1978 ◽  
Vol 87 (2) ◽  
pp. 233-241 ◽  
Author(s):  
A. Davey
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Cross Section ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Major Axis ◽  
Reynolds Numbers ◽  
Minor Axis ◽  
Critical Reynolds Numbers ◽  
The Stability

The linear stability of Poiseuille flow in an elliptic pipe which is nearly circular is examined by regarding the flow as a perturbation of Poiseuille flow in a circular pipe. We show that the temporal damping rates of non-axisymmetric infinitesimal disturbances which are concentrated near the wall of the pipe are decreased by the ellipticity. In particular we estimate that if the length of the minor axis of the cross-section of the pipe is less than about 96 ½% of that of the major axis then the flow will be unstable and a critical Reynolds number will exist. Also we calculate estimates of the ellipticities which will produce critical Reynolds numbers ranging from 1000 upwards.

Experimental Determination of the Critical Reynolds Number for Pulsating Poiseuille Flow

1966 ◽  
Vol 88 (3) ◽  
pp. 589-598 ◽  
Author(s):  
Turgut Sarpkaya
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Pulsating Flow ◽  
Mean Velocity ◽  
Cross Sectional ◽  
Pressure Transducers ◽  
Mean Pressure ◽  
The Stability

The stability of fully developed Poiseuille flow pulsating under a harmonically and a nonharmonically varying pressure gradient was studied experimentally. The characteristics of turbulent plugs were determined for both steady and pulsating flow by means of pressure transducers. It was found that (a) for oscillating, stable Poiseuille flow, the phase angles determined experimentally agree well with those determined theoretically; (b) for the same mean pressure gradient, pulsating flow is more stable than the corresponding steady Poiseuille flow; (c) in pulsating flow, the presence of one or more inflection points is necessary but not sufficient for instability; and (d) the curves of the critical Reynolds number versus the relative amplitude of the periodic component of the cross-sectional mean velocity reach their maximum when at least one inflection ring continues to exist a time period 53 percent of the period of oscillation.

On the stability of pipe-Poiseuille flow to finite-amplitude axisymmetric and non-axisymmetric disturbances

1985 ◽  
Vol 158 ◽  
pp. 289-316 ◽  
Author(s):  
P. K. Sen ◽  
D. Venkateswarlu ◽  
S. Maji
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Higher Order ◽  
Equilibrium States ◽  
Minimum Threshold ◽  
Threshold Amplitude ◽  
The Stability ◽  
The One ◽  
Axisymmetric Disturbances

The stability of fully developed pipe-Poiseuille flow to finite-amplitude axisymmetric and non-axisymmetric disturbances has been studied using the equilibrium-amplitude method of Reynolds & Potter (1967). In both the cases the least-stable centre-modes were investigated. Also, for the non-axisymmetric case the mode investigated was the one with azimuthal wavenumber equal to one. Many higher-order Landau coefficients were calculated, and the Stuart-Landau series was analysed by the Shanks (1955) method and by using Padé approximants to look for the existence of possible equilibrium states. The results show in both cases that, for each value of the Reynolds number R, there is a preferred band of spatial wavenumbers α in which equilibrium states are likely to exist. Moreover, in both cases it was found that the magnitude of the minimum threshold amplitude for a given R decreases with increasing R. The scales of the various quantities obtained agree very well with those deduced by Davey & Nguyen (1971).

On the instability of rapidly rotating shear flows to non-axisymmetric disturbances

1968 ◽  
Vol 31 (3) ◽  
pp. 603-607 ◽  
Author(s):  
T. J. Pedley
Keyword(s):  
Growth Rate ◽  
Poiseuille Flow ◽  
Shear Flows ◽  
The Stability ◽  
Axisymmetric Disturbances ◽  
Rotating Shear Flows

The stability is considered of the flow with velocity components \[ \{0,\Omega r[1+O(\epsilon^2)],\;2\epsilon\Omega r_0f(r/r_0)\} \] (where f(x) is a function of order one) in cylindrical polar co-ordinates (r, ϕ, z), bounded by the rigid cylinders r/r0 = x1 and r/r0 = 1 (0 [les ] x1 < 1). When ε [Lt ] 1, the flow is shown to be unstable to non-axisymmetric inviscid disturbances of sufficiently large axial wavelength. The case of Poiseuille flow in a rotating pipe is considered in more detail, and the growth rate of the most rapidly growing disturbance is found to be 2εΩ.

scholarly journals Study of the instability of the Poiseuille flow using a thermodynamic formalism

2015 ◽  
Vol 112 (31) ◽  
pp. 9518-9523 ◽  
Author(s):  
Jianchun Wang ◽  
Qianxiao Li ◽  
Weinan E
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Thermodynamic Formalism ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Action Function ◽  
Continuous Transition ◽  
Wide Range ◽  
The Stability

The stability of the plane Poiseuille flow is analyzed using a thermodynamic formalism by considering the deterministic Navier–Stokes equation with Gaussian random initial data. A unique critical Reynolds number, Rec≈2,332, at which the probability of observing puffs in the solution changes from 0 to 1, is numerically demonstrated to exist in the thermodynamic limit and is found to be independent of the noise amplitude. Using the puff density as the macrostate variable, the free energy of such a system is computed and analyzed. The puff density approaches zero as the critical Reynolds number is approached from above, signaling a continuous transition despite the fact that the bifurcation is subcritical for a finite-sized system. An action function is found for the probability of observing puffs in a small subregion of the flow, and this action function depends only on the Reynolds number. The strategy used here should be applicable to a wide range of other problems exhibiting subcritical instabilities.

An experimental investigation of the stability of plane Poiseuille flow

1975 ◽  
Vol 72 (4) ◽  
pp. 731-751 ◽  
Author(s):  
M. Nishioka ◽  
S. Iid A ◽  
Y. Ichikawa
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Rectangular Cross Section ◽  
Reynolds Numbers ◽  
Plane Poiseuille Flow ◽  
Background Turbulence ◽  
Sinusoidal Disturbance ◽  
The Stability ◽  
Long Channel

Stability experiments were made on plane Poiseuille flow generated in a long channel of a rectangular cross-section with a width-to-depth ratio of 27·4. By reducing the background turbulence down to a level of 0·05 %, we succeeded in maintaining the flow laminar at Reynolds numbers up to 8000, which is much larger than the critical Reynolds number of the linear theory, about 6000. The downstream development of the sinusoidal disturbance introduced by the vibrating ribbon technique was studied in detail at various frequencies in the range of Reynolds number from 3000 to 7500. This paper presents the experimental results and clarifies the linear stability, the nonlinear subcritical instability and the breakdown leading to the transition.

The effect of external forcing on the stability of plane Poiseuille flow

1978 ◽  
Vol 359 (1699) ◽  
pp. 453-478 ◽  
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Stable Solution ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Plane Poiseuille Flow ◽  
Critical Reynolds Numbers ◽  
Threshold Amplitude ◽  
The Stability

The stability of plane Poiseuille flow in a channel forced by a wavelike motion on one of the channel walls is investigated. The amplitude Є of this forcing is taken to be small. The most dangerous modes of forcing are identified and it is found in general the critical Reynolds number is changed by O (Є) 2 . However, we identify two particular modes of forcing which give rise to decrements of order Є 2/3 and Є in the critical Reynolds number. Some types of forcing are found to generate sub critical stable finite amplitude perturbations to plane Poiseuille flow. This contrasts with the unforced case where the only stable solution is the zero amplitude solution. The forcing also deforms the unstable subcritical limit cycle solution from its usual circular shape into a more complicated shape. This has an effect on the threshold amplitude ideas suggested by, for example, Meksyn & Stuart (1951). It is found that the phase of disturbances must also be considered when finding the amplitude dependent critical Reynolds numbers.

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