Weakly nonlinear stability of viscous flow past a flexible surface

The weakly nonlinear stability of viscous fluid flow past a flexible surface is analysed in the limit of zero Reynolds number. The system consists of a Couette flow of a Newtonian fluid past a viscoelastic medium of non-dimensional thickness H (the ratio of wall thickness to the fluid thickness), and viscosity ratio μr (ratio of the viscosities of wall and fluid media). The wall medium is bounded by the fluid at one surface and two different types of boundary conditions are considered at the other surface of the wall medium – for ‘grafted’ gels zero displacement conditions are applied while for ‘adsorbed’ gels the displacement normal to the surface is zero but the surface is permitted to move in the lateral direction. The linear stability analysis reveals that for grafted gels the most unstable modes have α ∼ O(1), while for adsorbed gels the most unstable modes have α → 0, where α is the wavenumber of the perturbations. The results from the weakly nonlinear analysis indicate that the nature of the bifurcation at the linear instability is qualitatively very different for grafted and absorbed gels. The bifurcation is always subcritical for the case of flow past grafted gels. It is found, however, that relatively weak but finite-amplitude disturbances do not significantly reduce the critical velocity required to destabilize the flow from the critical velocity predicted by the linear stability theory. For the case of adsorbed gels, it is found that a supercritical equilibrium state could exist in the limit of small wavenumber for a wide range of parameters μr and H, while the bifurcation becomes subcritical at larger values of the wavenumber and there is a transition from supercritical to subcritical bifurcation as the wavenumber is increased.

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The Linear and Weakly-Nonlinear Stability of a Core Annular Flow in a Corrugated Tube

10.1115/imece2000-2626 ◽

2000 ◽

Author(s):

Hsien-Hung Wei ◽

David S. Rumschitzki

Keyword(s):

Thin Film ◽

Nonlinear Stability ◽

Annular Flow ◽

Linear Instability ◽

Base Flow ◽

Nonlinear Regime ◽

Weakly Nonlinear ◽

Corrugated Tube

Abstract Both linear and weakly nonlinear stability of a core annular flow in a corrugated tube in the limit of thin film and small corrugation are examined. Asymptotic techniques are used to derive the corrugated base flow and corresponding linear and weakly nonlinear stability equations. Interesting features show that the corrugation interaction can excite linear instability, but the nonlinearity still can suppress such instability in the weakly nonlinear regime.

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Decomposition of the temporal growth rate in linear instability of compressible gas flows

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.380 ◽

2015 ◽

Vol 778 ◽

pp. 120-132 ◽

Cited By ~ 4

Author(s):

Mario Weder ◽

Michael Gloor ◽

Leonhard Kleiser

Keyword(s):

Growth Rate ◽

Energy Balance ◽

Linear Stability ◽

Couette Flow ◽

Stability Theory ◽

Compressible Flows ◽

Linear Instability ◽

Linear Stability Theory ◽

Gas Flows ◽

Temporal Growth Rate

We present a decomposition of the temporal growth rate ${\it\omega}_{i}$ which characterises the evolution of wave-like disturbances in linear stability theory for compressible flows. The decomposition is based on the disturbance energy balance by Chu (Acta Mech., vol. 1 (3), 1965, pp. 215–234) and provides terms for production, dissipation and flux of energy as components of ${\it\omega}_{i}$. The inclusion of flux terms makes our formulation applicable to unconfined flows and flows with permeable or vibrating boundaries. The decomposition sheds light on the fundamental mechanisms determining temporal growth or decay of disturbances. The additional insights gained by the proposed approach are demonstrated by an investigation of two model flows, namely compressible Couette flow and a plane compressible jet.

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Design and Operation of a Droplet Deposition System for Freeform Fabrication of Metal Parts

10.1115/imece1999-1252 ◽

1999 ◽

Author(s):

Ampere A. Tseng ◽

Munhee Lee ◽

Bosen Zhao

Keyword(s):

Linear Stability ◽

Stability Theory ◽

Linear Stability Theory ◽

Liquid Jets ◽

Droplet Generator ◽

Droplet Deposition ◽

Generator System ◽

Freeform Fabrication ◽

Wide Range ◽

Droplet Sizes

Abstract Based on a linear stability theory for liquid jets, the design of a droplet generator system is presented for freeform fabrication. The linear stability theory of forming droplets is first reviewed and its suitability to the present droplet deposition system is then discussed. The analytical formulae for predicting droplet size and break-up length at optimal conditions are developed. Based on the present formulation, a droplet generator is designed and built to make wax and tin alloy droplets. Experiments have been conducted at a wide range of jet velocities, frequencies, and droplet sizes. Good agreements are found between the analytical predictions based on the linear theory and experimental results. It has also been found that using the present design and procedure recommended, the droplet sizes can be controlled having a size deviation of less than 3 % and the variation of the deposited layer can be managed within 3% of its width deposited.

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Parabolized stability analysis of jets from serrated nozzles

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.719 ◽

2016 ◽

Vol 789 ◽

pp. 36-63 ◽

Cited By ~ 15

Author(s):

Aniruddha Sinha ◽

Kristján Gudmundsson ◽

Hao Xia ◽

Tim Colonius

Keyword(s):

Linear Stability ◽

Near Field ◽

Pressure Fluctuations ◽

Three Dimensional ◽

Hydrodynamic Pressure ◽

Linear Instability ◽

Base Flow ◽

Linear Stability Theory ◽

Instability Wave ◽

Mean Flow

We study the viscous spatial linear stability characteristics of the time-averaged flow in turbulent subsonic jets issuing from serrated (chevroned) nozzles, and compare them to analogous round jet results. Linear parabolized stability equations (PSE) are used in the calculations to account for the non-parallel base flow. By exploiting the symmetries of the mean flow due to the regular arrangement of serrations, we obtain a series of coupled two-dimensional PSE problems from the original three-dimensional problem. This reduces the solution cost and manifests the symmetries of the stability modes. In the parallel-flow linear stability theory (LST) calculations that are performed near the nozzle to initiate the PSE, we find that the serrated nozzle reduces the growth rates of the most unstable eigenmodes of the jet, but their phase speeds are approximately similar. We obtain encouraging validation of our linear PSE instability wave results vis-à-vis near-field hydrodynamic pressure data acquired on a phased microphone array in experiments, after filtering the latter with proper orthogonal decomposition (POD) to extract the energetically dominant coherent part. Additionally, a large-eddy simulation database of the same serrated jet is investigated, and its POD-filtered pressure field is found to compare favourably with the corresponding PSE solution within the jet plume. We conclude that the coherent hydrodynamic pressure fluctuations of jets from both round and serrated nozzles are reasonably consistent with the linear instability modes of the turbulent mean flow.

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Weakly nonlinear stability analysis of a flow past a neo-Hookean solid at arbitrary Reynolds numbers

Physics of Fluids ◽

10.1063/1.2982521 ◽

2008 ◽

Vol 20 (9) ◽

pp. 094109 ◽

Cited By ~ 12

Author(s):

Paresh Chokshi ◽

V. Kumaran

Keyword(s):

Stability Analysis ◽

Nonlinear Stability ◽

Reynolds Numbers ◽

Nonlinear Stability Analysis ◽

Weakly Nonlinear ◽

Flow Past

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Double-diffusive instability in core–annular pipe flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.760 ◽

2016 ◽

Vol 789 ◽

pp. 830-855 ◽

Cited By ~ 13

Author(s):

Kirti Chandra Sahu

Keyword(s):

Linear Stability ◽

Underlying Mechanism ◽

Linear Instability ◽

Single Component ◽

Nonlinear Regime ◽

Linear Stability Theory ◽

Cylindrical Pipe ◽

Miscible Fluids ◽

Diffusive Flow ◽

Double Diffusive

The instability in a pressure-driven core–annular flow of two miscible fluids having the same densities, but different viscosities, in the presence of two scalars diffusing at different rates (double-diffusive effect) is investigated via linear stability analysis and axisymmetric direct numerical simulation. It is found that the double-diffusive flow in a cylindrical pipe exhibits strikingly different stability characteristics compared to the double-diffusive flow in a planar channel and the equivalent single-component flow (wherein viscosity stratification is achieved due to the variation of one scalar) in a cylindrical pipe. The flow which is stable in the context of single-component systems now becomes unstable in the presence of two scalars diffusing at different rates. It is shown that increasing the diffusivity ratio enhances the instability. In contrast to the single fluid flow through a pipe (the Hagen–Poiseuille flow), the faster growing axisymmetric eigenmode is found to be more unstable than the corresponding corkscrew mode for the parameter values considered, for which the equivalent single-component flow is stable to both the axisymmetric and corkscrew modes. Unlike single-component flows of two miscible fluids in a cylindrical pipe, it is shown that the diffusivity and the radial location of the mixed layer have non-monotonic influences on the instability characteristics. An attempt is made to understand the underlying mechanism of this instability by conducting the energy budget and inviscid stability analyses. The investigation of linear instability due to the double-diffusive phenomenon is extended to the nonlinear regime via axisymmetric direct numerical simulations. It is found that in the nonlinear regime the flow becomes unstable in the presence of double-diffusive effect, which is consistent with the predictions of linear stability theory. A new type of instability pattern of an elliptical shape is observed in the nonlinear simulations in the presence of double-diffusive effect.

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On two approaches to the problem of instability of short-crested water waves

Journal of Fluid Mechanics ◽

10.1017/s0022112095004289 ◽

1995 ◽

Vol 303 ◽

pp. 297-326 ◽

Cited By ~ 21

Author(s):

Sergei I. Badulin ◽

Victor I. Shrira ◽

Christian Kharif ◽

Mansour Ioualalen

Keyword(s):

Water Waves ◽

A Priori ◽

Three Dimensional ◽

Wave Pattern ◽

Linear Instability ◽

Weakly Nonlinear ◽

Canonical Variables ◽

Wide Range ◽

The Matrix ◽

The Galerkin Method

The work is concerned with the problem of the linear instability of symmetric short-crested water waves, the simplest three-dimensional wave pattern. Two complementary basic approaches were used. The first, previously developed by Ioualalen & Kharif (1993, 1994), is based on the application of the Galerkin method to the set of Euler equations linearized around essentially nonlinear basic states calculated using the Stokes-like series for the short-crested waves with great precision. An alternative analytical approach starts with the so-called Zakharov equation, i.e. an integro-differential equation for potential water waves derived by means of an asymptotic procedure in powers of wave steepness. Both approaches lead to the analysis of an eigenvalue problem of the type {\rm det}|{\boldmath A}-\gamma{\boldmath B}|=0 where A and B are infinite square matrices. The first approach should deal with matrices of quite general form although the problem is tractable numerically. The use of the proper canonical variables in our second approach turns the matrix B into the unit one, while the matrix A gets a very specific ‘nearly diagonal’ structure with some additional (Hamiltonian) properties of symmetry. This enables us to formulate simple necessary and sufficient a priori criteria of instability and to find instability characteristics analytically through an asymptotic procedure avoiding a number of additional assumptions that other authors were forced to accept.A comparison of the two approaches is carried out. Surprisingly, the analytical results were found to hold their validity for rather steep waves (up to steepness 0.4) for a wide range of wave patterns. We have generalized the classical Phillips concept of weakly nonlinear wave instabilities by describing the interaction between the elementary classes of instabilities and have provided an understanding of when this interaction is essential. The mechanisms of the relatively high stability of short-crested waves are revealed and explained in terms of the interaction between different classes of instabilities. A helpful interpretation of the problem in terms of an infinite chain of interacting linear oscillators was developed.

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