Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; “modern” physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of such particles will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ... All are deformable. A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys.

The non-linear field theories of mechanics

The goal of this survey is to review recent developments in the hydro­ dynamic stability theory of spatially developing flows pertaining to absolute/convective and local/global instability concepts. We wish to dem­ onstrate how these notions can be used effectively to obtain a qualitative and quantitative description of the spatio-temporal dynamics of open shear flows, such as mixing layers, jets, wakes, boundary layers, plane Poiseuille flow, etc. In this review, we only consider open flows where fluid particles do not remain within the physical domain of interest but are advected through downstream flow boundaries. Thus, for the most part, flows in "boxes" (Rayleigh-Benard convection in finite-size cells, Taylor-Couette flow between concentric rotating cylinders, etc.) are not discussed. Further­ more, the implications of local/global and absolute/convective instability concepts for geophysical flows are only alluded to briefly. In many of the flows of interest here, the mean-velocity profile is non-

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Local and global instabilities in spatially developing flows

Fluid flows that are smooth at low speeds become unstable and then turbulent at higher speeds. This phenomenon has traditionally been investigated by linearizing the equations of flow and testing for unstable eigenvalues of the linearized problem, but the results of such investigations agree poorly in many cases with experiments. Nevertheless, linear effects play a central role in hydrodynamic instability. A reconciliation of these findings with the traditional analysis is presented based on the "pseudospectra" of the linearized problem, which imply that small perturbations to the smooth flow may be amplified by factors on the order of 105 by a linear mechanism even though all the eigenmodes decay monotonically. The methods suggested here apply also to other problems in the mathematical sciences that involve nonorthogonal eigenfunctions.

https://people.maths.ox.ac.uk/trefethen/publication/PDF/1993_57.pdf

Hydrodynamic Stability Without Eigenvalues

Perturbed Free Shear Layers

The Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the QR matrix eigenvalue algorithm. It is shown that results of great accuracy are obtained very economically. The method is applied to the stability of plane Poiseuille flow; it is found that the critical Reynolds number is 5772·22. It is explained why expansions in Chebyshev polynomials are better suited to the solution of hydrodynamic stability problems than expansions in other, seemingly more relevant, sets of orthogonal functions.

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Accurate solution of the Orr–Sommerfeld stability equation

Stability of parallel flows

New classes of exact solutions of the incompressible Navier-Stokes equations are presented. The method of solution has its origins in that first used by Kelvin (Phil. Mag. 24 (5), 188-196 (1887)) to solve the linearized equations governing small disturbances in unbounded plane Couette flow. The new solutions found describe arbitrarily large, spatially periodic disturbances within certain two- and three-dimensional 'basic' shear flows of unbounded extent. The admissible classes of basic flow possess spatially uniform strain rates; they include two- and three-dimensional stagnation point flows and two-dimensional flows with uniform vorticity. The disturbances, though spatially periodic, have time-dependent wavenumber and velocity components. It is found that solutions for the disturbance do not always decay to zero; but in some instances grow continuously in spite of viscous dissipation. This behaviour is explained in terms of vorticity dynamics.

Evolution of Wavelike Disturbances in Shear Flows: A Class of Exact Solutions of the Navier-Stokes Equations

The growth of small disturbances in a two-dimensional incompressible wake has been investigated theoretically and experimentally. The theoretical analysis is based upon inviscid stability theory wherein small disturbances are considered from both temporal and spatial reference frames. Through a combined stability analysis, in which small disturbances are permitted to amplify in both time and space, the relationship between the disturbance characteristics for the temporal and spatial reference frames is shown. In these analyses a quasi-uniform assumption is adopted to account for the continuously varying mean-velocity profiles that occur behind flat plates and thin airfoils. It is found that the most unstable disturbances in the wake produce transverse oscillations in the mean-velocity profile and correspond to growing waves that have a minimum group velocity.Experimentally, the downstream development of the wake of a thin airfoil and the wave characteristics of naturally amplifying small disturbances are investigated in a water tank. The disturbances that develop are found to produce transverse oscillations of the mean-velocity profile in agreement with the theoretical prediction. From the comparison of the experimental results with the predictions for the characteristics of the most unstable waves via the temporal and spatial analyses, it is concluded that the stability analysis for the wake is to be considered solely from the more realistic spatial viewpoint. Undoubtedly, this conclusion is also applicable to other highly unstable flows such as jets and free shear layers.In accordance with the disturbance vorticity distribution as determined from the spatial model, a description of the initial development of a vortex street is put forth that contrasts with the description given by Sato & Kuriki (1961).

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112072001193

The stability of an incompressible two-dimensional wake

The study of hydrodynamic stability is fundamental to many subjects, ranging from geophysics and meteorology through to engineering design. This treatise covers both classical and modern aspects of the subject, systematically developing it from the simplest physical problems, then progressing chapter by chapter to the most complex, considering linear and nonlinear situations, and analysing temporal and spatial stability. The authors examine each problem both analytically and numerically: many chapters end with an appendix outlining relevant numerical techniques. All relevant fluid flows are treated, including those where the fluid may be compressible, or those from geophysics, or those that require salient geometries for description. Details of initial-value problems are explored equally with those of stability. As a result, the early transient period as well as the asymptotic fate for perturbations for a flow can be assessed. The text is enriched with many exercises, copious illustrations and an extensive bibliography and the result is a book that can be used with courses on hydrodynamic stability or as an authoritative reference for researchers.

/pdf/theory-and-computation-of-hydrodynamic-stability-l3hlma2zve.pdf

Theory and Computation of Hydrodynamic Stability

In recent years, considerable effort has been expended in attempting to gain an understanding of the behavior of non-Newtonian fluids in shear. MARKOVITZ [1] recently attempted to collect and organize experimental data and to compare it with predictions of existing theories. One difficulty is that conclusions which can be drawn from different types of experiments seem to disagree. Theoretical work on the steady flow of non-Newtonian fluids has brought to light a new phenomenon. According to the linear theory of viscous fluids, it is always possible for a fluid flowing through a cylindrical tube, to which it adheres, to undergo steady rectilinear motion, each particle moving with constant speed in a straight line parallel to the generators of the cylinder. Analyses made by ERICKSEN [2] and by GREEN • RIVLm [31 show that, for many ideal fluids described by the REINER [41 -RIVLIN [5] theory, this simple type of motion is possible only for very special shapes of tubes and that, in general, it is replaced by a flow consisting of such a rectilinear motion combined with a secondary flow in cross-sectional planes. Using the more general theory of fluids proposed by RIVLIN & ERICKSEN [61, LANGLOIS [7] has obtained solutions involving secondary flows in tubes and other types of "boundary geometries. As has been discussed by RIVLIN [8], similar secondary flows have been observed in the mean flow of fluids through non-circular tubes when the motion is turbulent. To our knowledge, it is not known whether secondary flows occur in the steady flow of any real fluid through tubes. Our analysis indicates that, if the conclusions which ROBERTS [9] drew from his experiments ai-e correct, the fluids which he observed should be capable of undergoing rectilinear motionthrough tubes of any shape. MARKOVITZ [11 draws self-consistent conclusions from other experimental data which contradict some of ROBERTS' conclusions. Calculations made in accordance with these conclusions suggest that secondary flows are to be expected in some shapes of tubes. We offer a partial proof that if secondary flows do not occur in elliptical tubes, they will not occur in tubes of any other shape.

William O. Criminale

Papers

Stability of parallel flows

Evolution of Wavelike Disturbances in Shear Flows: A Class of Exact Solutions of the Navier-Stokes Equations

The stability of an incompressible two-dimensional wake

Theory and Computation of Hydrodynamic Stability

Steady shear flow of non-Newtonian fluids