Shear Flows In Channels Research Articles

Stability of flow through deformable channels and tubes: implications of consistent formulation

The present study is aimed at assessing the existing results concerning the stability of canonical shear flows in channels and tubes with deformable walls, in light of consistent formulations of the nonlinear solid constitutive model and linearised interface conditions at the fluid–solid interface. We show that a class of unstable shear-wave modes at low Reynolds number, predicted by previous studies for pressure-driven flows through neo-Hookean tubes and channels, is absent upon use of consistent interfacial conditions. Furthermore, we analyse the consequences of the change in solid model on the stability of the canonical shear flows by using both neo-Hookean and Mooney–Rivlin models. We show that the salient features of the stability of the system are adequately captured by a consistent formulation of the neo-Hookean solid model, thus precluding the need to employ more detailed solid models. The stability analysis of planar flows past a neo-Hookean solid subjected to three-dimensional disturbances showed that two-dimensional disturbances are more unstable than the corresponding three-dimensional disturbances within the consistent formulations. We show that prior inconsistent formulations of the solid constitutive equation predict a physically spurious spanwise instability in disagreement with experiments thereby demonstrating their inapplicability to predict instabilities in flow past deformable solid surfaces. Using the consistent formulation, the present work provides an accurate picture, over a range of Reynolds numbers, of the stability of canonical shear flows through deformable channels and tubes. Importantly, it is shown how inconsistencies in either the bulk constitutive relation or in the linearisation of the interface conditions can separately lead to physically spurious instabilities. The predictions of this work are relevant to experimental studies in flow through deformable tubes and channels in the low and moderate Reynolds number regime.

On Neutral and Unstable Modes and on the Role of Reynolds Stress in the Stability Problem of Density Stratified Shear Flows in Channels with Variable Bottom

The stability problem of inviscid incompressible stratified shear flows in sea straits of arbitrary cross sections is formulated without the Boussinesq approximation. Series solutions of the stability equation are found by the method of Frobenius and these solutions are used to study the existence of neutral modes when the Richardson number is larger than one quarter. The energy aspect of the problem is studied by discussing the role of the Reynolds stress and its variation in the vertical direction for both unstable modes and neutral modes adjacent to the unstable modes. The interfacial conditions at layers of discontinuity of the basic velocity, density or topography are derived and these are used in the study of instability of two examples of basic flows.

Stability of stratified shear flows in channels with variable cross sections

For the instability problem of density stratified shear flows in sea straits with variable cross sections, a new semielliptical instability region is found. Furthermore, the instability of the bounded shear layer is studied in two cases: (i) the density which takes two different constant values in two layers and (ii) the density which takes three different constant values in three layers. In both cases, the dispersion relation is found to be a quartic equation in the complex phase velocity. It is found that there are two unstable modes in a range of the wave numbers in the first case, whereas there is only one unstable mode in the second case.

Maximum or minimum entropy production? How to select a necessary criterion of stability for a dissipative fluid or plasma

Ten necessary criteria for stability of various dissipative fluids and plasmas are derived from the first and the second principle of thermodynamics applied to a generic small mass element of the system, under the assumption that local thermodynamic equilibrium holds everywhere at all times. We investigate the stability of steady states of a mixture of different chemical species at the same temperature against volume-preserving perturbations. We neglect both electric and magnetic polarization, and assume negligible net mass sources and particle diffusion. We assume that both conduction- and radiation-induced heat losses increase with increasing temperature. We invoke no Onsager symmetry, no detailed model of heat transport and production, no "Extended Thermodynamics," no "Maxent" method, and no "new" universal criterion of stability for steady states of systems with dissipation. Each criterion takes the form of--or is a consequence of--a variational principle. We retrieve maximization of entropy for isolated systems at thermodynamic equilibrium, as expected. If the boundary conditions keep the relaxed state far from thermodynamic equilibrium, the stability criterion we retrieve depends also on the detailed balance of momentum of a small mass element. This balance may include the nablap-related force, the Lorenz force of electromagnetism and the forces which are gradients of potentials. In order to be stable, the solution of the steady-state equations of motion for a given problem should satisfy the relevant stability criterion. Retrieved criteria include (among others) Taylor's minimization of magnetic energy with the constraint of given magnetic helicity in relaxed, turbulent plasmas, Rayleigh's criterion of stability in thermoacoustics, Paltridge 's maximum entropy production principle for Earth's atmosphere, Chandrasekhar' minimization of the adverse temperature gradient in Bénard's convective cells, and Malkus' maximization of viscous power with the constraint of given mean velocity for turbulent shear flow in channels. It turns out that characterization of systems far from equilibrium, e.g., by maximum entropy production is not a general property but--just like minimum entropy production--is reserved to special systems. A taxonomy of stability criteria is derived, which clarifies what is to be minimized, what is to be maximized and with which constraint for each problem.

A REINVESTIGATION OF LAMINAR CHANNEL FLOW PASSING OVER POROUS BED

ABSTRACT Laminar shear flows in channels with porous media bed are reinvestigated in the present study. The problem is similar to multiple-layered Poiseuille flow; the upper layer is homogeneous fluid, while the lower layer is rigid porous media flow. The conventional Navier-Stokes equation simplified by the fully developed channel flow condition is used to deal with the upper layer flow. The governing equation of laminar porous media flow derived in Song [7] is simplified and applied to the porous media bed. The analytical solution obtained in this study not only provides more reasonable fluid velocity inside the porous media bed than the one Darcy's law gives, but is also capable of clearly describing the partial-slip boundary condition given in Beavers and Joseph [1]. We further find that when the thinickness of the porous media bed divided by the square root of specific permeability approaches infinity, the nondimensional parameter α in the partial-slip boundary condition is only a function of porosity; i.e. .

Outline of a theory of turbulent shear flow

In this paper the spatial variations and spectral structure of steady-state turbulent shear flow in channels are investigated without the introduction of empirical parameters. This is made possible by the assumption that the non-linear momentum transport has only stabilizing effects on the mean field of flow. Two constraints on the possible momentum transport are drawn from this assumption: first, that the mean flow will be statistically stable if an Orr-Sommerfeld type equation is satisfied by fluctuations of the mean; second, that the smallest scale of motion that can be present in the spectrum of the momentum transport is the scale of the marginally stable fluctuations of the mean. Within these two constraints, and for a given mass transport, an upper limit is sought for the rate of dissipation of potential energy into heat. Solutions of the stability equation depend upon the shape of the mean velocity profile. In turn, the mean velocity profile depends upon the spatial spectrum of the momentum transport. A variational technique is used to determine that momentum transport spectrum which is both marginally stable and produces a maximum dissipation rate. The resulting spectrum determines the velocity profile and its dependence on the boundary conditions. Past experimental work has disclosed laminar, ‘transitional’, logarithmic and parabolic regions of the velocity profile. Several experimental laws and their accompanying constants relate the extent of these regions to the boundary conditions. The theoretical profile contains each feature and law that is observed. First approximations to the constants are found, and give, in particular, a value for the logarithmic slope (von Kármán's constant) which is within the experimental error. However, the theoretical boundary constant is smaller than the observed value. Turbulent channel flow seems to achieve the extreme state found here, but a more decisive quantitative comparison of theory and experiment requires improvement in the solutions of the classical laminar stability problem.