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1.
The system under study models unsteady, one-dimensional shear flow of a highly elastic and viscous incompressible non-Newtonian
fluid with fading memory under isothermal conditions. The flow, in a channel, is driven by a constant pressure gradient, is
symmetric about the center line, and satisfies a no-slip boundary condition at the wall. The non-Newtonian contribution to
the stress is assumed to obey a differential constitutive law (due to Oldroyd, Johnson & Segalman), the key feature of which is a non-monotone relation between the total steady shear stress and strain rate. In a regime
in which the Reynolds number is much smaller than the Deborah (or Weissenberg) number, one obtains a degenerate, singularly
perturbed system of nonlinear reaction-diffusion equations. It is shown that if the driving pressure gradient exceeds a critical
value (the local shear stress maximum of the steady stress vs. strain rate relation), then the solution to the governing system,
starting from rest at , tends as to a particular discontinuous steady state solution (the “top-jumping” steady state), except in a small neighborhood of the
discontinuity. This discontinuous steady state is shown to be nonlinearly stable in a precise sense with respect to perturbations
yielding smooth initial data. Such discontinuous steady states have been proposed to explain “spurting” flows, which exhibit
a large increase in mean flow rate when the driving pressure is raised above a critical value.
(Accepted April 22, 1996) 相似文献
2.
E. Magyari 《Transport in Porous Media》2009,80(2):389-395
In a recent article by Barletta and Nield (Transport in Porous Media, DOI , 2009), the title problem for the fully developed parallel flow regime was considered assuming isoflux/isothermal wall conditions.
For the limiting cases of the forced and the free convection, analytical solutions were reported; for the general case, numerical
solutions were reported. The aim of the present note is (i) to give an analytical solution for the full problem in terms of
the Weierstrass elliptic P-function, (ii) to illustrate this general approach by two easily manageable examples, and (iii) to rise a couple of questions
of basic physical interest concerning the interplay between the viscous dissipation and the pressure work. In this context,
the concept of “eigenflow” introduced by Barletta and Nield is discussed in some detail. 相似文献
3.
The apparent viscosities of purely viscous non-Newtonian fluids are shear rate dependent. At low shear rates, many of such
fluids exhibit Newtonian behaviour while at higher shear rates non-Newtonian, power law characteristics exist. Between these
two ranges, the fluid's viscous properties are neither Newtonian or power law. Utilizing an apparent viscosity constitutive
equation called the “Modified Power Law” which accounts for the above behavior, solutions have been obtained for forced convection
flows. A shear rate similarity parameter is identified which specifies both the shear rate range for a given fluid and set
of operating conditions and the appropriate solution for that range. The results of numerical solutions for the friction factor–Reynolds
number product and for the Nusselt number as a function of a dimensionless shear rate parameter have been presented for forced
fully developed laminer duct flows of different cross-sections with modified power law fluids. Experimental data is also presented
showing the suitability of the “Modified Power Law” constitutive equation to represent the apparent viscosity of various polymer
solutions.
Received on 21 August 2000 相似文献
4.
Buoyant flow is analysed for a vertical fluid saturated porous layer bounded by an isothermal plane and an isoflux plane in
the case of a fully developed flow with a parallel velocity field. The effects of viscous dissipation and pressure work are
taken into account in the framework of the Oberbeck–Boussinesq approximation scheme and of the Darcy flow model. Momentum
and energy balances are combined in a dimensionless nonlinear ordinary differential equation solved numerically by a Runge–Kutta
method. Both cases of upward pressure force (upward driven flows) and of downward pressure force (downward driven flows) are examined. The thermal behaviour for upward driven flows and downward driven flows is quite different. For upward driven
flows, the combined effects of viscous dissipation and pressure work may produce a net cooling of the fluid even in the case
of a positive heat input from the isoflux wall. For downward driven flows, viscous dissipation and pressure work yield a net
heating of the fluid. A general reflection on the roles played by the effects of viscous dissipation and pressure work with
respect to the Oberbeck–Boussinesq approximation is proposed. 相似文献
5.
A perturbation analysis is presented for the steady-state radial flow of a third-order fluid between two parallel disks. The
results include previous perturbation analyses in which various other rheological models were used. The pressure drop needed
to maintain the radial flow is less than that for the Newtonian creeping-flow solution because of fluid inertia and shear-thinning
viscosity, whereas the normal stresses have the opposite effect. Possible use of the “radial flow viscometer” for experimental
evaluation of second-order constants is also discussed. Finally, molecular stretching in the flow system is examined using
the elastic dumbbell model for a polymer molecule. 相似文献
6.
This paper mainly concerns the mathematical justification of a viscous compressible multi-fluid model linked to the Baer-Nunziato
model used by engineers, see for instance Ishii (Thermo-fluid dynamic theory of two-phase flow, Eyrolles, Paris, 1975), under a “stratification” assumption. More precisely, we show that some approximate finite-energy weak solutions of the
isentropic compressible Navier–Stokes equations converge, on a short time interval, to the strong solution of this viscous
compressible multi-fluid model, provided the initial density sequence is uniformly bounded with corresponding Young measures
which are linear convex combinations of m Dirac measures. To the authors’ knowledge, this provides, in the multidimensional in space case, a first positive answer
to an open question, see Hillairet (J Math Fluid Mech 9:343–376, 2007), with a stratification assumption. The proof is based on the weak solutions constructed by Desjardins (Commun Partial Differ Equ 22(5–6):977–1008, 1997) and on the existence and uniqueness of a local strong solution for the multi-fluid model established by Hillairet assuming initial density to be far from vacuum. In a first step, adapting the ideas from Hoff and Santos (Arch Ration Mech Anal 188:509–543, 2008), we prove that the sequence of weak solutions built by Desjardins has extra regularity linked to the divergence of the velocity without any relation assumption between λ and μ. Coupled with the uniform bound of the density property, this allows us to use appropriate defect measures and their nice
properties introduced and proved by Hillairet (Aspects interactifs de la m’ecanique des fluides, PhD Thesis, ENS Lyon, 2005) in order to prove that the Young measure associated to the weak limit is the convex combination of m Dirac measures. Finally, under a non-degeneracy assumption of this combination (“stratification” assumption), this provides
a multi-fluid system. Using a weak–strong uniqueness argument, we prove that this convex combination is the one corresponding
to the strong solution of the multi-fluid model built by Hillairet, if initial data are equal. We will briefly discuss this assumption. To complete the paper, we also present a blow-up criterion
for this multi-fluid system following (Huang et al. in Serrin type criterion for the three-dimensional viscous compressible flows, arXiv, 2010). 相似文献
7.
The authors have carried out a study to investigate and clarify the characteristics of purely oscillating pipe flows over
the developing region. The main objective of this study is to establish the method of time-dependent velocity profiles obtained
by the ultrasonic velocity profile (UVP) measurement method. First, the relationship between the test fluids and the microparticles,
as reflectors of ultrasonic pulses, was investigated. In addition, the relationship between the sound speeds of the test fluids
and the wall materials was studied. Second, the UVP was used to obtain the instantaneous velocity profiles in oscillating
pipe flows, and the developing characteristics of the flows were analyzed. Finally, the “entrance length” (by analogy with
a unidirectional pipe flow) required for oscillating pipe flows was analyzed by examining the amplitude of the harmonic spectral
components of the oscillating frequency. A fast Fourier transform (FFT) is proposed as the applicable method to estimate the
“entrance length”. From the Fourier transform of the velocity on the centerline, nonlinear oscillation of fluid occurs in
the “entrance length” of the oscillating flows, and the viscous dissipation of the higher-order velocity harmoncis determines
the entrance region. The “entrance length” can be obtained from the dissipation length of the third-order harmonic. These
results prove that the UVP method is highly applicable to carry out the flow measurement in the “entrance length” of oscillating
pipe flow.
Received: 20 March 2000 / Accepted: 10 August 2001 相似文献
8.
Klaus Zimmermann Igor Zeidis Michail Pivovarov Carsten Behn 《Archive of Applied Mechanics (Ingenieur Archiv)》2010,80(11):1317-1328
This paper deals with the analysis of a one-dimensional motion of two mass points in a resistive medium. The force of resistance
is described by small non-symmetric viscous friction acting on each mass point. The magnitude of this force depends on the
direction of motion. The mass points are interconnected with a kinematic constraint or with an elastic element. Using the
averaging method the expressions for the stationary “on the average” velocity of the systems’s motion as a single whole is
found. In case of a small degree of non-symmetry an explicit expression for the stationary “on the average” velocity of the
system is derived. For the other case we obtained algebraic equations for the corresponding stationary velocity. 相似文献
9.
F. Fleury 《Wave Motion》1980,2(1):39-50
A theory is developed for the propagation of waves in a suspension of elastic or rigid solid particles in a viscous or inviscid compressible fluid, using a homogenization process. We study the case where the characteristic length of the particles is small compared with the wave length. In the case of a viscous fluid, a law similar to Darcy's law for the average velocity of the suspension is established, and in the case of macroscopic homogeneity and isotropy, the propagation of a plane wave displays one dilatational, damped and dispersive wave. In the case of a barotropic inviscid fluid, the average acceleration of the suspension depends, in a linear way, on the mean pressure gradient and in the case of macroscopic homogeneity and isotropy, the propagation of a plane wave displays one dilatational, undamped and non dispersive wave. 相似文献
10.
The viscous fingering instability of miscible reactive–dispersive flows in a homogeneous porous media is investigated through
nonlinear numerical simulations. In particular, the role of velocity-dependent transverse and longitudinal dispersions as
well as the type and rate of auto-catalytic chemical reactions is analyzed. It is found that for a third-order auto-catalytic
reaction, the higher the reaction rate, the more complex the finger structures. Furthermore, major differences between the
flow development of third-order and second-order autocatalytic reactions are reported. In addition, the anisotropy and velocity
dependence of the dispersion tensor are found to have a more profound effect on the fingering instability in the case of reactive
flows than in the non-reactive ones. The qualitative characterization of the finger structures is explained by examining the
flow velocity field and further quantified through an analysis of the average concentration and relative contact area. 相似文献
11.
We study the stability and pointwise behavior of perturbed viscous shock waves for a general scalar conservation law with
constant diffusion and dispersion. Along with the usual Lax shocks, such equations are known to admit undercompressive shocks.
We unify the treatment of these two cases by introducing a new wave-tracking method based on “instantaneous projection”, giving
improved estimates even in the Lax case. Another important feature connected with the introduction of dispersion is the treatment
of a non-sectorial operator. An immediate consequence of our pointwise estimates is a simple spectral criterion for stability
in all L
p
norms, p≥ 1 for the Lax case and p > 1 for the undercompressive case.
Our approach extends immediately to the case of certain scalar equations of higher order, and would also appear suitable for
extension to systems.
Accepted May 29, 2000?Published online November 16, 2000 相似文献
12.
Robert M. Lewis 《Archive for Rational Mechanics and Analysis》1960,5(1):355-381
The basic formulas of classical equilibrium statistical mechanics are derived from well-known theorems in measure theory and
ergodic theory. The method used is a generalization of the methods of Khinchin and Grad and deals with several, in fact a “complete set”, of “invariants” or “integrals of the motion”. Most of the results are simple
corollaries of Birkhoff's ergodic theorem, and since time-averages are used, the whole approach is characterized by an absence of statistical “ensembles”
and probability notions. In the course of the development a “generalized temperature” is introduced, and a generalization
of the second law of thermodynamics is derived. Formulas for the “microcanonical”, “canonical”, and “grand canonical” distributions
appear as special cases of the general theory. 相似文献
13.
Antonio Barletta Eugenia Rossi di Schio Leiv Storesletten 《Transport in Porous Media》2010,81(3):461-477
The vertical throughflow with viscous dissipation in a horizontal porous layer is studied. The horizontal plane boundaries
are assumed to be isothermal with unequal temperatures and bottom heating. A basic stationary solution of the governing equations
with a uniform vertical velocity field (throughflow) is determined. The temperature field in the basic solution depends only
on the vertical coordinate. Departures from the linear heat conduction profile are displayed by the temperature distribution
due to the forced convection effect and to the viscous dissipation effect. A linear stability analysis of the basic solution
is carried out in order to determine the conditions for the onset of convective rolls. The critical values of the wave number
and of the Darcy–Rayleigh number are determined numerically by the fourth-order Runge–Kutta method. It is shown that, although
generally weak, the effect of viscous dissipation yields an increase of the critical value of the Darcy–Rayleigh number for
downward throughflow and a decrease in the case of upward throughflow. Finally, the limiting case of a vanishing boundary
temperature difference is discussed. 相似文献
14.
Thermodynamical considerations have largely been avoided in the modelling of complex fluids by invoking the assumption of incompressibility. This approximation allows pressure to be defined as a Lagrange multiplier, and therefore its natural connection with other thermodynamic variables such as density and temperature is irretrievably lost. Relaxing this condition to allow more realistic modelling involves much more than prescribing an equation of state. Even for a simple isothermal viscoelastic model, as explored in this paper, the transition to a compressible model is non-trivial. This paper shows that pressure enters the governing equations in a non-intuitive way. Furthermore, a fluid volume element, which is no longer constant, radically changes the way the basic element of the constitutive equations is viewed—stress is no longer the fundamental constitutive link between the momentum equations and velocity. The importance of geometry in fluid modelling is emphasised through the use of the Lie derivative, which is of a more fundamental character than the “upper” and “lower” convected derivatives prevalent in the literature and which are found to be almost redundant for a compressible fluid. There is now a strong body of non-equilibrium thermodynamics theory for flowing systems, which proves indispensible for this development. These fundamental principles are described herein using methodology and examples, that are sometimes conflicting, from the literature. The main conflict arises from the relationship between thermodynamic pressure and the trace of Cauchy stress, where the current preferred choice is (up to a constant) to set them equal—this is shown to be incorrect. Other issues such as the dependence of viscosity on density, bulk viscosity, integral modelling, the principle of objectivity and convected derivatives, are also clarified and resolved. 相似文献
15.
The rheological behaviour of Ca(OH)2 suspensions is investigated, predominantly at a solid volume fraction of 0.25. The influence of standing without being subject
to shear (“contact time”) is distinguished from that of being sheared (“shearing time”). The results are interpreted on the
basis of the “elastic floc” model of energy dissipation during flow, with a view to the problem whether, in addition to an
energy dissipation term related to the viscous drag experienced by particles moving within flocs, there should be an independent
energy dissipation term related to fluid movement in the flocs when they change volume or shape. It appears that this additional
energy dissipation term is not necessary, if the increase in viscous friction, experienced by two particles which are close
together, is taken into account.
Paper, presented at the First Conference of European Rheologists at Graz, April 14–16, 1982. A short version has been published
in [18]. 相似文献
16.
O. I. Kocherha 《Nonlinear Oscillations》2007,10(2):246-256
We prove the asymptotic character of a solution of the Cauchy problem for a singularly perturbed linear system of differential
equations with degenerate matrix of the coefficients of derivatives in the case where the limit matrix pencil is regular and
has multiple “finite” and “infinite” elementary divisors. We establish conditions under which the constructed formal solutions
are asymptotic expansions of the corresponding exact solutions.
__________
Translated from Neliniini Kolyvannya, Vol. 10, No. 2, pp. 247–257, April–June, 2007. 相似文献
17.
O. A. Shmagunov 《Journal of Applied Mechanics and Technical Physics》2012,53(1):20-26
Results obtained previously by the discrete vortex method with a “viscous” correction are generalized. The boundaries of applicability
of this method are determined. Previous results obtained for a flow past a flat plate are supplemented with solution convergence
estimates. Exhaustion of a plane jet of a viscous incompressible fluid into the ambient space is modeled. The geometric parameters
of the jet (its half-width, shapes of the streamwise velocity profiles, and intensity of oscillations) are analyzed. The calculated
results are found to agree well with experimental data and with results calculated by other methods. 相似文献
18.
Feimin Huang Jing Li Akitaka Matsumura 《Archive for Rational Mechanics and Analysis》2010,197(1):89-116
We are concerned with the large-time behavior of solutions of the Cauchy problem to the one-dimensional compressible Navier–Stokes
system for ideal polytropic fluids, where the far field states are prescribed. When the corresponding Riemann problem for
the compressible Euler system admits the solution consisting of contact discontinuity and rarefaction waves, it is proved
that for the one-dimensional compressible Navier–Stokes system, the combination wave of a “viscous contact wave”, which corresponds
to the contact discontinuity, with rarefaction waves is asymptotically stable, provided the strength of the combination wave
is suitably small. This result is proved by using elementary energy methods. 相似文献
19.
Fluid mechanical peristaltic transport through esophagus is studied in the paper.A mathematical model has been developed to study the peristaltic transport of a rheological fluid for arbitrary wave shapes and tube lengths.The Ostwald-de Waele power law of a viscous fluid is considered here to depict the non-Newtonian behaviour of the fluid.The model is formulated and analyzed specifically to explore some important information concerning the movement of food bolus through esophagus.The analysis is carried out by using the lubrication theory.The study is particularly suitable for the cases where the Reynolds number is small.The esophagus is treated as a circular tube through which the transport of food bolus takes place by periodic contraction of the esophageal wall.Variation of different variables concerned with the transport phenomena such as pressure,flow velocities,particle trajectory,and reflux is investigated for a single wave as well as a train of periodic peristaltic waves.The locally variable pressure is seen to be highly sensitive to the flow index "n".The study clearly shows that continuous fluid transport for Newtonian/rheological fluids by wave train propagation is more effective than widely spaced single wave propagation in the case of peristaltic movement of food bolus in the esophagus. 相似文献
20.
V. K. Kedrinskii 《Journal of Applied Mechanics and Technical Physics》2011,52(3):363-370
Specific features of the dynamics of the wave field structure and growth of a “collective” bubble behind the decompression
wave front in the “Lagrangian” section of the formed cavitation zone are numerically analyzed. Two cases are considered: with
no diffusion of the dissolved gas from the melt to cavitation nuclei and with the diffusion flux providing an increase in
the gas mass in the bubbles. In the first case, it is shown that an almost smooth decompression wave front approximately 100
m wide is formed, with minor perturbations that appear when the front of saturation of the cavitation zone with nuclei is
passed. In the case of the diffusion process, the melt state behind the saturation front is principally different: jumps in
mass velocity and viscosity are observed in the vicinity of the free surface, and the pressure in the “collective” cavitation
bubble remains unchanged for a sufficiently long time interval, despite the bubble growth and intense diffusion of the gas
from the melt. It is assumed that the diffusion process (and, therefore, viscosity) actually become factors determining the
dynamics of growth of cavitation bubbles beginning from this time interval. A pressure jump is demonstrated to form near the
free surface. 相似文献