The stability of plane Couette flow with viscous heating
01 Mar 1973-Journal of Fluid Mechanics (Cambridge University Press (CUP))-Vol. 57, Iss: 4, pp 651-670
TL;DR: In this paper, the stability of plane Couette flow with viscous heating of a Navier-Stokes-Pourier fluid with an exponential dependence of viscosity upon temperature is investigated.
Abstract: An investigation of the stability of plane Couette flow with viscous heating of a Navier–Stokes–Pourier fluid with an exponential dependence of viscosity upon temperature is presented. Using classical small perturbation theory, the stability of the flow can be described by a sixth-order set of coupled ordinary differential equations. Using Galerkin's method, these equations are reduced to an algebraic eigenvalue problem. An eigenvalue with a negative real part means that the flow is unstable.Neutral stability curves are determined at Brinkman numbers of 15, 19, 25, 30,40,80 and 600 for Prandtl numbers of 1, s and 50. A Brinkman number of 19 corresponds approximately to the maximum shear stress which can be applied to the system.The results indicate that four different modes of instability occur: one termed an inviscid mode, arising from an inflexion point in the primary flow; a viscous mode, due to the stratification of viscosity in the flow field and an associated diffusive mechanism; a coupling mode, resulting from the convective and viscous dissipation terms in the energy equation; and finally a purely thermal mode.
Citations
More filters
AT&T1
TL;DR: In this article, the authors present a review of the latest developments as well as earlier work in this area, organized into the following categories: Taylor-Couette flows, instabilities in cone and plate-and-plate flows, parallel shear flows, extrudate distortions and fracture, Instabilities in shear flow with interfaces, extensional flows, and thermohydrodynamic instabilities.
Abstract: Viscoelastic instabilities are of practical importance, and are the subject of growing interest. Reviewed here are the fresh developments as well as earlier work in this area, organized into the following categories: instabilities in Taylor-Couette flows, instabilities in cone-and-plate and plate-and-plate flows, instabilities in parallel shear flows, extrudate distortions and fracture, instabilities in shear flows with interfaces, instabilities in extensional flows, instabilities in multidimensional flows, and thermohydrodynamic instabilities. Emphasized in the review are comparisons between theory and experiment and suggested directions for future work.
883 citations
TL;DR: In this paper, the heat transfer in shear flow is analyzed and a large emphasis is laid on replacing the commonly used idealized boundary conditions–– constant wall temperature or constant wall heat flux by more general conditions.
Abstract: Publisher Summary Heat transfer in flowing molten polymers is largely influenced by rheology–– the rheological properties of the polymer and by the flow geometry. The rheology of steady shear flow can treat most of the heat transfer problems completely. This chapter discusses the heat transfer problem, and classifies the heat transfer and viscous dissipation in molten polymers. The heat transfer in shear flow is analyzed and a large emphasis is laid on replacing the commonly used idealized boundary conditions–– constant wall temperature or constant wall heat flux by more general conditions. The heat transfer at the wall is described by an outer temperature difference and the Biot number that is used successfully for describing the boundary conditions for temperature calculations in solids. The Biot number is appropriate for describing the boundary conditions between isothermal and adiabatical, as they occur in real processes. A unifying concept is developed that makes it possible to comprise the most important shear flow cases into a single one that can be solved with one numerical program. The nonviscometric flow in channels and flow with free boundaries is also discussed. An example of heat transfer in unsteady unidirectional shear flow is also provided.
146 citations
TL;DR: In this article, a two-dimensional high speed infrared (IR) camera was employed to observe the temperature field evolution during the initiation and propagation of dynamic shear bands in C300 maraging steel.
141 citations
Additional excerpts
...The linear stability of plane Couette ¯ow with viscous heating was numerically studied by Sukanek et al. (1973) and later with improved numerical accuracy by Yueh and Weng (1996)....
[...]
TL;DR: In this paper, the authors studied the thermal and mechanical effects caused by viscous heating in infinitely long tubes of finite lengths and found that viscous heat is responsible for the evolution from Poiseuille flow, with a uniform temperature distribution at the inlet, to a plug flow with a hotter layer near the walls.
Abstract: . Viscous heating plays an important role in the dynamics of fluids with strongly temperature-dependent viscosity because of the coupling between the energy and momentum equations. The heat generated by viscous friction produces a local temperature increase near the tube walls with a consequent decrease of the viscosity which may dramatically change the temperature and velocity profiles. These processes are mainly controlled by the Peclet number, the Nahme number, the flow rate and the thermal boundary conditions. The problem of viscous heating in fluids was investigated in the past for its practical interest in the polymer industry, and was invoked to explain some rheological behaviours of silicate melts, but was not completely applied to study magma flows. In this paper we focus on the thermal and mechanical effects caused by viscous heating in tubes of finite lengths. We find that in magma flows at high Nahme number and typical flow rates, viscous heating is responsible for the evolution from Poiseuille flow, with a uniform temperature distribution at the inlet, to a plug flow with a hotter layer near the walls. When the temperature gradients induced by viscous heating are very pronounced, local instabilities may occur and the triggering of secondary flows is possible. For completeness, this paper also describes magma flow in infinitely long tubes both at steady state and in transient phase.
99 citations
Cites result from "The stability of plane Couette flow..."
...The stability of the plane Couette flow was recently re-examined by Yueh and Weng (1996), who improve the results previously found by Sukanek et al. (1973)....
[...]
References
More filters
TL;DR: In this article, it was shown that the variation of viscosity in a fluid can cause instability, however small the Reynolds number is, and that the unstable modes are in the neighbourhood of a hidden neutral mode for the case of a single fluid, which is entirely ignored in the usual theory of hydrodynamic stability.
Abstract: The principal aim of this paper is to show that the variation of viscosity in a fluid can cause instability. Plane Couette-Poiseuille flow of two superposed layers of fluids of different viscosities between two horizontal plates is considered, and it is found that both plane Poiseuille flow and plane Couette flow can be unstable, however small the Reynolds number is. The unstable modes are in the neighbourhood of a hidden neutral mode for the case of a single fluid, which is entirely ignored in the usual theory of hydrodynamic stability, and are brought out by the viscosity stratification.
740 citations
TL;DR: In this article, it was shown that the breakdown of laminar flow depended primarily on the size of the initial disturbance, in agreement with the Reynold's view, and that the reoughness of the walls might not be the determining factor.
Abstract: The turbulence problem is still unsolved, through a number of valuable papers have been published on it comparatively recently. But, since Hopf and von Mises proved that uniform shearing motion between two parallel planes was stable for infinitesimal disturbances but unstable for disturbances of a finite size has become more and more widely held. Von mises suggested that the reoughness of the walls might be the determining factor, but the experiments of Schiller have shown that the degree of roughness of the walls is of negligible influence on the critical value of Reynold's number. He concluded that the breakdown of laminar flow depended primarily on the size of the initial disturbance, in agreement eith Osborne Reynold's view. Important papers have been published by Noether and Tollmien, whose conclusions are in contradiction to one another. On the one hand, Noether, by a formal investigation of the asymptotic solutions of the equation governing the two-dimensional disturbances of flow between parallel walls, claims to have proved that all velocity profiles are stable for all values of Reynolds' number. On the other hand, Tollmien has determined a critical value of Reynolds' number for the flow past a flat plate placed edgeways to the stream. This value is in good agreement with the experimental results. There are, however, certain points in his analysis which are not clear and it would be useful to know if the method gave results in agreement with those derived more strictly.
675 citations
TL;DR: The runner of a luggage case or the like provides a stud having shoulders engageable by a spring clip or slidable bolt member in a caster for enabling attachment of the caster to the runner.
338 citations
TL;DR: In this article, an unbounded parallel flow, consisting of a linear shear layer between uniform streams, is investigated for stability, and a conventional eigenvalue problem is formulated, and solved by both analytical and numerical methods.
Abstract: An unbounded parallel flow, consisting of a linear shear layer between uniform streams, is investigated for stability. A conventional eigenvalue problem is formulated, and solved by both analytical and numerical methods. The region of instability in the plane of Reynolds number R and disturbance wave number α is determined, and typical growth rates in the unstable region are computed.Unstable disturbances are found at all values of R. Results for αR > 100 are found to agree closely with inviscid theory results. An analytic method useful for αR < 1 is developed.The extent to which the present results can be applied to the laminar boundary layer between free streams is discussed.
83 citations