Nonlinear equilibrium solutions for the channel flow of fluid with temperature-dependent viscosity

2000 ◽  
Vol 406 ◽  
pp. 1-26 ◽  
Author(s):  
D. P. WALL ◽  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Secondary Flow ◽  
Critical Reynolds Number ◽  
Isothermal Flow ◽  
Basic Flow ◽  
Secondary Flows ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Viscosity Models ◽  
Dependent Viscosity

The nonlinear stability of the channel flow of fluid with temperature-dependent viscosity is considered for the case of vanishing Péclet number for two viscosity models, μ(T), which vary monotonically with temperature, T. In each case the basic state is found to lose stability from the linear critical point in a subcritical Hopf bifurcation. We find two-dimensional nonlinear time-periodic flows that arise from these bifurcations. The disturbance to the basic flow has wavy streamlines meandering between a sequence of triangular-shaped vortices, with this pattern skewing towards the channel wall which the basic flow skews towards. For each of these secondary flows we identify a nonlinear critical Reynolds number (based on half-channel width and viscosity at one of the fixed wall temperatures) which represents the minimum Reynolds number at which a secondary flow may exist. In contrast to the results for the linear critical Reynolds number, the precise form of μ(T) is not found to be qualitatively important in determining the stability of the thermal flow relative to the isothermal flow. For the viscosity models considered here, we find that the secondary flow is destabilized relative to the corresponding isothermal flow when μ(T) decreases and vice versa. However, if we remove the bulk effect of the non-uniform change in viscosity by introducing a Reynolds number based on average viscosity, it is found that the form of μ(T) is important in determining whether the thermal secondary flow is stabilized or destabilized relative to the corresponding isothermal flow. We also consider the linear stability of the secondary flows and find that the most unstable modes are either superharmonic or subharmonic. All secondary disturbance modes are ultimately damped as the Floquet parameter in the spanwise direction increases, and the last mode to be damped is always a phase-locked subharmonic mode. None of the secondary flows is found to be stable to all secondary disturbance modes. Possible bifurcation points for tertiary flows are also identified.

scholarly journals A Novel Viscosity-Temperature Model of Glass-Forming Liquids by Modifying the Eyring Viscosity Equation

2020 ◽  
Vol 10 (2) ◽  
pp. 428 ◽  
Author(s):  
Chunyu Chen ◽  
Huidan Zeng ◽  
Yifan Deng ◽  
Jingtao Yan ◽  
Yejia Jiang ◽  
...  
Keyword(s):  
Mathematical Expression ◽  
Impact Factors ◽  
Viscosity Data ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Regression Methods ◽  
Glass Forming ◽  
Viscosity Models ◽  
Polymer Liquids ◽  
Dependent Viscosity

Many models have been created and attempted to describe the temperature-dependent viscosity of glass-forming liquids, which is the foundational feature to lay out the mechanism of obtaining desired glass properties. Most viscosity models were generated along with several impact factors. The complex compositions of commercial glasses raise challenges to settle these parameters. Usually, this issue will lead to unsatisfactory predicted results when fitted to a real viscosity profile. In fact, the introduction of the reliable viscosity-temperature data to viscosity equations is an effective approach to obtain the accurate parameters. In this paper, the Eyring viscosity equation, which is widely adopted for molecular and polymer liquids, was applied in this case to calculate the viscosity of glass materials. On the basis of the linear variation of molar volume with temperature during glass cooling, a modified temperature-dependent Eyring viscosity equation was derived with a distinguished mathematical expression. By means of combining high-temperature viscosity data and the glass transition temperature (Tg), nonlinear regression analysis was employed to obtain the accurate parameters of the equation. In addition, we have demonstrated that the different regression methods exert a great effect on the final prediction results. The viscosity of a series of glasses across a wide temperature range was accurately predicted via the optimal regression method, which was further used to verify the reliability of the modified Eyring equation.

Developing laminar flow in a pipe of circular cross-section

1971 ◽  
Vol 321 (1547) ◽  
pp. 461-476 ◽  
Keyword(s):  
Laminar Flow ◽  
Cross Section ◽  
Analytical Approach ◽  
Circular Cross Section ◽  
Isothermal Flow ◽  
Experimental Measurements ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Flow Development ◽  
Dependent Viscosity

A method is described of predicting flow development and pressure drop in the inlet of a uniform pipe of circular cross-section for the incompressible isothermal laminar flow of a Newtonian fluid. Employment of both the differential and integral momentum equations leads to greater simplicity of expression than is found in other theoretical treatments, though the solutions obtained compare equally favourably with the limited experimental measurements available; those reported herein suggest the need for a more generalized analytical approach which covers non-isothermal flow and the effect on flow development of temperature-dependent viscosity.

scholarly journals Viscous heating in fluids with temperature-dependent viscosity: implications for magma flows

2003 ◽  
Vol 10 (6) ◽  
pp. 545-555 ◽  
Author(s):  
A. Costa ◽  
G. Macedonio
Keyword(s):  
Plug Flow ◽  
Practical Interest ◽  
Viscous Friction ◽  
Secondary Flows ◽  
Viscous Heating ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Magma Flow ◽  
Energy And Momentum ◽  
Dependent Viscosity

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 Peclét 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.

The linear stability of channel flow of fluid with temperature-dependent viscosity

1996 ◽  
Vol 323 ◽  
pp. 107-132 ◽  
Author(s):  
D. P. Wall ◽  
S. K. Wilson
Keyword(s):  
Linear Stability ◽  
Channel Flow ◽  
Finite Difference Methods ◽  
Fluid Viscosity ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Viscosity Models ◽  
Shape Effects ◽  
The Stability ◽  
Dependent Viscosity

The classical fourth-order Orr-Sommerfeld problem which arises from the study of the linear stability of channel flow of a viscous fluid is generalized to include the effects of a temperature-dependent fluid viscosity and heating of the channel walls. The resulting sixth-order eigenvalue problem is solved numerically using high-order finite-difference methods for four different viscosity models. It is found that temperature effects can have a significant influence on the stability of the flow. For all the viscosity models considered a non-uniform increase of the viscosity in the channel always stabilizes the flow whereas a non-uniform decrease of the viscosity in the channel may either destabilize or, more unexpectedly, stabilize the flow. In all the cases investigated the stability of the flow is found to be only weakly dependent on the value of the Péclet number. We discuss our results in terms of three physical effects, namely bulk effects, velocity-profile shape effects and thin-layer effects.

Turbulence and internal waves in stably-stratified channel flow with temperature-dependent fluid properties

2012 ◽  
Vol 697 ◽  
pp. 175-203 ◽  
Author(s):  
Francesco Zonta ◽  
Miguel Onorato ◽  
Alfredo Soldati
Keyword(s):  
Reynolds Number ◽  
Thermal Expansion ◽  
Thermal Expansion Coefficient ◽  
Channel Flow ◽  
Expansion Coefficient ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Fluid Properties ◽  
Effects Of Temperature ◽  
Dependent Viscosity

AbstractDirect numerical simulation (DNS) is used to study the behaviour of stably-stratified turbulent channel flow with temperature-dependent fluid properties: specifically, viscosity ($\ensuremath{\mu} $) and thermal expansion coefficient ($\ensuremath{\beta} $). The governing equations are solved using a pseudo-spectral method for the case of turbulent water flow in a channel. A systematic campaign of simulations is performed in the shear Richardson number parameter space (${\mathit{Ri}}_{\tau } = \mathit{Gr}/ {\mathit{Re}}_{\tau } $, where $\mathit{Gr}$ is the Grashof number and ${\mathit{Re}}_{\tau } $ the shear Reynolds number), imposing constant-temperature boundary conditions. Variations of ${\mathit{Ri}}_{\tau } $ are obtained by changing ${\mathit{Re}}_{\tau } $ and keeping $\mathit{Gr}$ constant. Independently of the value of ${\mathit{Ri}}_{\tau } $, all cases exhibit an initial transition from turbulent to laminar flow. A return transition to turbulence is observed only if ${\mathit{Ri}}_{\tau } $ is below a threshold value (which depends also on the flow Reynolds number). After the transient evolution of the flow, a statistically-stationary condition occurs, in which active turbulence and internal gravity waves (IGW) coexist. In this condition, the transport efficiency of momentum and heat is reduced considerably compared to the condition of non-stratified turbulence. The crucial role of temperature-dependent viscosity and thermal expansion coefficient is directly demonstrated. The most striking feature produced by the temperature dependence of viscosity is flow relaminarization in the cold side of the channel (where viscosity is higher). The opposite behaviour, with flow relaminarization occurring in the hot side of the channel, is observed when a temperature-dependent thermal expansion coefficient is considered. We observe qualitative and quantitative modifications of structure and wall-normal position of internal waves compared to previous results obtained for uniform or quasi-uniform fluid properties. From the trend we observe in the investigated low-Reynolds-number range, we can hypothesize that, whereas the effects of temperature-dependent viscosity may be masked at higher Reynolds number, the effects of temperature-dependent thermal expansion coefficient will persist.

Extension of a viscous thread with temperature-dependent viscosity and surface tension

2016 ◽  
Vol 800 ◽  
pp. 720-752 ◽  
Author(s):  
Dongdong He ◽  
Jonathan J. Wylie ◽  
Huaxiong Huang ◽  
Robert M. Miura
Keyword(s):  
Surface Tension ◽  
Reynolds Number ◽  
Temperature Profile ◽  
Solid Wall ◽  
Analytic Solutions ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Initial Shape ◽  
Vertically Aligned ◽  
Dependent Viscosity

We consider the evolution of a long and thin vertically aligned axisymmetric viscous thread that is composed of an incompressible fluid. The thread is attached to a solid wall at its upper end, experiences gravity and is pulled at its lower end by a fixed force. As the thread evolves, it experiences either heating or cooling by its environment. The heating affects the evolution of the thread because both the viscosity and surface tension of the thread are assumed to be functions of the temperature. We develop a framework that can deal with threads that have arbitrary initial shape, are non-uniformly preheated and experience spatially non-uniform heating or cooling from the environment during the pulling process. When inertia is completely neglected and the temperature of the environment is spatially uniform, we obtain analytic solutions for an arbitrary initial shape and temperature profile. In addition, we determine the criteria for whether the cross-section of a given fluid element will ever become zero and hence determine the minimum stretching force that is required for pinching. We further show that the dynamics can be quite subtle and leads to surprising behaviour, such as non-monotonic behaviour in time and space. We also consider the effects of non-zero Reynolds number. If the temperature of the environment is spatially uniform, we show that the dynamics is subtly influenced by inertia and that the location at which the thread will pinch is selected by a competition between three distinct mechanisms. In particular, for a thread with initially uniform radius and a spatially uniform environment but with a non-uniform initial temperature profile, pinching can occur either at the hottest point, at the points near large thermal gradients or at the pulled end, depending on the Reynolds number. Finally, we show that similar results can be obtained for a thread with initially uniform radius and uniform temperature profile but exposed to a spatially non-uniform environment.

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