Nahme Griffith Number Research Articles

Viscous Heating in Compressed Liquid Films

The greatest challenge in high-shear viscometry is the recognition of the difference between constitutive behavior and thermal feedback. The failure to realize this distinction was the cause of a serious mistake which has had a profound effect upon the EHL field for four decades. A traction curve is not a rheological flow curve. The classical approach to EHL has ignored the measurement of viscosity in viscometers at elevated pressures. As a consequence, the field has not benefitted from the understanding gained from directly observing the thermal softening effects of viscous heating. The Eyring sinh-law appears in high-shear viscometry when viscous heating causes thermal softening. The thermal limit of pressurized thin film Couette viscometers with high thermal conductivity cylinders is well-defined by the Nahme–Griffith number. The Spikes and Jie equation overcorrects at low values of the Nahme–Griffith number and undercorrects at high values. The viscous heating in a shear-thinning liquid may cause an apparent transition from power-law to sinh-law above some viscous power. Surprisingly, the sinh-law in this instance extrapolates back to the correct low-shear viscosity. The same response has been observed in one set of NEMD simulations.

Synchronized molecular-dynamics simulation for the thermal lubrication of a polymeric liquid between parallel plates

The Synchronized Molecular-Dynamics simulation, which was recently proposed by authors (Yasuda and Yamamoto, 2014), is applied to the analysis of polymer lubrication between parallel plates. The changes in rheological properties, conformational change of polymer chains, and temperature rise due to the viscous heating are investigated with varying values of thermal conductivity of the polymeric liquid. It is found that with a small applied shear stress on the plate, the temperature of the polymeric liquid only slightly increases in inverse proportion to the thermal conductivity; the apparent viscosity of the polymeric liquid is little affected by changing the thermal conductivity. However, at a large shear stress the transitional behaviors of the polymeric liquid are observed due to the interplay of the shear deformation and viscous heating by changing the thermal conductivity. This transition is characterized by the Nahme–Griffith number Na, which is defined as the ratio of the viscous heating to the thermal conduction at a characteristic temperature. When the Nahme–Griffith number exceeds unity, the temperature of the polymeric liquid increases rapidly and the apparent viscosity also exponentially decreases as the thermal conductivity decreases. The conformation of polymer chains is stretched and aligned by the shear flow when Na < 1, but the coherent structure becomes disturbed by the thermal motion of molecules when Na > 1.

Synchronized Molecular-Dynamics Simulation via Macroscopic Heat and Momentum Transfer: An Application to Polymer Lubrication

The Synchronized Molecular-Dynamics simulation which was recently proposed by authors [Phys. Rev. X {\bf 4}, 041011 (2014)] is applied to the analysis of polymer lubrication between parallel plates. The rheological properties, conformational change of polymer chains, and temperature rise due to the viscous heating are investigated with changing the values of thermal conductivity of the polymeric liquid. It is found that at a small applied shear stress on the plate, the temperature of polymeric liquid only slightly increases in inverse proportion to the thermal conductivity and the apparent viscosity of polymeric liquid is not much affected by changing the thermal conductivity. However, at a large shear stress, the transitional behaviors of the polymeric liquid occur due to the interplay of the shear deformation and viscous heating by changing the thermal conductivity. This transition is characterized by the Nahme-Griffith number $Na$ which is defined as the ratio of the viscous heating to the thermal conduction at a characteristic temperature. When the Nahme-Griffith number exceeds the unity, the temperature of polymeric liquid increases rapidly and the apparent viscosity also exponentially decreases as the thermal conductivity decreases. The conformation of polymer chains is stretched and aligned by the shear flow for $Na<1$, but the coherent structure becomes disturbed by the thermal motion of molecules for $Na>1$.

Multiscale simulation for thermo-hydrodynamic lubrication of a polymeric liquid between parallel plates

Thermo-hydrodynamic lubrication of a polymeric liquid composed of short chains between parallel plates is analysed by a multi-scale simulation, i.e. the synchronised molecular dynamics simulation via macroscopic heat and momentum transfer, which has been recently developed by us. The rheological properties and conformation of polymer chains coupled with the temperature rise caused by local viscous heating are investigated with a non-dimensional parameter, i.e. the Nahme–Griffith number, which is defined by the ratio of the viscous heating to the thermal conduction at the characteristic temperature required to sufficiently change the viscosity. The present simulation demonstrates that strong shear thinning and transitional behaviour of the conformation of the polymer chains occurs with a rapid temperature rise when the Nahme–Griffith number exceeds unity.

Sensitivity of calendered thickness to temperature variations for Newtonian fluids

In this work we treat theoretically the calendering process of Newtonian fluids with finite sheet initial thickness, taking into account that the viscosity of the fluid is a well-defined function of the temperature. We predict that the inclusion of temperature-dependent viscosity influences significantly on the exiting sheet thickness in the calendering process. The mass, momentum and energy balance equations, based on the lubrication theory, were nondimensionalized and solved for the velocity, pressure and temperature fields by using perturbation and numerical techniques, where the exiting sheet thickness represents an eigenvalue of the mathematical problem. When the above variables were obtained, the exiting sheet thickness in the calendering process was determined, considering the influence of the temperature variations in the process. The mentioned governing equations contain basically two dimensionless parameters: the well-known Graetz number, Gz; and a parameter that takes into account the effect of the variable viscosity as a function of the temperature, ϵ, defined as the ratio of the Nahme–Griffith number, Na, to the Graetz number, Gz. Using the limit of ϵ≪1, the dimensionless exiting sheet thickness of the calendering process has been obtained as a function of the involved dimensionless parameters. The numerical results show that the inclusion of temperature-dependent viscosity effect reduces about 5.89% the dimensionless exiting sheet thickness or 20.1% the leave-off distance in comparison with the case of temperature-independent viscosity.

Effect of temperature-dependent consistency index on the exiting sheet thickness in the calendering of power-law fluids

In this work we treat theoretically the calendering process of inelastic (power-law fluid) sheets of finite initial thickness, taking into account that the consistency index of the fluid is a well-defined function of the temperature. We predict the influence of the temperature-dependent consistency index on the exiting sheet thickness in the calendering process. The mass, momentum and energy balance equations, based on the lubrication theory, were nondimensionalized and solved for the velocity, pressure and temperature fields by using perturbation and numerical techniques, where the exiting sheet thickness represents an eigenvalue of the mathematical problem. When the above variables were obtained, the exiting sheet thickness in the calendering process was determined, considering the influence of the temperature variations in the process. The mentioned governing equations contain four dimensionless parameters: the Graetz number, Gz, a geometrical aspect ratio, β, the power law index of the fluid, n, and a parameter that takes into account the effect of the variable consistency index as a function of the temperature and ϵ, defined as the ratio of the Nahme–Griffith number, Na, to the Graetz number. Using the limit of ϵ ≪ 1, the dimensionless exiting sheet thickness of the calendering process have been obtained as a function of the involved dimensionless parameters. The numerical results show that the inclusion of temperature-dependent consistency index effect modifies the dimensionless exiting sheet thickness in about 6.91%.

Analysis of viscoelastic fluid flow with temperature dependent properties in plane Couette flow and thin annuli

The steady state flow in very thin annuli has been studied analytically for the case where the annular gap is much smaller than the radius of the inner cylinder and for the outer cylinder rotating at constant angular speed and the inner cylinder at rest. The cylinders were subjected to two different thermal boundary conditions. The exponential effect of temperature on the relaxation time and the viscosity coefficient was accounted into the governing differential equations using Nahme’s law. Effects of viscous dissipation as well as εDe 2 (viscoelastic index for SPTT constitutive equation) on the dimensionless velocity and temperature profiles have been investigated. Results show that while the properties of the fluid depend on temperature, the velocity and temperature profiles are different compared to those obtained with constant physical properties. The Nahme–Griffith number increases whereas εDe 2 as a viscoelastic index decreases when temperature dependent physical properties are considered. In addition, the results indicate that the viscous dissipation has a sensible effect on heat transfer and the Nusselt number decreases with an increase in the Nahme–Griffith number.

Analytical and Numerical Solutions for Torsional Flow between Coaxial Discs with Heat Transfer

We consider nonisothermal torsional flow between two coaxial parallel plates with heat transfer at the upper rotating plate, constant temperature on the lower stationary plate, and no heat loss at the fluid-air interface. Viscous heating is modelled by a Nahme law with exponential dependence on temperature. Due to the highly nonlinear nature of the governing equations an exact solution is not feasible. Therefore we solve the problem using both numerical and perturbation methods. Specifically, analytical solutions are obtained using asymptotic expansions based on the aspect ratio and the Nahme–Griffith number, a measure of viscous heating, as perturbation parameters. The numerical solutions are obtained by a finite element method. Good agreement is found between the analytical and numerical solutions in the appropriate parameter range. In viscometric applications the torque exerted by the fluid on the lower plate is an important quantity. For isothermal flow the dimensionless torque can easily be calculate...

Analytical solutions for non-isothermal viscoelastic torsional flow in a bounded domain

The effects of viscous heating on torsional flow of viscoelastic fluids in the parallel-plate viscometer are considered. The fluid is modelled by the Oldroyd-B constitutive equation with temperature-dependent viscosities and relaxation time which obey a Nahme type law. Analytic solutions are obtained in the limit of small Nahme–Griffith number which are uniformly valid up to the fluid–air interface. If the aspect ratio, α is small then there is a thermal boundary layer of width α at the free surface.

Experimental studies on the stability of Newtonian Taylor–Couette flow in the presence of viscous heating

The dramatic effects of viscous dissipation on the stability of Newtonian Taylor–Couette (TC) flows are studied experimentally using flow visualization techniques. Viscous heating, parameterized by the Nahme–Griffith number Na, drives a transition to a new, oscillatory mode of instability when coupled with the effects of centrifugal destabilization. This instability, consisting of travelling axisymmetric vortices, only occurs when viscous heating and centrifugal destabilization are both present. Step tests in cylinder velocity show that the time following initiation of shearing required for onset of instability scales well with the time for the fluid to reach a steady temperature profile under the action of viscous heating. The onset time can be dramatically reduced at fixed Na by increasing the centrifugal destabilization through the addition of co-rotation of the outer cylinder. The onset time can also be reduced while holding the centrifugal destabilization constant by increasing the amount of viscous heating (i.e. holding Reynolds number Re constant while increasing Na). The effects of viscous heating on the critical conditions of Newtonian TC flows are also quantified using ramp tests in cylinder velocity. These tests reveal the large extent to which viscous heating is destabilizing; at Na ≈ 2, a transition occurs at a critical Re that is less than 5% of the isothermal value.

Non-isothermal modification of purely elastic flow instabilities in torsional flows of polymeric fluids

Previous experimental measurements and linear stability analyses of curvilinear shearing flows of viscoelastic fluids have shown that the combination of streamwise curvature and elastic normal stresses can lead to flow destabilization. Torsional shear flows of highly elastic fluids with closed streamlines can also accumulate heat from viscous dissipation resulting in nonuniformity in the temperature profile within the flow and nonlinearity in the viscometric properties of the fluid. Recently, it has been shown by Al-Mubaiyedh et al. [Phys. Fluids 11, 3217 (1999)] that the inclusion of energetics in the linear stability analysis of viscoelastic Taylor–Couette flow can change the dominant mode of the purely elastic instability from a nonaxisymmetric and time-dependent secondary flow to an axisymmetric stationary Taylor-type toroidal vortex that more closely agrees with the stability characteristics observed experimentally. In this work, we present a detailed experimental study of the effect of viscous heating on the torsional steady shearing of elastic fluids between a rotating cone and plate and between two rotating coaxial parallel plates. Elastic effects in the flow are characterized by the Deborah number, De, while the magnitude of the viscous heating is characterized by the Nahme–Griffith number, Na. We show that the relative importance of these two competing effects can be quantified by a new dimensionless thermoelastic parameter, Θ=Na1/2/De, which is a material property of a given viscoelastic fluid independent of the rate of deformation. By utilizing this thermoelastic number, experimental observations of viscoelastic flow stability in three different fluids and two different geometries over a range of temperatures can be rationalized and the critical conditions unified into a single flow stability diagram. The thermoelastic number is a function of the molecular weight of the polymer, the flow geometry, and the temperature of the test fluid. The experiments presented here were performed using test fluids consisting of three different high molecular weight monodisperse polystyrene solutions in various flow geometries and over a large range of temperatures. By systematically varying the temperature of the test fluid or the configuration of the test geometry, the thermoelastic number can be adjusted appreciably. When the characteristic time scale for viscous heating is much longer than the relaxation time of the test fluid (Θ≪1) the critical conditions for the onset of the elastic instability are in good agreement with the predictions of isothermal linear stability analyses. As the thermoelastic number approaches a critical value, the strong temperature gradients induced by viscous heating reduce the elasticity of the test fluid and delay the onset of the instability. At even larger values of the thermoelastic parameter, viscous heating stabilizes the flow completely.

A NUMERICAL INVESTIGATION OF NON-ISOTHERMAL EXTRUSION THROUGH ANNULAR DIES

The process of non-isothermal extrusion of Newtonian fluids through annular dies is studied numerically using a finite element method; in particular the phenomenon of extrudate swell is considered. It is shown that swelling of the extrudate thickness increases with increasing Nahme–Griffith number, Na, but decreases with higher Péclet number, Pe, in qualitative agreement with an extrapolated interpretation of a previous theory. In particular a combination of high Na and low Pe values, or equivalently in the present context high sensitivity of the fluid viscosity to temperature change and narrow annular gap, can result in very large extrudate thickness swell; here a maximum swell value of about 108% relative to the annular gap size has occurred. The large swell values obtained are much higher than those reported in the literature, and show that while geometry and temperature have been recognised as having influence on extrudate swell, the extent to which their combination can affect this phenomenon can be much more than perhaps so far realised. Changes in the extrudate's outer and inner radii, on the other hand, are much smaller, with the outer radius generally following thickness swell's trend, whereas the inner radius varies in the opposite direction. The study also shows that a natural tendency for the extrudate is to swing outward. The very similar behaviour of the maximum pressure and maximum temperature of the flow field is also demonstrated.

Viscous heating correction for thermally developing flows in slit die viscometry

In the thermally developing region, dπ yy /dx| y=h varies along the flow direction x, where π yy denotes the component of stress normal to the y-plane; y = ±h at the die walls. A finite element method for two-dimensional Newtonian flow in a parallel slit was used to obtain an equation relating dπ yy /dx/ y=h and the wall shear stress σω0 at the inlet; isothermal slit walls were used for the calculation and the inlet liquid temperature T0 was assumed to be equal to the wall temperature. For a temperature-viscosity relation η/η0 = [1+β(T−T0]−1, a simple expression [(hdπ yy /dx/ y=h )/σ w0] = 1−[1-F c(Na)] [M(χ)+P(Pr) ·Q(Gz −1)] was found to hold over the practical range of parameters involved, where Na, Gz, and Pr denote the Nahme-Griffith number, Graetz number, and Prandtl number; χ is a dimensionless variable which depends on Na and Gz. An order-of-magnitude analysis for momentum and energy equations supports the validity of this expression. The function F c(Na) was obtained from an analytical solution for thermally developed flow; F c(Na) = 1 for isothermal flow. M(χ), P(Pr), and Q(Gz) were obtained by fitting numerical results with simple equations. The wall shear rate $$\dot \gamma _{w0} $$ at the inlet can be calculated from the flow rate Q using the isothermal equation.

Some finite elements studies of extrusion

Some finite element results of the non-isothermal extrusion process are presented. For Newtonian fluids, increasing the Nahme-Griffith number Na is found to increase the swelling ratio ϵ. The temperature distribution near the exit plane is shown to be a critical factor determining whether the extrudate would swell or contract relative to the isothermal, Newtonian case. In this waya recent theory by Tanner has been partly verified. For non-Newtonian fluids of the power-law type, where the viscosity is an exponential function of the temperature, increasing the exponential coefficient α (or decreasing the Péclet number Pe causes an increase in ϵ when the power-law index n and the (modified) Brinkman number Br are kept fixed. However, the effects are small for shear-thinning fluids (n<1), but become stronger as n increases. On the other hand, at fixed (Br) and (Pe) (thus fixing α) increasing n causes rapid increases in ϵ

Channel flow with temperature-dependent viscosity and internal viscous dissipation

This paper uses asymptotic methods to analyse the flow in a narrow channel of a fluid with temperature-dependent viscosity and internal viscous dissipation. When the Nahme–Griffith number is large we show how the flow evolves from Poiseuille flow with a uniform temperature distribution to a plug flow with hot boundary layers on the walls. An asymptotic solution is obtained for the flow in the region of transition from Poiseuille to plug flow and an explicit equation is derived for the pressure gradient in terms of the local downstream co-ordinate in this transition region.

Variable-viscosity flows in channels with high heat generation

This paper presents a similarity solution for plane channel flow of a very viscous fluid, whose viscosity is exponentially dependent upon temperature, when heat generation is very large. A dimensionless formulation of the problem involves two length scales (the depthhand lengthl, respectively, of the channel), one velocity scale (the mean velocityVof the fluid along the channel), the thermal conductivityk, thermal diffusivitykand viscosityVof the fluid, and the temperature coefficientbof the viscosity. From these, two important dimensionless groups arise, the Graetz number (Gz = Vh2/kl) and the Nahme–Griffith number (G= μV2b/k). In the case of steady flow withG−1[Lt ]Gz−1[Lt ] 1 a thin thermal boundary layer of thickness proportional toGz−½arises at each wall with an even thinner shear layer, detached from the wall and embedded in the thermal boundary layer, of thickness proportional toGz−½(lnG)−1, coinciding with the region of maximum temperature (lnG)/b. The similarity variable is (Pe½y/x½) wherePeis the Péclet number (Vh/k) andyandxare measured away from and along (either) boundary wall. The analogous unsteady uniform flow solution is also given.