Steady shear viscosity profile

Interestingly, it has been shown experimentally for many systems (Cox, 1958) that the profile of the complex viscosity ( / ) as a function of the dynamic frequency (o) is equivalent to the profile of the steady-shear viscosity tj) with respect to the shear rate (y ) for the same system. That is. 

The relationship between steady-shear viscosity and dynamic-shear viscosity is also a common fundamental rheological relationship to be examined. The Cox-Merz empirical rule (Cox, 1958) showed for most materials that the steady-shear-viscosity-shear-rate relationship was numerically identical to the dynamic-viscosity-frequency profile, or r] y ) = r] m). Subsequently, modified Cox-Merz rules have been developed for more complex systems (Gleissle and Hochstein, 2003, Doraiswamy et al., 1991). For example Doriswamy et al. (1991) have shown that a modified Cox-Merz relationship holds for filled polymer systems for which r](y ) = t] (yco), where y is the strain amplitude in dynamic shear. 

Derive the equation for the steady state temperature profile in a simple shear flow with viscous dissipation. Assume a Newtonian viscosity model. 

In Figure 2.3a, we have a fluid between two large parallel plates separated by a distance H. This system is initially at rest however, at time t = 0, the lower plate is set in motion by a constant force F in the positive x-direction at a constant velocity v. As time proceeds, the fluid gains momentum, and achieves a linear steady-state velocity profile. Newton s law of viscosity relates the shear stress to the velocity gradient in a Newtonian fluid for a one-dimensional flow we have... 

Figure 4.7 shows the steady- and dynamic-viscosity profiles as functions of shear rate for a filled reactive epoxy-resin moulding compound. Here, interestingly, the Cox-Merz rule provides a better correlation than does the modified Cox-Merz rule. 

Section 4.2.1) gives a good description of the steady-shear-dynamic-shear relationship. They found that the following general relationship is a good representation of the chemo-viscosity profile ... 

The MRI-based viscosity measurement relies on local calculations of the velocity gradient based on using MRI velocity profiles to provide a wide range of shear viscosity-shear rate data. Two observations are used to characterize fully developed and steady laminar flow in a MRI-based viscometer the velocity profile and the pressure drop per unit tube length measurements. Viscosity measurements were acquired on strawberry milk samples using H P LG tubing for the sample channel. 

Low-shear-rate solution viscosity was measured on a Couette-type rheometer (Contraves LS 30) with a No. 1 bob and cup. The viscosity-shear rate profile was determined fi om 10" to 10 s" at 25 BC. The system was allowed to reach steady state at each shear rate before the measured viscosity was recorded. 

Optohydrodynamics Fluid Actuation by Light, Fig. 5 (a) Fluid velocity profile in a two-layer system in presence of an interfacial tension gradient notations are also illustrated, (b) Analytical resolution of the steady flow pattern and the interface deformation in a double-layer conflguration with same shear viscosities and Hi = 5 2 = 5w. (c) Interface deformation in a three-layer system composed of a thin Aim of thickness 2Hi bounded by two external liquid layers of same thickness H2 for Hi = O.lw and H2 = 2w. Top. The dald T > 0 case leads to the formation of a dimple. Bottom. The dat I < 0 case leads to the formation of a nose... 

The velocity profile for steady, fully developed, laminar flow in a pipe can be determined easily by the same method as that used in Example 1.9 but using the equation of a power law fluid instead of Newton s law of viscosity. The shear stress distribution is given by... 

Fluid Kinematics. Water flowing at a steady rate in a constant-diameter pipe has a constant averse velocity. The viscosity of water introduces shear stresses between particles that move at different velocities. The velocity of the particle adjacent to the wall of the pipe is zero. The velocity increases for particles away from the wall, and it reaches its maximum at the center of the pipe for a particular flow rate or pipe discharge. The velocity profile in a pipe has a parabolic shape. Hydraulic engineers use the average velocity of the velocity profile distribution, which is the flow rate over the cross-sectional area of the pipe. 

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