Timing is critical to assessing fluid performance in applications where streamlines converge, such as in spraying, rolling, mixing and squeezing. Because of extremely short relaxation times, Newtonian fluids do not store measurable elastic energy in shear or extensional flow, and the extensional viscosity is simply three times the shear viscosity. But non-Newtonian, viscoelastic fluids possess a microstructure that requires measurable time to relax and consequently can store appreciable elastic energy as well as dissipate viscous energy. Such structured fluids can show extensional viscosity and elasticity values that are many times larger than shear values. Current methods of measuring the extensional properties of fluids reveal only the effects of energy dissipation and miss the elastic properties that are needed to yield information on the timing of extensional structure formation and decay. Vilastic now provides a method to quantify these important extensional properties of liquids.

Flow entering an orifice from a larger tube produces both shear and extensional deformation of the fluid [1-9] where elastic energy is stored and viscous energy is dissipated in the fluid structure. Though both shear and extensional effects are present, the pressure across the orifice is often dominated by extensional effects. With the Vilastic VE System, Elongational Flow (Orifice) Attachment, VMax^© and Extensor^© software, then the shear and extensional components of the pressure can be resolved and viscous and elastic properties determined.

ORIFICE IMPEDANCE

Figure 1 shows an orifice plate partition in a large tube. In region (A) the flow lines converge, producing both shear and extensional deformation of the fluid but extensional effects dominate. In region (B) the shear effects dominate. Particularly at low amplitudes of oscillatory flow, the fluid movement in the region (B) is like an oscillating plane piston and can be described as shear flow in a tube with special end corrections. With increasing amplitude the piston-like profile becomes an undulating profile. As the amplitude is increased, the undulations develop into a vortex at the exit of the orifice. With a still greater increase in amplitude this vortex detaches from the orifice and is ejected into region (C) [10].

Figure 1. Orifice Section

The instantaneous pressure drop P can be separated into three components. Extensional effects in region (A) contribute a  pressure P_E. The shear effects in region (B) contribute a pressure P_S and the kinetic energy of fluid ejected into region (C) contribute a pressure P_K. So, 

The complex representation of the three components of the pressure (1) can be divided by the volume flow to obtain three complex impedances. Then the orifice impedance is

where R and X are the resistance and reactance. Thus the three parts of the R and X are

IMPEDANCES FOR POLYETHYLENE OXIDE

The impedance (resistance R and reactance X) of a non-Newtonian solution of 3333 ppm polyethylene oxide (Union Carbide, Mw = 5x10⁶) in distilled water was measured in an orifice with a radius = 0.0172 cm and effective length of 0.076 cm. The measurements were made at 2 Hz and at 22°C. Also, the shear impedance was measured in a long cylindrical tube with a radius = 0.0524 cm and effective length = 6.134 cm and were used to calculate R_S and X_S for the orifice.

 
Figure 2 shows the orifice impedance (R and X) together with shear impedance (R_S and X_S) and the kinetic energy impedance (R_K and X_K). At low volume flows the values of R and X of the orifice approximate R_S and X_S. When volume flows exceeds 0.0005 cm³/sec the orifice values become larger than the shear values and for volume flows above 0.01cm³/sec there is a dramatic increase in the orifice impedance. Because both the shear and kinetic resistance and reactance are well below the orifice values, the rising R and X are due to the extensional properties of the fluid.

Figure 2. Polyethylene Oxide in an Orifice.

The presence of NaCl changes the molecular conformation of xanthan gum (Keltrol from Kelco, San Diego, CA) from a random coil to helical [13]. The extensional impedance is sensitive to this conformational change. This is seen in measurements made at 2 Hz and at 22°C for two xanthan gum solutions at a concentration of 1000 ppm. One solution contained distilled water and the other 0.5% NaCl. The orifice radius was 0.0172 cm and effective length = 0.076 cm. Figure 3 shows that the resistance and reactance of xanthan gum in distilled water is higher than in 0.5% NaCl. The behavior of the two solutions in the orifice has a complex dependence on volume flow and this cannot be predicted from the shear properties alone.

Figure 3. Two Xanthan Gum Solutions in an Orifice

The extensional impedance component of the measured impedance of a fluid in an orifice can be used to calculate extensional viscosity and elasticity. For a non-Newtonian fluid, the extensional components are determined using equations (3) and (4) and the following procedures. First the shear viscosity, h '_S, and shear elasticity, h"_S, are measured in pure shear using a long cylindrical tube with the Vilastic VE System and VMax^© software. Then measurements are made of the impedance components R and X versus volume flow for the test fluid in the orifice. With the density, shear viscosity and shear elasticity Extensor^© calculates the impedance components R_S and X_S of the orifice [12]. Extensor^© also calculates the kinetic impedance terms R_K and X_K [11]. The calculated shear impedance and kinetic impedance are subtracted from the measured orifice impedance to yield the extensional impedance.

The components of the extensional impedance, R_E and X_E are used to calculate the extensional components of the orifice pressure drop

where U is the volume flow, is the pressure component in phase with U (due to viscous energy loss) and is the pressure in quadrature with U (due to elastic energy storage).

To determine the extension rate sink flow analysis is used [2]

where a is the radius of the orifice, U is the volume flow and f is the half angle of convergence of the flow approaching the orifice. Flow visualization studies have shown that the entry flow geometry varies with volume flow [5,10]. However, lacking precise information on convergence angle, a constant reference angle can be used. Here, an angle of 15° is selected [2]. Changing this angle will shift the values of the extension rate and the extensional viscosity and elasticity. However, the shift does not change the characteristic shape of the relationship between viscoelasticity and extension rate.

The extensional viscosity and elasticity are related to the pressure P'_E and P"_E

Figure 4 shows the calculated extensional viscosity and elasticity for the PEO together with published steady flow extensional viscosity data [12]. The two extensional viscosities display similar character. No prior extensional elasticity data exists for comparison.

Figure 4. Extensional Viscosity and Elasticity of PEO

Since both an extensional viscosity and extensional elasticity are obtained, an apparent extensional relaxation time can be calculated

where w is the radian frequency of oscillation equal to 2*p*frequency. For the example in Figure 4 at an extension rate of 2049/sec the extensional relaxation time is 0.055 sec. The Trouton ratios for viscosity and elasticity are

For the example in Figure 4 at an extension rate of 2049/sec (orifice shear rate 16100/sec), TR_viscous = 72.4 and TR_elastic=48400.

Orifice resistance and reactance are precise indices for quantifying the rheological properties of a fluid in elongational flow. The orifice impedance is a direct reflection of extensional effects and is not subject to the approximations needed for calculation of extensional viscoelasticity. However, by assuming an angle of convergence of the fluid entering the orifice and separating out the shear effects, then the extensional viscosity and elasticity can be determined by using the measured orifice impedance. From this relaxation times and Trouton ratios are also obtained and can be correlated with fluid performance in an application.

[14] P. Dontula, M. Pasquali, L. E. Scriven and C. W. Macosko, "Can extensional viscosity be measured with opposed nozzle devices?" Rheologica Acta, Vol. 36, 429-448 (1997).