Eiler equation

If fat globules are present as separate particles, the fat content is <40% and the milk fat completely molten, milk and cream behave as Newtonian fluids at intermediate and high shear rates (Phipps, 1969 McCarthy, 2003), i.e., its viscosity is not influenced by shear rate (t = rj / y, where r is the shear stress [Pa], 17 is the viscosity [Pa s] and y is the shear rate[l/s]). For a Newtonian fluid, Eilers equation (Eilers, 1941) is generally obeyed (Walstra, 1995) ... 

The relative viscosity in equation 19 is obtained from the following modified Eiler equation ... 

See also the Eilers equation and Frankel-Acrivos equation. 

Eilers Equation n A modification of the Einstein Equation relating the viscosity ryyof a Newtonian liquid filled with spherical particles to the viscosity of the pure liquid 7]o and extending to concentrations above 10% of filler. It is ... 

Many equations similar to Eilers equation have been proposed for example, Maron s [21] equation is,... 

The Newtonian coefficient of viscosity at a given temperature for milk, creams and some concentrated milk products is related to the concentration of individual components by Eiler s equation ... 

Of a slightly different form is the empirical equation of Eilers [22] ... 

Eilers 49, 50) proposed the following empirical equation to fit his data on polydisperse bitumen emulsions ... 

Tests were conducted with a rotational viscometer with 30, 40, and 60 lb HPG/1000 gal fluids containing neutrally buoyant 60-100 mesh styrene divinylbenzene beads at concentrations up to 12 lb/gal and temperatures up to 65.5 °C. Data were gathered only at three shear rates 5, 170, and 1000 s-1. Their modified Eiler s equation was based on correlating relative viscosity as a function of clean fluid n values, solids concentration, and fracture shear rate. The gel concentration and temperature effects were incorporated into n. Figure 1 depicts the effect of polymer concentration on the relative viscosity of suspension at a shear rate of 170 s-1 and 23.9 °C. It can be seen that the lower polymer concentration has the greater viscosity ratio than the higher concentrations and that the difference between these increases with volume fraction solids. 

The Eilers (1941) equation has been frequently used to describe the behavior of such filled elastomeric systems, e.g., by Blatz (1956), Schwarzl et al (1965), and Landel (1958) and Fedors and Landel (1967) ... 

Filers 1941) (sometimes quoted as the Eilers-Van Dyck equation)... 

Values predicted by other equations such as those proposed by Kemer, Nielsen, Van der Poel, Grezczuk, Sato and Furukawa, Eilers-Van Dijck, and others fall between the bounds set by Eqs. 2 and 3. 

The reservations about determining equilibrium moduli and compliances for unloaded polymers, occasioned by slow approaches to elastic equilibrium, apply also to filled materials, but by making measurements at rather long times (or even dynamic measurements at low frequency, at elevated temperatures) the modulus or compliance can be obtained with sufficient accuracy to study its dependence on filler content and size. For noninteracting rigid fillers, the shear modulus appears to be independent of the particle size and increases with volume fraction of filler in accordance with an empirical equation of Eilers and van Dijk - ... 

In the smdy of mechanical properties of particulate filled polymers, numerous models were developed to predict the effect of the particles on tensile or shear modulus. Most of these were derived from rheological models such as Einstein s, Eilers and Mooney s equations. A strong relationship exists between rheology and mechanical properties measurements and such correlations were studied by Gahleitner et al [66], as well as by Pukansky and Tudos [67]. There seems to have a direct relation between viscosity and shear modulus [59]. However, compensation has to be taken for matrix s Poisson ratio which is lower than 0.5 as shown by Nielsen and Landel [59]. Nevertheless, these equations on modulus predictions can be broadly classified under two groups. 

In general the concentration dependence of viscosity of dispersed spheres may be described with a form of equation given by Eilers [19]. 

Eilers (87) proposed an empirical equation for the same purpose... 

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