A simple maxwell based model in order to represent the frequency-dependent viscosity measured by ultrasound

A previous work showed that viscosity values measured at high frequency by ultrasound agreed with the values measured at low frequency by the rotational viscometer when special conditions are met, such as relatively low frequency and viscosity. However, these conditions strongly reduce the useful range of the measurement cell. In order to obtain a better measurement range and sensitivity high frequency must be used, but it causes a frequency-dependent decrease on the obtained viscosity values. This work introduces a new simple model in order to represent this frequency-dependent behavior. The model is based on the Maxwell model for viscoelastic materials, but using a variable parameter. This parameter has a physical meaning because it represents the linear behavior of the apparent elasticity measured along with the viscosity by ultrasound. Automotive oils SAE 90 and SAE 250 at 22.5plusmn0.5degC with viscosities at low frequency of 0.6 and 6.7 Pa.s, respectively, were tested in the range of 1-5 MHz. The model was used in order to fit the obtained data using an algorithm of non-linear regression in Matlab. By including the viscosity at low frequency as an unknown fitting parameter, it is possible to extrapolate its value. Relative deviations between the values measured by the viscometer and extrapolated using the model for the SAE 90 and SAE 250 oils were 5.0% and 15.7%, respectively.