However, the shear-rate dependent equation 30 is the five-parameter model, while the shear-stress dependent model (Eq.
The corresponding viscosity curves are equally well described by the inverse three-constant shear-stress dependent Kohlrausch model and by the four-constant shear-rate dependent equation 28.
For the description of more complicated viscosity curves by means of shear-rate dependent equations the multimode models can also be applied.
The increase in mobility of the charge carriers with temperature is governed by the following Arrhenius dependent equation [5, 6]:
The relation between temperature and resistivity maybe governed by the following Arrhenius dependent equation [6, 7, 14, 15, 18]:
The integration of equation (4) is the dependent equation
of crack length function to the cycles number.
The structural function is related to the structural parameter of stress dependent equation written as:
As can be seen for all three samples, the energy dependent equation can better predict the overshoot in early stage of experiment.
Results show the stress dependent equation has higher contribution to stress growth behavior of the neat blend and the sample containing 2% hydrophobic nanorods (samples phase separating under VPS mechanism) while the energy dependent equation controls the flow behavior of the sample containing 2% hydrophilic nanorods (sample phase separating under SD mechanism).
In essence, for nonlinear time dependent equations
, the ETD scheme provides a systematic coupling of the explicit treatment of nonlinearities and the implicit and possibly exact integration of the stiff linear part of the equation, while achieving high accuracy and maintaining good stability.
As it was shown above in the equations (16)-(17) the time dependent equations
are nonlinear and their Laplace transformation could be produced only by changing the equations the equivalent, which could be transformed.
The temporal problem of equation 2 was solved using standard finite difference techniques for time dependent equations