Abstract:
The linear viscoelastic material functions of complex fluids relate stress and strain
when these assume sufficiently small values and are used to simulate fluid flow using
more sophisticated constitutive equations in complex flow regimes. The standard
method of determination is to apply a sinusoidal torque at discrete frequencies to obtain
the complex modulus at those frequency values. In this thesis, the implications of
using a completely arbitrary applied torque are investigated. Recent research has concentrated
on Fourier transform methods, but here the problem is analyzed in the timedomain
in terms of the relaxation modulus, which allows questions of well-posedness
to be more easily addressed.
The work falls into two main parts. The first part is concerned with the analysis of
the relationship between the applied torque and observed strain response. A variety of
candidate torque functions are considered and analytical expressions are obtained for
the simulated response using Laplace transform techniques, assuming known material
properties. The second part addresses questions concerning stability of the solution
of the Volterra integro-differential equation and methods of numerical solution. It is
demonstrated that the process of obtaining a solution for the relaxation modulus is
equivalent to solving a Volterra integral equation of the first kind, which is known to
be an ill-posed problem. Considering the governing equations in such a form allows
existing methods involving perturbed solutions to be adapted to provide estimates of
bounds on the error level in the data such that a stable solution can exist. It is shown
that the applied torque function which minimizes the ill-posedness of the problem is one
that corresponds to a kernel with one-smoothing characteristics. Finally, discretization
and regularization schemes for numerical solution of the problem are discussed and an
existing predictor-corrector regularization scheme is implemented which preserves the
Volterra (causal) nature of the problem and allows near real-time solution.
