AbstractThe 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 time domain 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.
|Date of Award||28 Oct 2009|
|Supervisor||David Binding (Supervisor), Andrew Robert Breen (Supervisor) & Russell Davies (Supervisor)|