Controllability of the Stresses in Multimode Viscoelastic Fluid of Upper Convected Maxwell Type
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Viscoelastic fluids, or Non-Newtonian fluids, are those that do not have a linear algebraic relation between the velocity field and the stresses arising in the media. Such fluids exhibit properties of both solids and liquids, and therefore cannot be modeled with methods of elasticity or Newtonian fluid mechanics. The popular models of viscoelasticity differ from each other only by the differential equation that describes the constitutive law for the fluid. Also, the media can have several relaxation modes, such as fluid mixes. This means that the stresses are determined as the sum of the stresses for each individual relaxation mode, which are described by corresponding differential equations evolving independently. The question of controllability of the equations that describe the evolution of viscoelastic fluids is largely open. The presence of the non-algebraic constitutive relation makes the analysis unfeasible in general setup. The presence of several relaxation modes makes the problem even more complicated. Another issue is the necessity of controlling the stresses, since they are not determined by the momentary velocity field, thus they need to be included as the controlled states. In this work we are concentrating on the controllability of the stresses arising in the viscoelastic fluid that has its motion constrained to be of the shearing type. This restriction allows us to concentrate on the stresses only and assign the shearing rate to be the control. We consider only the Upper Convected Maxwell fluid which has several relaxation modes present. The results demonstrate that contrary to the one relaxation mode case the normal stresses cannot be driven arbitrary close to the exponentially decaying regime, unless the shearing stresses satisfy certain requirements, while the shear stresses remain exactly controllable.
- Doctoral Dissertations