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A finite element code based on the level-set method is used to perform direct numerical simulations (DNS) of the transient and steady-state motion of bubbles rising in a viscoelastic liquid modelled by the Oldroyd-B constitutive equation. The role of the governing dimensionless parameters, the capillary number (Ca), the Deborah number (De) and the polymer concentration parameter c, in both the rising speed and the deformation of the bubbles is studied. Simulations show that there exists a critical bubble volume at which there is a sharp increase in the terminal velocity with increasing bubble volume, similar to the behaviour observed in experiments, and that the shape of both the bubble and its wake structure changes fundamentally at that critical volume value. The bubbles with volumes smaller than the critical volume are prolate shaped while those with volumes larger than the critical volume have cusp-like trailing ends. In the latter situation, we show that there is a net force in the upward direction because the surface tension no longer integrates to zero. In addition, the structure of the wake of a bubble with a volume smaller than the critical volume is similar to that of a bubble rising in a Newtonian fluid, whereas the wake structure of a bubble with a volume larger than the critical value is strikingly different. Specifically, in addition to the vortex ring located at the equator of the bubble similar to the one present for a Newtonian fluid, a vortex ring is also present in the wake of a larger bubble, with a circulation of opposite sign, thus corresponding to the formation of a negative wake. This not only coincides with the appearance of a cusp-like trailing end of the rising bubble but also propels the bubble, the direction of the fluid velocity behind the bubble being in the opposite direction to that of the bubble. These DNS results are in agreement with experiments.

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A three-dimensional finite element based numerical method is used to simulate the rise of a bubble in a viscoelastic fluid modeled by the Oldroyd-B model. The rise velocity is studied as a function of the bubble volume on a log-log plot. The dependence of rise velocity on the bubble shape and the viscoelastic properties of the ambient fluid are also investigated. In simulations, rather than a jump in the rise velocity at a critical volume as observed in experiments, we find that there is a steep, but continuous, change in the rise velocity over a very small range of bubble volumes. Interestingly, this steep increase in the rise velocity is exaggerated when a parameter, which is a measure of the polymer concentration, is increased, while keeping the zeroshear viscosity fixed. The wake is ‘negative’ in the sense that the direction of fluid velocity behind the bubble for this parameter range is the opposite of that for a Newtonian fluid.

Under normal conditions, Cerebral Blood Flow (CBF) is related to the metabolism of the cerebral tissue. Three factors that contribute significantly to the regulation of CBF include the carbon dioxide and hydrogen ion concentration, oxygen deficiency and the level of cerebral activity. These regulatory mechanisms ensure a constant CBF of 50 to 55 ml per 100g of brain per minute for mean arterial blood pressure between 60–180 mm Hg. Under severe conditions when the autoregulatory mechanism fails to compensate, sympathetic nervous system constricts the large and intermediate sized arteries and prevents very high pressure from ever reaching the smaller blood vessels, preventing the occurrence of vascular hemorrhage. Several invasive and non-invasive techniques such as pressure and thermoelectric effect sensors to Positron Emission Tomography (PET) and magnetic resonance imaging (MRI) based profusion techniques have been used to quantify CBF. However, the effects of the non-Newtonian properties of blood, i.e., shear thinning and viscoelasticity, can have a significant influence on the distribution of CBF in the human brain and are poorly understood. The aim of this work is to quantify the role played by the non-Newtonian nature of blood on CBF. We have developed mathematical models of CBF that use direct numerical simulations (DNS) for the individual capillaries along with the experimental data in a one-dimensional model to determine the flow rate and the methods for regulating CBF. The model also allows us to determine which regions of the brain would be affected more severely under these conditions.