Session R37: Non-Newtonian Flows: Theory

The shape of a maximal pile of yield-stress fluid supported by an obstruction

Viscoplastic fluid flows that occur in nature, such as lava, mud or debris flows, frequently interact with natural or constructed barriers leaving static mounds of material. Using Charpit's method, we present an analytical solution to the shape of the static mound supported by a square barrier on an inclined plane in the lubrication approximation. Expressions for the physical characteristics of the mound, such as its maximal height and total weight, are found from the solution. The calculated shape represents the maximal static mound supported by the barrier, and thus is independent of the original upstream flux source. A single dimensionless parameter describes the shape of the mound, the ratio of the yield stress to the slope-induced stress, and hence it is independent of the constitutive relation beyond the yield stress. A new rheometric technique is suggested based on our solution, where the yield stress is determined using a simple experimental setup. We demonstrate that the method generalises to other barrier shapes, such as circular and rhomboidal barriers. Comparison with preliminary experimental results are discussed.   

Shallow viscoplastic flow around the inside of a rotating cylinder

Lubrication theory is presented for the flow of a viscoplastic fluid (described by the Herschel-Bulkley law) around the inside surface of a rotating cylinder. The analysis predicts the steady states typically reached after a small mumber of rotations, at speeds for which the fluid largely collects in a prominent pool in the lower part of the cylinder. The analysis also captures the drainage of the film around the inside of a stationary cylinder. Specific consideration is given to the plastic limit, arising for arbitrarily small rotation rates where the fluid behaves like a perfectly plastic material satisfying the von Mises yield condition. Theoretical predictions are compared with experiments using a Carbopol suspension.   

Similarity relations for laminar pipe flows of Bingham fluids in friction coordinates

We introduce novel similarity relations, along with their corresponding symmetry groups, for examining the velocity profile and friction factor of laminar Bingham fluid flows using friction coordinates. Specifically, we provide a valuable new expression for calculating the friction factor of Bingham plastic fluids when pressure gradient data is accessible. Our findings reveal that there are no clear similarity relations for the mean velocity profile (MVP) and friction factor formulas in bulk coordinates. Consequently, friction coordinates serve as the most suitable framework for describing this problem.   

Quasi-steady sedimentation of a sphere in an Oldroyd-B fluid

We investigate the quasi-steady sedimentation of a sphere perpendicular

to a rigid plane in an Oldroyd-B fluid at low Deborah numbers. Utilizing

lubrication approximations, we derive analytical expressions for

resistive forces and pressures. Additionally, we introduce alternative

methods using a reciprocal theorem to improve perturbation corrections

for forces and consider different local geometries near the sphere's

lowest point to study their impact on viscoelasticity. Our results are

validated against numerical simulations using Basilisk.   

Theoretical Model for Velocity Profile and Pressure Drop in Perfect Core-annular Flow of Immiscible Herschel–Bulkley Fluids in a Horizontal Pipe

This study investigates an axisymmetric perfect core-annular flow (PCAF) scenario where two incompressible and immiscible Herschel–Bulkley fluids flow in a horizontal pipe. A theoretical model is developed for steady and fully developed PCAF to predict the velocity profiles and pressure drops for different yield stress and power law characteristics of both the fluids. The model presented in this study clearly demonstrates that the velocity profiles and pressure drops in a PCAF arrangement depend on the power law characteristics of both the core and annular fluids, as well as on the respective yield stress values. As the consistency index of the core fluid increases, it behaves more like a solid, leading to a reduced dependency on its flow behaviour index. However, when there are smaller viscosity differences between the core and annular fluids, the flow behaviour index of the core fluid plays a significant role in determining velocity profiles. The yield stresses of the core and annular fluids are also found to significantly affect the velocity profiles and pressure drop in the PCAF arrangement, even at comparable values of actual shear stresses of the co-flow system. Lastly, the validation of the theoretical model is presented through experimental data and numerical simulations available in the literature.   

Global nonlinear stability of Oldroyd-B fluids in plane Couette flow

This talk will give a rigorous proof of global nonlinear stability for Oldroyd-B fluids in plane Couette flow. Specifically, we derive conditions under which perturbations of arbitrary magnitude must decay asymptotically, which naturally extend classical results for Newtonian fluids to the non-Newtonian setting. The challenge of performing nonlinear stability analysis for viscoeasltic flows is in finding an appropriate quantification of both perturbations from a base-state. To this end, we introduce a new measure, the perturbation entropy, which generalises the notion of relative entropy and gives a probabilistic measure of the deviation of the polymer configuration tensor from its steady distribution. We will show that at any Reynolds number lower than the Newtonian energy stability limit, there exists a range of polymer densities and Weissenberg numbers for which the Oldroyd-B viscoelastic model is also nonlinearly stable.   

Viscoelastic contributions to the infinite length journal bearing

Viscoelastic fluids in shear produce normal stresses. In rotating devices, these normal stresses produce a non-zero pressure field that would not exist in the same flow conditions with purely viscous fluids. This additional elastic pressure field has been used to increase the load carrying capacity of thrust bearings, and that same analysis can be extended to journal bearings. Here, the Cauchy momentum equations in polar coordinates are solved in the thin film limit using a perturbation expansion in the Deborah number (De). Viscoelasticity is included through an Upper Convected Maxwell model with a solvent viscosity (Upper Convected Jeffreys model). When De=0, the model resembles the Reynolds equation (a restatement of conservation of mass and momentum for a purely viscous fluid) for an infinite length journal bearing. The De=0 results match the predictions of the Reynolds equation over all eccentricity ratios, validating the model. As the De increases, the load carrying capacity and altitude angle for a given eccentricity ratio also increase. This suggests that viscoelasticity is beneficial in journal bearing applications and provides designers another mechanism for decreasing friction in lubricated journal bearings.