We consider the slow motion of a gas bubble in a cylindrical column filled with a viscoplastic fluid, modeled here as a Herschel–Bulkley fluid. Because of the yield stress of the fluid, it is possible that a bubble will remain trapped in the fluid indefinitely. We adapt Prager’s two variational principles to our problem. From these variational principles we develop two general stopping conditions, i.e., for a given bubble we can calculate a critical Bingham number above which the bubble will not move. The first condition is derived by bounding the velocity field and the second condition by bounding the stress field. We illustrate these conditions by considering specific bubble shapes, e.g., axisymmetric bubbles. We also develop a condition for bubble motion.

Conditions for static bubbles in viscoplastic fluids

The motion of a spherical drop with a bulk-insoluble surfactant immersed in a background flow in the limits of low surface Peclet number and low Reynolds number is investigated. We develop a reciprocal theorem that applies to any prescribed background flow and provide a specific example of an unbounded Poiseuille flow. Analytical formulas for the migration velocity of the drop are obtained perturbatively in powers of the surface Peclet number. We show that the redistribution of surfactant due to the background flow acts to retard the motion of the drop, with the magnitude of this slip velocity being independent of the drop’s position in the Poiseuille flow. Moreover, a surfactant-induced cross-streamline migration of the drop occurs towards the centre of the Poiseuille flow, with its magnitude depending linearly on the distance of the drop from the centre of the Poiseuille flow.

Viscous Marangoni migration of a drop in a Poiseuille flow at low surface Péclet numbers

A general non-axisymmetric Stokes flow past a porous sphere in a viscous, incompressible fluid is considered. The flow inside the sphere is governed by Brinkman's equations. A representation for velocity and pressure for the Brinkman's equations is suggested and a method of finding the flow quantities is given. Faxen's laws for drag and torque for the flow past a porous sphere are also given.

Stokes flow past a porous sphere using Brinkman's model

This Brief Communication derives a more compact form of the fundamental solution to the Stokes equations for a point force (Stokeslet) in a fluid domain bounded by a rigid spherical container. Although equivalent to Oseen’s (1927) solution, the present form is more useful in subsequent applications in boundary integral equation methods.

Image systems for a stokeslet inside a rigid spherical container

Multiscale simulation of water flow past a C540 fullerene

The velocity and pressure in Stokes flow are written in terms of two functions A and B, where A is biharmonic and B is harmonic. Lamb's (1) general solution of Stokes's equations and Oseen's (2) solution due to a Stokeslet in the presence of a no-slip spherical boundary have the same structure as our representation. Ranger's (3) representation follows as a special case of our result. A sphere theorem for non-axisymmetric flow outside or inside a sphere is stated and proved. Collins's theorem (4) for axisymmetric flow follows as a special case of our theorem. A few illustrative examples are given and in each case the drag and torque on the sphere are calculated.

Lamb's solution of stokes's equations: a sphere theorem

https://iopscience.iop.org/article/10.1016/0169-5983(93)90113-O/pdf

S D Nigam

Papers

Lamb's solution of stokes's equations: a sphere theorem

Stokes flow past a sphere with mixed slip-stick boundary conditions

Flow in a spherical cavity due to a stokeslet