The quasi-steady power-law Stokes flow of a mixture of incompressible fluids with shear-dependent viscosity is studied. The fluids are immiscible and have constant densities. Existence results are presented for both the no-slip and the no-stick boundary value conditions. Use is made of Schauder's fixed-point theorem, compactness arguments, and DiPerna–Lions renormalized solutions.

This article presents a method for the numerical quadrature of highly oscillatory integrals with stationary points. We begin with the derivation of a new asymptotic expansion, which has the property that the accuracy improves as the frequency of oscillations increases. This asymptotic expansion is closely related to the method of stationary phase, but presented in a way that allows the derivation of an alternate approximation method that has similar asymptotic behaviour, but with significantly greater accuracy. This approximation method does not require moments.

We generalise the notion of fractal interpolation functions (FIFs) to allow data sets of the form where I=[0,1]n. We introduce recurrent iterated function systems whose attractors G are graphs of continuous functions f:I→, which interpolate the data. We show that the proposed constructions generalise the previously existed ones on . We also present some relations between FIFs and the Laplace partial differential equation with Dirichlet boundary conditions. Finally, the fractal dimensions of a class of FIFs are derived and some methods for the construction of functions of class Cp using recurrent iterated function systems are presented.

We present an unsteady Hele–Shaw model of the fluid–fluid displacements that take place during primary cementing of an oil well, focusing on the case where one Herschel–Bulkley fluid displaces another along a long uniform section of the annulus. Such unsteady models consist of an advection equation for a fluid concentration field coupled to a third-order non-linear PDE (Partial differential equation) for the stream function, with a free boundary at the boundary of regions of stagnant fluid. These models, although complex, are necessary for the study of interfacial instability and the effects of flow pulsation, and remain considerably simpler and more efficient than computationally solving three-dimensional Navier–Stokes type models. Using methods from gradient flows, we demonstrate that our unsteady evolution equation for the stream function has a unique solution. The solution is continuous with respect to variations in the model physical data and will decay exponentially to a steady-state distribution if the data do not change with time. In the event that density differences between the fluids are small and that the fluids have a yield stress, then if the flow rate is decreased suddenly to zero, the stream function (hence velocity) decays to zero in a finite time. We verify these decay properties, using a numerical solution. We then use the numerical solution to study the effects of pulsating the flow rate on a typical displacement.

Amortization of a shock in an electro-rheological shock absorber is carried out in the motion of a piston in an electrorheological fluid. The drag force acting on the piston is regulated by varying the voltage applied to electrodes. A model of an electrorheological shock absorber is constructed. A problem on shock absorber reduces to the solution of a coupled problem for motion equation of the piston and non-linear equations of fluid flow in an unknown domain that varies with the time. A method of semi-discretization for approximate solution of the coupled problem is considered. Results on the existence and on the uniqueness of the solution of the coupled problem are obtained. Convergence of approximate solutions to the exact solution is proved. Numerical simulation of the operation of the shock absorber is performed.