Pipe flow of fluids with time-dependent rheological behaviour

Several technical applications make use of fluids whose rheological characteristics change with time owing to the external shearing stress, even if it is steady. They are the so called rheopectic or thixotropic fluids, depending on whether the fluid viscosity increases or decreases significantly with time owing to an imposed shear stress or shear rate. In thixotropic fluids the applied stress breaks gradually the molecular structure decreasing the viscosity, which, as the applied stress is removed and the structure rebuilds, increases again. The time-scales can range from many minutes in the case of breakdown to many hours in rebuilding. Thixotropy has been widely studied; even recently several models have been proposed, e.g. by Dullaert and Mewis: the latter produced in 1979 a well informed review which is still worth consulting. A more recent detailed description of the phenomenon, an extensive review of time effects on viscosity and of the thixotropic materials is due to Barnes which gives an exhaustive list of references. Nakaishi and Yasutomi examine theoretically and experimentally the thixotropic properties of clay-water systems, showing that they depend on the length of the measurement time. A typical application of thixotropic fluids is electronic packaging (Chen et al.) where epoxy and adhesive are used for encapsulation and surface mounting; the dispensed amounts of materials have to be well controlled, and generally vary with time: typically in a first time interval the fluid is delivered, and then the motion is stopped; the fluid viscosity changes during the delivery time and when the force inducing the flow is removed. Other remarkable applications concern the behaviour of fluids used in paint industry: the rheological properties of paints depend on the polymer matrix; the thixotropy plays a fundamental role in sagging and levelling during and after the application process (Armelin et al.); or fluids used in the drill industry, which are subjected to cyclic pressure and temperature loads when circulating in the bore; many of these fluids (e.g. xanthan) exhibit a viscosity degradation and are not able to re-acquire the original viscosity (Fiber et al., Billingham and Ferguson). Many organic fluids too present thixotropic characteristics; the rheological properties of human blood depend on its time-variable microstructure: the aggregation and the disaggregation process of red cells, which collide rarely, take place over different time scales, giving rise to thixotropic behaviour (Owens). Fluids used in food industry such as yogurt, butter and mayonnaise have often a gel structure where the micelles are linked to each other in the form of a tridimensional matrix which includes the liquid phase; the application of a shear modifies the structure and gives rise to a time-dependent viscosity (O’Donnel and Butler, Gopalakrishnan et al.). The aim of this paper is to solve the equation of the unsteady motion in circular pipes of a Newtonian fluid with time-dependent viscosity. Two different situations are considered: in the first the hydraulic slope is applied for a finite time interval whereas in the second the time interval is infinite and an asymptotic steady flow can be reached. Simple expressions are given, suitable for the numerical calculation of velocity and flow discharge in both cases.