Entry, start up and stability effects in visco-plastically lubricated pipe flows

Abstract

Interfacial instabilities of multi-layer shear flows may be eliminated by astute positioning of yield stress fluid layers that remain unyielded at the interface(s). We study the initiation, development lengths and temporal stability of such flows in the setting of a Newtonian core fluid surrounded by a Bingham lubricated fluid, within a pipe. Flow initiation is effected by starting the flow with a pipe full of stationary Bingham fluid and injecting both inner and outer fluids simultaneously. Initial instability and dispersive mixing at the front remains localised and is advected from the pipe leaving behind a stable multi-layer configuration, found for moderate Reynolds numbers (Re), for a broad range of interface radii (ri) and for different inlet diameters (Ri), whenever the base flow parameters admit a multi-layer flow with unyielded interface. The established flows have three distinct entry lengths. These relate to: (i) establishment of the first unyielded plug close to the interface (shortest); (ii) establishment of the interface radius; (iii) establishment of the velocity profile (longest). The three entry lengths increase with Re and decrease with both the Bingham number (B) and the viscosity ratio (m). Nonlinear temporal stability to axisymmetric perturbations is studied numerically, considering initial perturbations that are either localised in yielded parts of the flow or that initially break the unyielded plug regions. The aim is to understand structural aspects of the flow stability, not easily extracted from the energy stability results of Moyers-Gonzalez, Frigaard & Nouar (J. Fluid Mech., vol. 506, 2004, p. 117). The initial stages of a stable perturbed flow are characterised by a very rapid decay of the perturbation kinetic energy during which time the unyielded plug reforms (or breaks and reforms). This is followed by slower exponential decay on a viscous timescale (t ~ Re). For smaller Re and moderate initial amplitudes A, the perturbations decay to the numerical tolerance. As either Re or A is increased sufficiently, a number of interesting phenomena arise. The amount of dispersion increases, making the interfacial region increasingly diffuse and limiting the final decay. At larger Re or A, we find secondary flow structures that persist. A first example of these is when the shear stress decays below the yield stress before the velocity perturbation has decayed, leading to freezing in of the interface shape. This can lead to flows with a rigid wavy interface. Secondly, depending on the core fluid radius and thickness of the surrounding plug region, we may observe a range of dispersive structures akin to the pearls and mushrooms of d'Olce et al. (Phys. Fluids, vol. 20, 2008, art. 024104).