Boundary layer (shear-band) in frustrated viscoplastic flows

We show that frustrated creep flows of yield stress fluids give rise to a boundary layer, which takes the form of a liquid region of uniform significant thickness separating two solid regions. In this boundary layer the shear rate is approximately constant for a given flow rate and the layer thickness extremely slowly varies with the flow rate. Yield stress fluids such as concentrated colloids, emulsions, foams, are jammed systems which behave as liquids if they are submitted to a sufficiently large stress, and as solids otherwise [1-2]. In usual situations the stress is heterogeneous throughout the material so that “solid” regions coexist with “liquid” regions. In simple uniform flows (e.g. in straight conduits or open channels, in Couette geometry, etc) the solid-liquid interface reaches the boundaries when the velocity tends to zero, so that the liquid region tends to disappear for creep flows [3-4]. A similar result is obtained for thixotropic yield stress fluids which develop shear-banding in channelized or concentric cylinder flows [5-6]. Another type of shear-banding was also observed in starting (Couette) flow of simple yield stress fluids [7]. The recent generalized theoretical approach of shearbanding [8] for time-dependent flows also considered uniform flows. On the contrary, in pure solids although shear-banding can also occur in simple shear [9] it is currently observed in more complex deformation fields [10]. A wide range of practical operations in civil engineering (paints, mortars, concrete, drilling fluids) in food industry and cooking (purees, sauces, dough), or in cosmetic applications (gels, foams, creams), involve flows of yield stress fluids initially at rest and partially removed by some flow or by the displacement of an object through it: mixing, injection, formwork filling, coating on walls or skins, etc. Such a problematic is typical of yield stress fluids and to what extent the fluid initially in its solid regime is liquefied and/or removed is a crucial question to address in order to minimize the required energy and improve the efficiency of the process. However our knowledge of these flows is extremely poor and essentially concerns the displacement of solid spheres or cylinders through yield stress fluids [11]. Even in that case the complexity of the problem and the difficulty to measure internal flow characteristics in usual pasty materials precluded a full understanding. For example it was shown that in some cases the elasticity of the material in its solid regime can allow the sphere to displace through it by liquefying only an extremely small volume of material [12]. As a consequence one needs to have a detailed information on these frustrated viscoplastic flows, in which due to the geometrical conditions and the yielding character of the fluid there necessarily exists a region which will not flow. Here we focus on two typical simple situations of that type, i.e. (i) the injection of a yield stress fluid through a box initially fulfilled with the same fluid and (ii) the displacement of a long object through a yield stress fluid lying in a container. We show that in both cases the motion relies on the existence of a boundary layer, which takes the form of a liquid region of uniform significant thickness separating two solid regions. Surprisingly, in this boundary layer the shear rate is approximately constant and the layer thickness extremely slowly varies with the flow rate. Our first experiment consists in the injection of a yield stress fluid in a cylindrical box (diameter and length: 7cm) from a coaxial smaller conduit of radius mm 5 . 17  R . The fluid is a concentrated inverse emulsion exhibiting a yield stress ( c  ) of 74 Pa, and with a simple shear flow curve well represented by a HerschelBulkley (HB) model ( c , in which n k      is the shear stress and  the shear rate) with and n k 5 . 13  Pa.s 4 . 0  n . The material was prepared by dispersing a water + CaCl2 (150g/l) salt solution in a surfactant solution (dodecane (Acros organics) + 7.5%wt of Span 80 surfactant) in a Silverson mixer. For the tests the material is supplied via an extrusion syringe (8 cm diameter, 1 m long) to the small conduit in which a uniform flow is established at a significant distance from the entrance