J. Non-Newtonian Fluid Mech. 164 (2009) 9–16 Contents lists available at ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: www.elsevier.com/locate/jnnfm Drag of an isolated cylinder and interactions between two cylinders in yield stress fluids Laurent Jossic a,b , Albert Magnin a,b,∗ a Laboratoire de Rhéologie, Domaine Universitaire, B.P. 53, 38041 Grenoble cedex 9, France b Université Joseph Fourier Grenoble 1, Grenoble Institut Polytechnique, CNRS UMR 5520, France article info Article history: Received 3 June 2009 Received in revised form 1 July 2009 Accepted 2 July 2009 Keywords: Yield stress Cylinder Drag coefficient Interactions Viscoplasticity abstract The drag force of two interacting cylinders moving at very low controlled velocity in yield stress fluids was measured. The influence of the distance separating them was evaluated. The two cylinders are parallel, one in the wake of the other. The drag force on an isolated cylinder was studied and is used as reference. Correlations for predicting the drag coefficient and stability criterion are proposed. Special attention was paid to the fluid–cylinder interfaces, as they govern the adherence and slip of the fluid at the wall. Their influence on the drag coefficient was quantified. The experimental results are compared with the theoretical or numerical predictions for a yield stress fluid. These results show that the yield stress reduces the extent of the interactions in comparison with the Newtonian fluid case. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Understanding the flows of yield-stress fluids around obstacles is necessary for monitoring and controlling numerous industrial applications. Nevertheless, the data found in the literature concern- ing this field of yield-stress fluid mechanics are still incomplete. Let us examine the case of isolated objects and then interacting objects. Spheres have been studied most. The experimental and numer- ical results about the drag force on a sphere in a yield stress fluid reveal disagreements between authors [1–3]. Recently, Tabuteau et al. [4] obtained experimental results that agree well with the theoretical predictions found in the literature. The literature con- tains fewer data concerning the drag of objects of more complex shape. The first experimental results [5–9] are affected by con- siderable uncertainties regarding the rheometrical properties of the fluids used. Jossic and Magnin [10] proposed a new approach to measure the drag force involving carefully characterised model yield-stress fluids and flows at very low controlled velocities where plastic effects are predominant. They thus removed the experi- mental uncertainties that were the cause of disagreements in the literature. Even though this geometry is of great interest, results concern- ing the flow of viscoplastic fluids around a cylinder are still quite ∗ Corresponding author at: Laboratoire de Rhéologie, Domaine Universitaire, B.P. 53, 38041 Grenoble cedex 9, France. E-mail address: magnin@ujf-grenoble.fr (A. Magnin). fragmentary, the results with respect to an isolated cylinder in vis- coplastic fluid and those of the present study are summarised in Table 1. To our knowledge, no complete analytical solution has yet been formulated for this problem. First of all, it should be recalled that in the case of 2D flow around a cylinder in an infinite medium, there is no solution for a Newtonian fluid. This result is commonly known as the Stokes paradox [11]. However, the paradox no longer applies in the presence of walls and the drag coefficient can then be calculated [12]. The non-Newtonian behaviour of the fluid also enables the paradox to be removed, for example with a constitutive equation of the power-law type [13]. In the case of viscoplastic flu- ids, Yoshioka and Adachi [14] used energy principles to study the creeping flow of a Bingham fluid around an infinite cylinder. Iner- tia effects are neglected and the medium surrounding the cylinder is considered to be infinite. The velocity at the wall is nil and the velocity far from the cylinder becomes uniform. Slip line analysis represents the limiting case. The shape of the sheared envelope is deduced from assumptions concerning the slip line fields defined by the stresses imposed in cavities of isotropic plastic material. They deduce a viscoplastic drag coefficient Cd* = 2( + 2) ≈ 10.3 where Cd* represents the drag force on the cylinder made dimensionless by the product of the yield stress and the frontal area. Also on the basis of the slip line theory, Randolph and Houlsby [15] take into account the influence of roughness and propose Cd* =9.14 in cases where the cylinder surface is smooth and Cd* = 10.94 in cases where it is rough. Numerical solutions have been proposed for the creeping flow of a viscoplastic fluid around a cylinder, taking into account wall 0377-0257/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2009.07.002