The creeping transverse flow of incompressible Carreau model fluids past an array of infinitely long cylinders has been analysed theoretically. The hydrodynamic interactions between cylinders have been simulated using a simple cylinder-in-cylinder free surface cell model. In this formalism, the overall mean porosity characterises an array of cylinders without any assumption regarding the actual geometrical arrangement of the individual cylinders. The resulting non-linear field equations have been solved approximately by employing the well known velocity and stress variational principles. The resulting upper and lower bounds are non-coincident and diverge increasingly with the rising extent of non-Newtonian behaviour of the liquid medium. However, the mean value of the drag coefficient deviates from the individual bounds by no more than 22% and therefore, the use of the arithmetic mean of the upper and lower bounds is suggested. The theoretical predictions reported herein encompass wide ranges of physical and kinematic conditions as follows: 1 ≥ n ≥ 0.2; 0.9 ≥ ε ≥ 0.3 and ∧ ≤ 500. The paper is concluded by presenting comparisons between the present predictions and the scant experimental results for two limiting cases, namely, for the flow of Newtonian fluids (n = 1) through assemblages of solid rods and random fibrous beds, and for the flow of power law liquids (i.e., for large values of ∧) through banks of rods. The correspondence is found to be satisfactory.