A finite-volume-based numerical approach has been used to solve the equations of motion for the steady and incompressible power-law fluid past a sphere in the two-dimensional symmetric range of conditions. The simulations are verified against previous numerical and experimental results available in the literature. The friction and pressure drag profiles and streamline plots showing the nature of flow and the wake structure are presented. The computed results cover the Reynolds number range of 5 ≤ Re ≤ 500 and the power-law index range of 0.5 ≤ n ≤ 2 (covering both shear-thinning and shear-thickening behavior). Based on the present numerical results, simple predictive correlations, in terms of Re and n values, are proposed to calculate the values of the total drag coefficient (C D) and the ratio of the pressure drag coefficient to the friction drag coefficient (C Dp/C Df) in a new application.

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a b s t r a c tWe develop a single segment differential tube model including interchain tube pressure effect (ITPE) [G. Marrucci, G. Ianniruberto, Interchain pressure effect in extensional flows of entangled polymers, Macromolecules 36 (2004) 3934-3942], able to describe the non-linear behaviour of entangled linear polymers. The model accounts for the effect of flow on the tube length and diameter. It is presented in two versions, depending on which tube dimension is assumed to deform affinely. The classical relaxation mechanisms, i.e., reptation, stretch dynamics, convective constraint release (CCR), as well as finite extensibility, are incorporated in a simple manner; hence the model allows an explicit comparison of the relative importance of various effects. A striking result is the insignificance of finite extensibility and the detrimental influence of CCR for moderately entangled systems when ITPE is taken into account. For highly entangled systems, CCR regains importance to avoid the well-known shear stress instability. The proposed model is able to make quantitative predictions of steady elongational and shear data for monodisperse melts, while transient values are less accurate but within experimental errors.

The steady rise of a spherical bubble through an incompressible quiescent power law fluid has been studied numerically in the 2-D axisymmetric flow regime using the finite volume method. Based on the present numerical results, a predictive correlation in terms of Reynolds number (Re) and power law index (n) is proposed which enables the prediction of the total drag coefficient for the ranges of conditions as 5 ≤ Re ≤ 500 and 0.5 ≤ n ≤ 2 (hence covering both shear-thinning and shear-thickening type of fluid behavior). For n < 1, the drag is reduced below the corresponding Newtonian value, whereas it increases above its Newtonian value in shear-thickening fluids (n > 1). Thus, the drag coefficient increases with the increasing power law index for all values of the Reynolds number. The contribution of the pressure drag also increases with the increasing Reynolds number, though the shear-thickening behavior (n > 1) seems to suppress this tendency. In addition to the drag behavior, streamline and constant vorticity plots are presented to show the detailed nature of the flow and the effect of Reynolds number and power law index on the flow characteristics.