Flow regime analysis of non-Newtonian duct flows

Abstract

Reoriented duct flows of generalized Newtonian fluids are an idealization of non-Newtonian fluid flow in industrial in-line mixers. Based on scaling analysis and computation we find that non-Newtonian duct flows have several limit behaviors, in the sense that such flows can become ͑nearly͒ independent of one or more of the rheological and dynamical control parameters, simplifying the general flow and mixing problem. These limit flows give several levels of modeling complexity to the full problem of non-Newtonian duct flow. We describe the sets of simplified flow models and their corresponding regions of validity. This flow-model decomposition captures the essential rheological and dynamical characteristics of the reoriented duct flows and enables a more efficient and systematic study and design of flow and mixing of non-Newtonian fluids in ducts. Key aspects of the flow-model decomposition are demonstrated via a specific, but representative, duct flow.

Topological mixing study of non-Newtonian duct flows

“…Previous analysis 18 showed that shear-thinning ͑n͒ and yield ͑ ͒ effects lead to comparable flow fields with progressively more solid-like fluid motion, the more non-Newtonian the fluid. Numerical experiments 27 suggest that the velocity field similarity carries over to the advection fields in HB duct flows and that the primary effect of rheology is to delay onset ͑increase ␤͒ relative to Newtonian fluids.…”

“…In particular limits of the control parameters non-Newtonian behavior may be dominated by shear-thinning ͑through n͒ or yielding ͑through ͒ only and/or depend on the transverse or axial flow ͑through D͒ only; there are rheological simplifications and dynamical simplifications. 18 For n Ն n min or Յ max the HB rheology simplifies to a Bingham fluid ͑B͒ or power-law fluid ͑P͒, and for n Ն n min ഫ Յ max to a Newtonian fluid ͑N͒. As D → 0, the shear rate becomes a function of only the transverse field ␥ ͑u xy ͒ = ␥ xy ; as D → ϱ, the shear rate becomes a function of only the axial field ␥ ͑u z ͒ = ␥ z .…”

“…18 Figure 3 shows a typical HB flow field. Panels a and b show the transverse stream function ⌿ and velocity magnitude ͉u xy ͉ for the RAM device; the moving boundary sets up a counterclockwise circulation with large velocities near the window.…”

“…4 are topologically similar for any solution to the cellular flow model, regardless of rheology. 18 Due to the transverse flow having reflection symmetry, the qualitative response to ⌰ depends only on the direction of reorientation ͑i.e., ⌰Ͻ0 or ⌰Ͼ0͒. As changes due to non-Newtonian effects are entirely quantitative, there are only two breakdown scenarios for the integrable state.…”

“…One spatial period is either N =5 ͑even ͉m͉͒ or N =10 ͑odd ͉m͉͒ cells. Calculation of the non-Newtonian velocity and advection fields follows Speetjens et al 18 …”