Quantifying mixing processes Rob Sturman Department of Mathematics - - PowerPoint PPT Presentation

Quantifying mixing processes

Rob Sturman

Department of Mathematics University of Leeds

Graduate course, Spring 2007 Leeds

Rob Sturman Quantifying mixing processes

Aim of a mixing process

To mix:

• 1. trans. a. To put together or combine (two or more

substances or things) so that the constituents or particles of each are interspersed or diffused more or less evenly among those of the rest; to unite (one or more substances or things) in this manner with another or others; to make a mixture of, to mingle,

• blend. (OED)

To produce a mixture: Substances that are mixed, but not chemically

• combined. Mixtures are nonhomogeneous, and may

be separated mechanically. (Hackh’s Chemical Dictionary)

Rob Sturman Quantifying mixing processes

Mixing is intuitive

Rob Sturman Quantifying mixing processes

Mixing via stretching and folding

"None other than Osborne Reynolds advocated in a 1894 lecture demonstration that, when stripped of details, mixing was essentially stretching and folding and went on to propose experiments to visualize internal motions of flows." [Ottino, Jana, Chakravarthy, 1994]

Rob Sturman Quantifying mixing processes

Mixing via chaotic advection

Mixing of two highly viscous fluids between eccentric cylinders [Ottino, 1989]

Rob Sturman Quantifying mixing processes

An mixer based on a chaotic dynamical system

The Kenics R Mixer

(a) (b) (c) (d) (e) (f) (g) (h)

from [Galaktionov et al., Int. Polymer Proc. 18 138–50 (2003)]

Rob Sturman Quantifying mixing processes

Mixing via MHD

A magnetohydrodynamic chaotic stirrer, [Qian & Bau, 2002]

Rob Sturman Quantifying mixing processes

Oceanographic Mixing

Plankton bloom at the Shetland islands. [NASA]

Rob Sturman Quantifying mixing processes

Duct flows

Schematic view of a duct flow with concatenated mixing elements Red and blue blobs of fluid mix well under a small number of applications Changing only the position

• f the centres of rotation

can have a marked effect

• n the quality of mixing

Rob Sturman Quantifying mixing processes

Mixing through turbulence

Scalar concentration distribution from a high resolution numerical simulation of a turbulent flow in a two-dimensional plane for a Schmidt number of 144 and a Reynolds number of

• 22. (Courtesy of G. Brethouwer and F

. Nieuwstadt)

Rob Sturman Quantifying mixing processes

Mixing via dynamos

magnetic field generated by inductive processes by the motions

• f a highly conducting fluid. The prescribed velocity is of a type

known to be a fast dynamo, i.e., capable of field amplification in the limit of infinite conductivity (Cattaneo et al. 1995).

Rob Sturman Quantifying mixing processes

A good definition...

THOMASINA: When you stir you rice pudding, Septimus, the spoonful of jam spreads itself round making red trails like the picture of a meteor in my astronomical atlas. But if you stir backward, the jam will not come together

• again. Indeed, the pudding does not notice and continues to turn pink just

as before. Do you think this odd? SEPTIMUS: No. THOMASINA: Well, I do. You cannot stir things apart. SEPTIMUS: No more you can, time must needs run backward, and since it will not, we must stir our way onward mixing as we go, disorder out of disorder into disorder until pink is complete, unchanging and unchangeable, and we are done with it for ever. Arcadia, Tom Stoppard

Rob Sturman Quantifying mixing processes

How to quantify all this...

Danckwerts (1952): "...two distinct parameters are required to characterize the ’goodness of mixing’...the scale of segregation...and the intensity of segregation" [Denbigh, 1986]

How big? How wide? How does

• this

compare with

• this?

Rob Sturman Quantifying mixing processes