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Researchers in the College of Engineering at Oregon State University have advanced a new theory in fluid mechanics, cracking a fundamental problem that has vexed scientific minds for nearly seven decades.
The problem relates to our basic understanding of how chemicals mix in fluids. The new theory has practical implications in a host of application areas — from how pollutants spread in the atmosphere to how drugs perfuse tissues within the human body.
The research, funded in part by the National Science Foundation, was led by chemical engineering doctoral candidate Ehsan Taghizadeh, his advisor, Brian D. Wood, and collaborator Francisco José Valdés-Parada.
The team set out to bridge a longstanding knowledge gap in what is known as Taylor dispersion theory. Named for scientist G.I. Taylor, who published a seminal paper on the topic in 1953, the theory concerns phenomena in which ﬂuctuations in velocity ﬁelds cause the spreading of chemicals.
“Researchers over the years had noted that the process of dispersive spreading tends to increase over time until it reaches a steady level,” Wood said. “You can think of it as analogous to investment in a startup, in which the rates of return are initiallyvery large before settling to a more sustainable level that is close to constant.”
Taylor’s theory was the first to allow researchers to predict thatsteady level of dispersion. The result, known as the macroscopic dispersion equation, describes the net movement of a chemical species with high ﬁdelity — provided that enough time has elapsed from the chemical injection.
“This was a signiﬁcant revelation at the time,” Wood said. “It was on par with what researchers were doing theoretically in other disciplines, like quantum mechanics.”
While Taylor’s theory was successful, researchers still struggled with the problem of how dispersive spreading evolves from its dynamic, early behavior (near what is known as its initial condition), to when it attains the more constant value predicted by Taylor’s theory.
Dispersion theory became a signiﬁcant area of research in its own right in the 1950s and 1960s. One of Oregon State’s best known alumni, Octave Levenspiel (’52 Ph.D., Chemical Engineering) — who taught at the university for decades until his retirement in 1991 — published one of his most-cited papers on the topic of dispersion in chemical reactors in 1957.
Some success was attained using a time-dependent dispersion coeﬃcient, which continued to be developed as one of the main theoretical approaches through the 1970s and 1980s. But there were problems with that approach, the primary one being that it led to paradoxes.
“For example, if chemical solutes injected at two diﬀerent times overlap, which time do you assign to the dispersion coeﬃcient?” Wood said. “Taylor himself understood that, where a time-dependent dispersion coeﬃcient was adopted, contemporary theories violated basic notions of causality in physics.”
Using results from the theory of partial diﬀerential equations, the research team showed that problems with the time-dependent dispersion coefficient arose from neglecting a term that represented the solute’s relaxation from its initial condition.
“When chemical species are ﬁrst injected, their behavior is not necessarily consistent with a dispersion-type equation,” Wood explained. “Rather, the initial condition first has to ‘relax.’ During this time, there is an additional term to account for that was missing in Taylor’s macroscale dispersion equation.”
This additional term corrects the dispersion equation to account for the initial spatial conﬁguration of the chemical species being transported. Somewhat surprisingly, Wood says, the theory also resolves paradoxes in theories with time-dependent dispersion coeﬃcients.
“In the new theory, there is never a question about what dispersion coeﬃcient should be used when chemical solutes overlap,” he said. “The adjustment to the spreading process is accounted for automatically by the presence of the additional term.”
Working alongside Taghizadeh and Wood was Francisco José Valdés-Parada, a professor of chemical engineering at Metropolitan Autonomous University in Mexico City. Wood and Valdés-Parada started this work in 2009, when Valdés-Parada was a postdoctoral researcher at Oregon State.
The paper, titled “Preasymptotic Taylor dispersion: evolution from the initial condition,” appeared in the April 25 edition of Journal of Fluid Mechanics. The article was first published online in February and has remained on the journal’s “most read” list for many weeks.