Abstract
In this chapter, we learn how to constitutive model an important class of viscoelastic fluids, polymer solutions, by a microstructure method. The constitutive equation is derived by modeling the microstructure directly, leading to the Oldroyd-B model and its varieties.
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- 1.
A stochastic process is a family of random variables X(t), where t is the time, X is a random variable, and X(t) is the value observed at time t. The totality of {X(t),t∈ℝ} is said to be a random function or a stochastic process.
- 2.
W. Kuhn (1899–1963) was a Professor at the Technische Hochschule in Karlruhe, and later on, in Basel, Switzerland. He is most famous for the f-summation theorem in quantum mechanics.
- 3.
P. Langevin (1872–1940) introduced the stochastic DE (7.20) in 1908, and showed that the particle obeys the same diffusion equation as described by Einstein (1905).
- 4.
The random zig-zag motion of small particles (less than about 10 μm) is named after R. Brown (1773–1858), an English botanist, who mistook this as a sign of life. He traveled with Matthew Flinders to Australia in 1801 on the ship Investigator as a naturalist. The correct explanation of the phenomenon was given by Perrin (Fig. 7.3). Brownian particles are those undergoing a random walk, or Brownian motion.
- 5.
This approximation is called white noise, i.e., Gaussian noise of all possible frequencies uniformly distributed. Sometimes it is called “rain-on-the-roof” approximation: two (or more) rain drops do not fall on the same spot on the roof.
- 6.
A.D. Fokker derived the diffusion equation for a Brownian particle in velocity space in 1914. The general case was considered by M. Planck (1858–1947) in 1917.
- 7.
The general solution to the random walk problem in one dimension was obtained by M. von Smoluchowski in 1906.
- 8.
Dilute solutions of polymers in highly viscous solvents [6].
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Phan-Thien, N. (2013). Polymer Solutions. In: Understanding Viscoelasticity. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32958-6_7
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