Page 63 - Essentials of physical chemistry

2 Viscosity of Laminar Flow

INTRODUCTION

Continuing our appeal to phenomenological derivations, we come to an experimental technique that
is very simple to use and has a clean calculus derivation. Despite the simplicity of the measurement
of viscosity, it is very useful in several areas of chemistry (polymers), aerodynamics (airplane wing
design), hydrodynamics (boat hull design), pharmaceutical delivery (oral delivery in syrups),
biophysics (blood flow), and material science (polymers). We are mainly motivated by a need to
support the revolutionary Boltzmann’s kinetic molecular theory of gases (KMTG; in Chapter 3)
with some experimental method. The Boltzmann KMTG can be treated in a cyclical set of self-
fulfilling equations (perhaps because it is true!), but a skeptic would require some sort of measure-
ment, mainly because it assumes the existence of very small atoms=molecules never seen individu-
ally. Even today there are only a few ‘‘pictures’’ of fat Au atoms on surfaces and x-ray diffraction
structures of molecules in crystals. The preponderance of evidence for the size and structure of
molecules is firm but indirect. Here, we want to discuss Poiseuille’s (Pwaz-e-ay’s) law of viscosity
for laminar flow [1,2] because it offers several useful applications, but primarily it will be a way to
verify Boltzmann’s KMTG.
Another modern application is an extension of Einstein’s thesis work [1,3,4] on viscous flow of
sugar-water solutions applied to the determination of the molecular weight of giant molecules now
called polymers. Polymers are typically the result of organic compounds which have both ‘‘head’’
and ‘‘tail’’ reactive groups that can react repeatedly to form large chains, sheets, or bulk materials,
which are really a single large molecule. Following WWII, there was a worldwide surge of research
in how to make, characterize, and develop application of polymers. This effort continues today in a
now mature branch of chemical research, and some amazing properties of specialized polymers
have been developed such as high-temperature stability approaching that of metals (polyimides),
polymers that change color with temperature, and dry lubricants such as perfluorocarbons. Let us
not forget the ubiquitous polystyrene coffee cup. When new polymeric materials are developed, one
of the foremost characteristics is the intrinsic viscosity of the polymer, and this is measured in a
simple way with a pipette, a viscometer, and a stopwatch!
A third motivation is that physical chemistry enters into some aspects of biomedical science, and
blood viscosity is a minor diagnostic parameter related to blood-thinning treatment of stroke
prevention. Poiseuille’s law for laminar flow is a beautiful example of the clean application of
calculus to a phenomenological equation which supports Boltzmann’s KMTG and is an important
method used in polymer science, but we have searched out some biological applications as well.
Here, we want to give a foundation to the experimental ideal of laminar flow of fluids, which can be
modeled using calculus incremental layers sliding over one another (Figure 2.1). We will see that
once we can relate gas viscosity to KMTG, we gain a number of important concepts related to gas-
phase chemical reactions, such as binary collision number and the mean free path. We just need
some physical data to tie the theory to laboratory reality!
Consider a model of two parallel sheets with one sliding over the other. Common sense tells us
that there is some sort of ‘‘friction’’ opposing the sliding motion. Viscosity is a drag, literally!
We can develop the idea of a laminar ‘‘coefficient of viscosity’’ from common experience. First,
the force required to move one sheet over the other is proportional to the area A of contact between
the sheets. Second, more force will be required to move the upper sheet faster. Third, the actual
contact between the sheets depends inversely on the contact distance between the sheets since all

25