Motion in a Viscous Medium
1. Introduction
Fluid resistance is defined as the force that a fluid (i.e. a gas or a liquid) exerts on a
body moving through it. The moving body exerts a force on the fluid to push it out of the
way. By Newton’s third law, the fluid pushes back on the body with an equal and opposite
force. The objectives of this experiment are to study the viscous drag on a body falling in a
viscous medium and verify Stokes’ law for such motion.
2. Theory
Suppose we consider a sphere falling in a viscous liquid. The
forces acting on the sphere comprises of the force due to gravity (mg), the force due to the
buoyancy1 of the liquid (B) and the force due to the fluid resistance acting on the sphere (f).
The direction of the fluid resistance force acting on the sphere is always opposite to the
direction of the sphere’s velocity relative to the fluid.
The magnitude of the fluid resistance force usually increases with the speed of the sphere
through the fluid. At low speeds, the magnitude f of the resisting force of the liquid is
approximately proportional to the sphere’s speed v:
f = kv (fluid resistance at low speed) - (1)
where k is a proportionality constant that depends on the shape and size of the body and the
properties of the fluid. Equation (1) is known as Stokes’ law and the proportionality constant
k = 6phr where h is the viscosity of the liquid, and r is the radius of the sphere.
2.2 Limitations of Stokes’ law
At a first glance of these equations (Equations (1) – (6)), it would seem that
application of these expressions would seem obvious. In many textbooks, the application of
Stokes’ Law is often described in a simple experiment where ball bearings are timed as they
fall through a column of liquid about a meter deep. It is important to realize that the
conditions under which Stokes’ law is valid are: the flow be slow and steady in an unbounded
incompressible Newtonian2 medium. The first two conditions imply that the viscous or the
frictional forces are much smaller that the inertial forces, i.e. Reynolds’ number3Re=1 (see
http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/Reynolds.htm). The unbounded
condition assumes that the sphere falls through a fluid of infinite extend. However, when
falling through a column of restricted width, a sphere experiences a greater drag than is by
Stokes’ law because the fluid cannot be so easily displaced. It has to “squeeze” between the
spheres and the walls of the column as the sphere descends. This is also known as the “wall
effect”.
Newtonian and Non-Newtonian Fluids
Fluids are either Newtonian or Non-Newtonian. The simplest are the Newtonian ones,
like water, dilute suspensions, aqueous solutions, and emulsions. Viscosity is temperature
dependent and typically decreases as the temperature rises. Other examples of Newtonian
fluids include some motor oils, most mineral oils, gasoline, kerosene, most salt solutions in
water.
Non-Newtonians are a group of liquids that change viscosity when they are stirred,
shaken, or otherwise agitated. Ketchup becomes thicker, or more viscous, when it sits still. If
you stir it up or shake it, it becomes thinner, or less viscous. Ketchup is a thixotropic liquid. It
becomes less viscous when agitated. It is similar to Visplex (a drilling fluid). This fluid is a
liquid while in motion, but when at rest it turns into a thick gel. This makes it useful because
when the circulation of the drilling fluid stops, the gel suspends the rock cuttings and
prevents them form sinking to the bottom of the borehole. Other thixotropic liquids include
most paints, silica gel, greases, inks, milk, mayonnaise, asphalt, glues, molasses, starch, lard,
and fruit juice concentrates).
Theory: Ketchup that has been standing still is thicker (more viscous) than ketchup
that has been stirred or shaken. Part of this comes from the nature of the macerated tomatoes.
The solid part of the fruit must form suspended microfibers when ground up. On standing still
the fibers in such a suspension increasingly make contact with each other and stick together.
This forms a 3-D structure or gel throughout the fluid, the strength of which increases with
time. The gel structure is broken by agitation, reducing the viscosity.
(Thay cai' topic hay wa, tao post them cai' nay` de moi ng xem choi cho vui ^^. Cac ban co' len ah, mai mot wa day hoc phys chung w tui cho vui ah... Hua' la` neu mai mot co ai wa day hoc phys thi` N se cho muon toan` bo text book, khoi phai chay mua lung tung, ton money ah )
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Mon Sep 08, 2008 9:18 am
