Download Demonstration 1: Fluid Properties, Viscosity

GO WITH THE FLOW
Introduction to Fluid Dynamics
Summer 1999
Demonstration 1: Fluid Properties, Viscosity
Brian M. Argrow
Aerospace Engineering Sciences
OBJECTIVES
 Definition a fluid
 Fluid Properties
 Fluid viscosity
 Measurement of fluid viscosity
BACKGROUND
States and Properties of Substances
A fluid will continuously deform under any applied stress, i.e., a fluid cannot hold a shape unless
supported by some external container. This is in contrast to a solid, which offers resistance to
deformation when a stress is applied. The two familiar types of fluids are liquids and gases. The
primary distinction between a liquid and a gas is that a liquid can support a free surface, i.e., a
boundary at which the liquid is on one side and some other fluid is on the other. A gas will not
support a free surface, it will continue to expand until it is confined by some external boundary
such as the walls of a container.
Properties are the characteristics used to describe the state of a substance. While there are a
number of defined properties, generally there are a select few used in thermodynamics (the
science of energy) and fluid dynamics (the study of fluid motion) that are of particular
importance. One of these properties is the mass density  (usually just referred to as the density)
of a substance is defined as
density   
amount of substance
mass
 kg 

,
unit volum e of substance unit volum e  m 3 
where the SI units associated with the property are included in brackets.
The definition of a fluid above included a reference to stress applied on the fluid. The stress at a
point is the force per unit area applied at that point. Depending on the orientation of the surface
on which the force is applied, the force may be decomposed into a part that is parallel
(tangential) to the surface, and a part that is perpendicular (normal) to the surface. A fluid at rest
can only support a perpendicular force since any tangential force will cause it to move.
force
normal force component
tangential force component
GO WITH THE FLOW
Introduction to Fluid Dynamics
Summer 1999
The fluid pressure p is defined by,
pressure  p 
normal force
[Pa] or [N/m 2 ]
unit area
Atoms that comprise a solid always possess some degree of random motion. The energy
associated with this random motion is often referred to as "heat" or "thermal energy." In a fluid,
groups of atoms, i.e. molecules, are not confined to specific positions with respect to one
another. For a liquid the molecules are generally in contact with each for the most part, but for a
gas the molecules are generally widely spaced and spend most of the time in motion and only
come in contact via intermolecular collisions. We often use imprecise adjectives such as hot,
cold, warm, etc. in reference to the property called temperature. Of course these descriptors are
quite useless as quantitative measures. A precise definition of temperature is
temperatur e  T  measure of thermal energy [K] .
Keeping all properties other than temperature fixed, fluid states have more collective random
motion than the solid state, and thus the fluid states possess more thermal energy and are
observed at correspondingly higher temperatures. As an example, water at 1-atmosphere
(101,325 Pa) pressure is a solid at the freezing point T = 273 K. As the temperature increases
above this value the water melts into a fluid mixture of liquid and gaseous water, finally
becoming totally gaseous as it is heated beyond the boiling point.
An important result from thermodynamics is that all the basic thermodynamic properties of a
substance can be determined if any pair of special state variables (this includes , p, and T) are
specified. So, for instance if p and T are specified, there is a unique corresponding value of .
Fluid Viscosity
Even though we described solids and fluids in terms of atoms and molecules, except for the
special fields of rarefied gas dynamics and molecular dynamics, fluid dynamics is usually
developed in terms of a continuum. A continuum description assumes that even the smallest
piece of the fluid, a fluid element, is large enough that the behavior of individual molecules is
unimportant and the bulk fluid properties are representative of an average of the molecular
properties. In the definition of a fluid from the previous section, it was stated that a fluid
continuously deforms under an applied stress. The normal component of this stress was defined
to be the pressure p. Since the pressure is always in the normal direction, it does not act to
deform the fluid element, it only acts to change the volume. The tangential stress component
called the shear stress, however acts to deform the fluid element as shown in the figure below.
The figure shows these effects on a two-dimensional fluid element
pressure
shear
GO WITH THE FLOW
Introduction to Fluid Dynamics
Summer 1999
Rub your hands together and you feel the familiar sensation of friction. Because your skin is a
solid, it resists the deforming shearing motion between your hands. You will also notice that if
your rub your hands together quickly, you will begin to feel warmth. The friction converts some
of the energy of motion between your hands into thermal energy (heat). Friction is also present
in all real fluids. For most fluids, the friction, or resistance to deformation, is proportional to the
rate at which the fluid is deformed (from the shear stress). This resistance is generally dependent
upon the temperature of the fluid and is characterized by a property called the dynamic viscosity
 [Ns/m2]. This property provides a measure of how much the fluid resists the shearing motion.
A large value of  is characteristic of a fluid with large resistance to shearing motions relative to
the low resistance of a fluid with a small  value
The simplest mathematical description of fluid motion around immersed bodies, for which some
of the forces between the fluid and the immersed body can be computed, is called hydrodynamic
theory. This theory describes a “perfect fluid,” i.e. one without viscosity (inviscid). When
hydrodynamic theory was first developed it provided a means for computing the “lift force” on
an immersed body. By definition, the lift is the force on a body that is perpendicular to the flow
direction, so it is associated with the fluid pressure. Hydrodynamic theory fails completely
however, for computing the “drag force,” that is the forced defined to be parallel and in the
direction of the flow. A complete description of fluid flow and the forces on an immersed body
must include the effects of viscosity.
Measuring Fluid Viscosity from Drag on an Immersed Body
As described, the drag force on an immersed body is in the direction of the flow; thus it works to
retard the motion of a body through a fluid. The diagram below is a schematic of a sphere of
radius a falling freely in a fluid. The weight of the sphere is W   b gV , the buoyancy force is
FB  gV , and D represents the drag force acting on the sphere. Here  is again the density of
the fluid, b is the density of the sphere, and V is the volume of the sphere. In the schematic, the
sphere is assumed to have reached its terminal velocity Ut. When it is released into the fluid, it
accelerates to the terminal velocity. Once this velocity is reached, it no longer accelerates and all
the forces on the sphere are in equilibrium.
D
FB
a
Ut
W
The drag force on immersed bodies with simple shapes can be correlated to the speed with which
the body moves through the fluid. This is achieved by specifying the drag coefficient CD defined
by
CD 
drag
D
1
,
inertial force 2  U 2 S
GO WITH THE FLOW
Introduction to Fluid Dynamics
Summer 1999
where D is the drag,  is the density of the fluid, U is the speed of the fluid approaching the
body, and S is the projected frontal area, i.e., the maximum area perpendicular to the flow
direction. The  subscript indicates “freestream” quantities, i.e. quantities that are measured in
the undisturbed fluid far upstream of the body. In general, the overall drag force is composed of
a component purely from friction and another component, called profile drag that results from
the finite size and shape of the body. A number of experiments have been performed to
determine CD for several geometries. These experiments show that the variation of CD depends
primarily on a parameter called the Reynolds number Re, defined by
Re 
inertial force  U  L
,

viscous force

where L is some characteristic length (diameter in the case of the sphere) and the other quantities
are as defined earlier. A flow with a relatively large value for Re is dominated by inertial forces,
thus appears nearly inviscid. In the case of a very low-Re flow, called creeping flow or Stokes’
flow, the inertial forces can be neglected and Newton’s second law of motion reduces to Stokes’
equation for a sphere, valid for Re < 1,
D  6 Ua .
If the velocity (speed) V in this equation is the terminal velocity Ut of the sphere of radius a, it
provides a means for computing the viscosity  by writing the equation for the balance of forces
on the sphere,
D  FB  W .
Or substituting with Stokes’ equation, we have finally,

W  FB W  FB

,
6 U t a 3 U t d
where d is the sphere diameter. In the following experiment, use this relation to compute and
compare the viscosities of a few common liquids.
Experimental Procedure
1. Measure the diameter and weight of the test sphere, and then compute the volume and
density.
2. Measure the density of the test fluid.
a) Measure the weight of an empty beaker.
b) Fill the beaker to some prescribed volume with the test fluid and record the weight of the
beaker/fluid combination.
c) Compute the fluid density by converting the weight of the liquid to mass and dividing by
the volume of liquid.
3. Place timing marks on the cylinders containing the fluids. The speed of the falling sphere is
determined by dividing the distance between the marks by the time it takes the sphere to fall
that distance.
4. Compute the coefficient of viscosity 