Mechanics is the
study of the
movement and
distortion of
matter under the
influence of
external forces. In general, a distinction is made between
solids and
fluids (lumping the
gasses in with the
liquids). The distinction lies in the fact that a
shear stress applied to a solid results in a distinct, or measurable, distortion, whereas a
shear stress applied to a fluid results in a continuing distortion, albeit one with a distinct, again measurable,
velocity.
For a large part the study of the mechanics of liquids and gasses are identical. One major difference is that only liquids have a
free surface level. Another difference is that gasses are generally more
compressible than liquids.
Fluid characteristics
When dealing with fluid mechanics, one uses a
continuum model to model the
characteristics of the fluid, thereby bypassing the intricacies of the
material on a lower level.
Density
Mass per unit of volume gives the
density of the material under study, generally denoted with the symbol ρ (ρ, or rho, for those that don't get these
HTML symbols to work). The
dimension for density follows from the definition, as given by the
SI:
[ρ] = M L-3; the SI unit is 1 [kg/m3]
The density of
pure water (and other pure liquids) is only dependent on
temperature and
pressure. In most cases where one uses fluid mechanics (like in
civil engineering) the density can be approximated to a
discreet value instead of including the influence of temperature and pressure in the calculations.
Values normally used are:
Constitutive equations
The
constitutive equations give the
relationship between the
stress in a material and the resulting distortion of
volume and
shape. The three areas for which we need equations are:
- Stress
- Compressibility
- Fluidity
Stress
In liquids and
gasses tensile stress is a very rare occurrence, and therefore the definition
pressure (
p) is introduced, which is equivalent to the
isotropic part of the
compressive stress in a liquid or gas:
p = - σ0 = - 1/3 (σxx + σyy + σzz)
Variations in the isotropic part of the stress (tension or pressure) result in changes in the
volume of the fluid, while variations in the deviator stress (the remainder) result in changes in the
shape of the fluid (usually resulting in the fluid being in motion).
Compressibility
If the pressure increases, fluids become compressed. The relation between volume (
V) and pressure (
p) for fluids under the idealization of
elasticity is expressed using the
compressibility modulus K.
The value of
K increases with increasing pressure. However, for a large
range of pressures the value of
K for
water (without gas
bubbles!) is practically constant, namely equal to roughly 2.2 x 10
9 [Pa].
Fluidity
The
dynamic viscosity η determines the
fluidity of a liquid or gas, and has the dimension (according to the
SI) of M L
-1 T
-1 (which is 1 [Pa s] = 1 [kg m
-1 s
-1]).
Usually the
kinematic viscosity ν is used in calculations, which is defined as follows:
This is just the dynamic viscosity divided by the density of the material in question.
The rest
The above information is the main background needed to understand fluid mechanics, at least in the context of situations in the size range usually encountered in
civil engineering. One of the things still left out here is
capillary attraction, which is a
phenomenon that is only of
importance in very
slow and
tiny distortions.
Another
omission in the above is the importance of
dimensionless parameters in fluid mechanics. For example, the discussion of the
compressibility modulus K can be followed further to the definition of the
Mach number, which is the
ratio of the
velocity of the
flow of a medium to the velocity of
sound in that medium (which is linked to the compressibility of the medium → sound == compression
waves).
The
Reynolds number is another such dimensionless parameter, which can be arrived at by following the discussion of the fluidity further. The Reynolds number gives an indication whether a certain flow situation is
turbulent or
laminar. It is also used to do
simulations on differing
scales, the idea being that if the Reynolds number is kept constant the simulation is
dynamically similar to the original.
Sources:
An adaptation of one of my college textbooks - node your homework
My first nodeshell rescue
August 16, 2001