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 109 [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
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