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/m^{3}]

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* = - σ_{0} = - 1/3 (σ_{xx} + σ_{yy} + σ_{zz})

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