The Nature of Fluid Flow - Fluid Properties at Nano/Meso Scale: A Numerical Treatment - page 11

If these conditions are met, the fluid system can be considered as a continuum.
This is an important classification, as it means the flow can be approximated using
continuum laws.
The continuum laws can be applied in both simple analytical form, as in the
Bernoulli equation (inviscid flows),

P
ρ

+ v

2

+ gh = constant , (1.22)

2

or for more complex situations that require numerical solution. For cases such as
simple pipe flows, the Bernoulli equation can be of use where little information
is required. However, in complex systems or geometries, a more detailed analysis
and interrogation is required. In this case, fluid behaviour can be simulated using
a set of conservative governing equations solved numerically. These simulations,
based on the continuum assumptions and continuum scale observations and laws,
provide a detailed and accurate model of fluid behaviour, where experiments are
difficult or expensive, or a greater amount of information is needed.

1.2.3 Continuum Scale Simulation

Both simple and complex fluid systems can be investigated, within the limits of
the continuum assumptions, by sets of governing differential equations that de-
scribe fluid behaviour. The mathematical solution of these equations throughout
a fluid domain is known as computational fluid dynamics (CFD). The governing
equations describe the mathematical representation of a physical model that is
derived from experimental flow measurements and observations. These represen-
tative equations are then replaced with an equivalent numerical description, which
is solved using numerical techniques for the dependent variables of velocity, den-
sity, pressure and temperature. One of the most widely used sets of governing
equations are the Navier-Stokes equations.

1.2.3.1 Navier-Stokes governing equations

The Navier-Stokes equations are a set of governing equations that describe the
behaviour of fluids in terms of continuous functions of space and time. They
state that changes of momentum in the fluid are based on the product of the change
in pressure and internal viscous dissipation forces acting internally. The scheme
works by not considering instantaneous values of the dependent variables, but
their flux, which in mathematical terms is interpreted as the derivative of the vari-
ables. The equation set is separated into three conservation laws for mass, energy
and momentum.