2.2 Velocity

Like force, velocity eqn is a vector with direction and magnitude, with SI units of eqn. Using the vector eqn to denote the position of a particle of fluid, its velocity is

 dx x u = ---= lim ---: dt t!0 t \relax \special {t4ht=

PICT\relax \special {t4ht=

Vector fields

While eqn can be used to denote a single velocity with magnitude and direction, it can also denote a vector field of velocity which varies from point to point across a spatial domain. A vector is represented by 3 numbers, relating to the co-ordinate system being used, e.g. eqn, eqn, eqn, in the Cartesian system.

PICT\relax \special {t4ht=

While the magnitude and direction of a vector is fixed, it is not invariant since the 3 values depend on the co-ordinate system used. We represent a vector without reference to the co-ordinate system by bold text, e.g.eqn” (compared to scalar “eqn”).

Basic vector algebra

Addition and subtraction of 2 vectors is performed by operating on respective components. Subtraction of two vectors eqn and eqn is performed by

a b = (ax bx;ay by;az bz): \relax \special {t4ht=
Multiplication of any vector eqn by a scalar eqn is performed by multiplying all the components by the scalar, e.g.
sa = (sa ;sa ;sa ) x y z \relax \special {t4ht=
Addition and multiplication are commutative, i.e. variables can be in any order (eqn). Subtraction is not commutative. Products between scalars and vectors are distributive, i.e.
s(a + b) = sa + sb; \relax \special {t4ht=
and with additional scalar eqn,
(s+ q)a = sa + qa: \relax \special {t4ht=
Division between a vector eqn and a scalar is only relevant when the scalar is the second argument of the operation, i.e.
a=s = (ax=s;ay=s;az=s): \relax \special {t4ht=
Notes on CFD: General Principles - 2.2 Velocity