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Review of Hydrodynamic Theory Chapter | 2 33

Some functions are frequently used in indicial notation. The Kronecker delta
δ, which is used later in this chapter, is defined as follows

δ i,j = 1 if i = j (2.9)
δ i,j = 0 if i = j

2.1.2 More Examples of Indicial Notation
Using the basic rules that we described previously, we are able to write some
equations very efficiently. For instance, the mass balance equation (which is
discussed later) can simply be written as

∂u i ∂u 1 ∂u 2 ∂u 3 ∂u ∂v ∂w
= 0or + + = 0or + + = 0 (2.10)
∂x i ∂x 1 ∂x 2 ∂x 3 ∂x ∂y ∂z
For the Laplace operator, which is used later to quantify shear stress, we have
3
2
2
2
2
2
2
2
2
∂ u ∂ u ∂ u ∂ u ∂ u ∂ u ∂ u ∂ u 2
= = + + = + + =∇ u (2.11)
∂x k ∂x k ∂x k ∂x k ∂x 2 ∂x 2 ∂x 2 ∂x 2 ∂y 2 ∂z 2
k=1 1 2 3
2.2 REYNOLDS TRANSPORT THEOREM
In mechanics, the equations of motion can be described for either a system
or a control volume. In particular, in fluid mechanics, a fixed volume of
space (control volume) is, usually, more suitable than a definite mass of the
fluid (system). Nevertheless, the basic laws of continuum mechanics, such as
Newton’s second law, are applied to mass (i.e. system). Therefore, we need to
set the relation between system and control volume. Reynolds transport theorem
describes the relationship between a system and a control volume, as follows
(Fig. 2.1)
dN d
= (ρβ)dV = (ρβ)dV + (ρβ)u · dS (2.12)
dt dt S
V system V CV
where V CV is the fixed control volume in space, which we hereafter call V;
V system is the system volume, which is moving as it has a constant mass; ρ
is the fluid density; S is the control surface; u is the fluid velocity; S = Sn
is the control surface; and n is the unit vector normal to the control surface.
β is defined as dN/dm, where N is any fluid property (e.g. mass, momentum,
energy), and m is mass. This theory simply indicates that the net flux of a fluid
property (outflux minus influx) through a control volume plus changes in the
control volume equals the system change of that property. For instance, consider
a pond or a lagoon as a fixed control volume. If the inflow Q in and outflow Q out
of water are equal, the amount of water in the pond (ρV pond ) does not change