Page 103 - Aeronautical Engineer Data Book

Basic fluid mechanics 81
The equations for 1D flow are derived by
considering flow along a straight stream tube
(see Figure 5.2). Table 5.2 shows the principles,
and their resulting equations.

5.2.2 2D Flow
2D flow (as in the space between two parallel
flat plates) is that in which all velocities are
parallel to a given plane. Either rectangular (x,y)
or polar (r, ) co-ordinates may be used to
describe the characteristics of 2D flow. Table 5.3
and Figure 5.3 show the fundamental equations.

Rectangular co-ordinates
y ∂ v δ y
v +
∂ y 2

v
P u δ y u + ∂ u δ x
∂ u δ x ∂ x 2
u –
∂ x 2
δ x
∂ v δ y
v –
∂ y 2 x
Unit thickness
Polar co-ordinates
δθ
+ ∂θ 2 δ r
∂ q t
∂ q n
q t (r + δr + ∂ r 2
2
q n
)δθ
q t
q n
P(r,θ )
δ r
∂ q n
– ∂ r 2
(r – δr
2
)δθ
q n
δ r
δθ
∂ q t
– ∂θ 2
q t
Fig. 5.3 The continuity equation basis in 2-D