Page 109 - Aeronautical Engineer Data Book

Basic fluid mechanics 85
5.2.3 The Navier-Stokes equations
The Navier-Stokes equations are written as:
� ∂u ∂u ∂u � ∂p � ∂ u ∂ 2 u �
2
+u +v = X– +µ +
2
2
∂x
∂y
∂y
∂x
∂x
∂t
�
∂v ∂v ∂v � ∂p � ∂ v ∂ v �
2
2
+u +v = Y– +µ +
2
2
∂y
∂x
∂y
∂x
∂t
�
�
�
�
∂y
Inertia Body Pressure Viscous
term force term term
term
Source y
ψ = constant, i.e. streamlines
radiating from the origin O.
φ = constant,
i.e. equipotential
lines centred at
the origin O.
x
O
If q>O this is a source of strength |q|
If q<O this is a sink of strength |q|
Flow due to a combination y
of source and sink

ψ = constant

φ = constant

x
A O B

Fig. 5.7 Sources, sinks and combination