The vector acceleration, a, has components both tangential and normal to the streamline, the
tangential component embodying the change in magnitude of the velocity, and the normal
component reflecting the change in direction
dv s ∂v s 1 ∂(v 2 )
as =
=
+
s
(2.3)
∂t
2 ∂s
dt
dv n ∂v n v 2
+
an =
=
(2.4)
∂t
dt
r
The first terms in Equations 2.3 and 2.4 represent the change in velocity, both magnitude and
direction, with time at a given point. This is called the local acceleration. The second term in
each equation is the change in velocity, both magnitude and direction, with distance. This is
called convective acceleration.
Uniform flow: In uniform flow the convective acceleration terms are zero.
∂v 2
v2
= 0 and
=0
s
(2.5)
∂s
r
Nonuniform flow: In nonuniform flow, the convective acceleration terms are not equal to zero.
∂v s
v2
≠ 0 and
≠0
(2.6)
∂s
r
∂v
Flow around a bend (v 2 / r ≠ 0) and flow in expansions or contractions
≠ 0 are examples of
∂s
nonuniform flow.
Steady flow: In steady flow, the velocity at a point does not change with time
∂v s
∂v
= 0 and n = 0
(2.7)
∂t
∂t
Unsteady flow: In unsteady flow, the velocity at a point varies with time
∂v s
∂v
≠ 0 and n ≠ 0
(2.8)
∂t
∂t
Examples of unsteady flow are channel flows with waves, flood hydrographs, and surges.
Laminar flow: In laminar flow, the mixing of the fluid and momentum transfer is by molecular
activity.
Turbulent flow: In turbulent flow the mixing of the fluid and momentum transfer is related to
random velocity fluctuations. The flow is laminar or turbulent depending on the value of the
Reynolds number (Re = ρVL/), which is a dimensionless ratio of the inertial forces to the viscous
forces. Here ρ and are the density and dynamic viscosity of the fluid, V is the fluid velocity, and
2.2
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