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Fluid Dynamics and the Navier-Stokes Equations

Steady Flow

The steady flow in a pipe is a very simple solvable case. Since the left-hand side of Eq.(2) vanishes for steady flow, and there is no external force, it can be written in cylindrical coordinates (Figure 03a) as:
Cylinderical Coordinates 0 = - dp / dz + (1/r) d (r d v / dr) / dr ---------- (4)

where v denotes the velocity field in the z direction as shown in Figure 02. For a constant pressure drop (per unit length)
d p / dz is a negative constant, the solution becomes:

v (r) = vm ( 1 - r2 / R2 ) ---------- (5)

where vm = (R2 / 4 ) (- dp / dz) ---------- (6)

Figure 03a Cylindrical Co- ordinates [view large image]

is the velocity at the centre, and R is the radius of the pipe. The boundary conditions are
v = vm at r = 0 and v = 0 at r = R.

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