# Interdisciplinary Applied Mathematics

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Perturbation solutions for the Dean flow have been derived in (Jones et al., 1989). Referring to the sketch of Figure 9.1 and using polar coordinates (r,    ф)    in the    transverse    xy    plane,    we    follow    Dean’s    formulation    for

FIGURE 9.1. Sketch    of    a twisted    pipe    and    notation used    in    the    Dean    flow solution. This basic    unit is composed    of    two    180°    curved    pipe    segments    of    constant

curvature.

the streamfunction ф and axial velocity w:

V2w

У2ф

1 ( дф dw r dr дф

1 f дф d

r dr дф

дф dw дф dr

дф d 5

дф dr j

)- C

^2ф + 2Dew

sin ф dw

r дф

cos ф

dw dr )

Here De = W2a3/(Rv2) is the Dean number with W the average axial velocity, a the pipe radius, and R the radius of curvature of the bend. Therefore, the Dean number is proportional to the square of the Reynolds number. Also, C is a nondimensional pressure gradient defined by

a2dp RWp ~дв’

The perturbation solution is obtained in a power series in De; at the lowest order the standard Poiseuille flow for a straight pipe is recovered. The first-order equations give the following equations for the particle motion

x

a

1152

h(r) + y2

/?/(?’)

r

У

a xy 1152 ~r~

h'(r),

^/3(1 ~r2),

(9.2)

where a = DeC2, в = DeC/Re, and

h(r) = -(4 — r2)(l — r2)2.

The angle в is used to describe the three-dimensional motion of the flow along the curved pipe together with the coordinates (x,y).

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