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The effect of non-steady wall shear on pressure surges in conduits carrying viscous liquids

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Summary

A study is made of the attenuation of pressure surges in a two-dimension a channel carrying a viscous liquid when a valve at the downstream end is suddenly closed. The analysis differs from previous work in this area by taking into account the transient nature of the wall shear, which in the past has been assumed as equivalent to that existing in steady flow. For large values of the frictional resistance parameter the transient wall shear analysis results in less attenuation than given by the steady wall shear assumption.

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Abbreviations

c :

√β/ρ, ft/sec

e :

base of natural logarithms

F(x′, y):

integration function, equation (38)

(x′):

mean value of F(x′, y)

g :

local acceleration of gravity, ft/sec2

h :

width of conduit, ft

k :

(2m−1)2 π 2 νL/h 2 c, equation (35)

k*:

12νL/h 2 c, frictional resistance parameter, equation (46)

L :

length of conduit, ft

m :

positive integer

n :

positive integer

p :

pressure, lb/ft2

p 0 :

constant pressure at inlet of conduit, lb/ft2

P :

pressure plus elevation head, p+ρgz, equation (4)

\(\bar P\) :

mean value of P over the conduit width h

P 0 :

p 0+ρgz 0, lbs/ft2

R :

frictional resistance coefficient for steady state wall shear, lb sec/ft4

s :

positive integer; also, condensation, equation (6)

t :

time, sec

t′ :

ct/L, dimensionless time

u :

x component of fluid velocity, ft/sec

u m :

mean velocity in conduit, equation (12), ft/sec

u 0(y):

velocity profile in Poiseuille flow, equation (19), ft/sec

ū :

transformed velocity

U :

initial mean velocity in conduit

x :

distance along conduit, measured from valve (fig. 1), ft

x′ :

x/L, dimensionless distance

y :

distance normal to conduit wall (fig. 1), ft

y′ :

yπ/h, equation (25)

z :

elevation, measured from arbitrary datum, ft

z 0 :

elevation of constant pressure source, ft

β :

isothermal bulk compression modulus, lbs/ft2

β n :

\(\tfrac{1}{2}\sqrt {k^2 - (2n - 1)^2 \pi ^2 }\), equation (37)

γ n :

(2n−1)π/2, equation (36)

μ :

viscosity, slugs/ft sec

ν :

μ/ρ = kinematic viscosity, ft2/sec

ρ :

density of fluid, slugs/ft3

ρ 0 :

density of undisturbed fluid, slugs/ft3

ø :

angle between conduit and vertical (fig. 1)

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Authors and Affiliations

  1. Mechanical Engineering Department, Carnegie Institute of Technology, Pittsburgh, Pennsylvania, USA

    W. T. Rouleau

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  1. W. T. Rouleau
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Additional information

The research upon which this paper is based was supported by a grant from the National Science Foundation.

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Rouleau, W.T. The effect of non-steady wall shear on pressure surges in conduits carrying viscous liquids. Appl. sci. Res. 13, 16–28 (1964). https://doi.org/10.1007/BF00382032

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  • DOI: https://doi.org/10.1007/BF00382032

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