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Stability of plane Poiseuille flow of a slightly viscoelastic fluid

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Abstract

Consideration is given to the stability of planePoiseuille flow of a slightly viscoelastic fluid which has a constant viscosity and normal stress differences varying nearly with the shear rate. It is shown that the presence of elasticity lowers the criticalReynolds number at which instability occurs.

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Abbreviations

B s :

= acceleration tensors, defined by eq. [2] of (5)

c :

= wave velocity of disturbance

D :

\(\frac{d}{{d\bar y}}\)

h :

= half width of channel

I :

= unit matrix

L :

= typical length

P :

= pressure gradient =\( - \frac{{\partial p}}{{\partial x}}\)

p :

= isotropic pressure

R :

=Reynolds number =\(\frac{{VL\varrho }}{\mu }\)

S :

=\(\frac{{\varrho LV}}{{w_3 }}\)

t :

= time

\(\bar u,\bar v\) :

= wave velocity

U 0 ′ :

=DU, evaluated aty 0

V :

= typical velocity

x,y,z :

= distance variables

\(\bar y\) :

=y/L

y 0 :

= critical layer, whereU=c

α :

= wave number

ε :

=\((\alpha RU'_0 ) - {\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 3$}}\)

ϱ :

= density

λ0 :

\(\frac{{(\alpha RU'_0 )^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} }}{{S\sqrt {2U'_0 + w_1 ^2 } }}\)

η :

= stretched coordinate system\(\eta = \frac{{y - y_0 }}{\varepsilon }\)

\(\bar \mu ,\bar w_2 ,\bar w_3 \) :

= coefficients of acceleration tensors (material coefficients)

μ, w 1,w 2,w 3 :

= material constants

References

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Author notes

  1. C. F. Chan Man Fong

    Present address: Department of Mathematics, University of Reading, Reading, England

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    1. C. F. Chan Man Fong
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    Chan Man Fong, C.F. Stability of plane Poiseuille flow of a slightly viscoelastic fluid. Rheol Acta 7, 324–326 (1968). https://doi.org/10.1007/BF01984845

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    • DOI: https://doi.org/10.1007/BF01984845

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