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Nonlinear Taylor stability of viscoelastic fluids

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Abstract

Using Stuart's energy method, the torque on the inner cylinder, for a second order fluid, in the supercritical regime is calculated. It is found that when the second normal stress difference is negative, the flow is more stable than for a Newtonian fluid and the torque is reduced. If the second normal stress difference is positive, then the flow is more stable and there is no torque reduction. Experimental data related to the present work are discussed.

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Abbreviations

a :

amplitude of the fundamentals

A (1) ij , A (2) ij :

first and second Rivlin-Ericksen tensors

d :

r 2−r 1

D:

d/dx

E :

\(\frac{{2\pi r_1 r_o \eta _0^2 h}}{{\rho d^2 }}\left( {1 - \frac{{I_1 I_4 }}{{I_3 }}} \right)\)

F :

\(\frac{{2\pi r_1^2 r_o \eta _0^2 T_c h}}{{\rho d^3 }}\frac{{I_1 I_4 }}{{I_3 }}\)

g ij :

metric tensor

G :

torque on the inner cylinder in the supercritical regime

h :

height of the cylinders

k 0 :

β/ρd 2

k 1 :

γ/ρd 2

I 1 :

\(\mathop \smallint \limits_{ - \tfrac{1}{2}}^{\tfrac{1}{2}} - u\upsilon dx + \left( {k_o + 2k_1 } \right)r_1 /d\mathop \smallint \limits_{ - \tfrac{1}{2}}^{\tfrac{1}{2}} \upsilon (D^2 - \lambda ^2 ) u dx\)

I 2 :

\(\mathop \smallint \limits_{ - \tfrac{1}{2}}^{\tfrac{1}{2}} [(D^2 - \lambda ^2 ) u]^2 dx\)

I 3 :

\(\begin{gathered} (1 + 2k_0 \lambda ^2 )(\mathop \smallint \limits_{ - \tfrac{1}{2}}^{\tfrac{1}{2}} - uv dx)^2 - (1 + k_0 \lambda ^2 )\mathop \smallint \limits_{ - \tfrac{1}{2}}^{\tfrac{1}{2}} u^2 v^2 dx + \hfill \\ + \left( {k_0 + 3k_1 } \right)\mathop \smallint \limits_{ - \tfrac{1}{2}}^{\tfrac{1}{2}} - uv dx\mathop \smallint \limits_{ - \tfrac{1}{2}}^{\tfrac{1}{2}} u^2 v dx + \hfill \\ + k_0 \mathop \smallint \limits_{ - \tfrac{1}{2}}^{\tfrac{1}{2}} (2u^3 v + 4uv Du Dv + 2u^2 (Dv)^2 + \hfill \\ + v^2 (Du)^2 )dx + k_1 \mathop \smallint \limits_{ - \tfrac{1}{2}}^{\tfrac{1}{2}} (u^2 (Dv)^2 + v^2 (Du)^2 + 2uv Dv Dv + 3u^3 v) dx \hfill \\ \end{gathered}\)

I 4 :

\((1 - 2k_0 \lambda ^2 )\mathop \smallint \limits_{ - \tfrac{1}{2}}^{\tfrac{1}{2}} - uv dx + (k_0 - 2k_1 )\mathop \smallint \limits_{ - \tfrac{1}{2}}^{\tfrac{1}{2}} u^2 dx\)

r 1, r 2 :

radii of inner and outer cylinders respectively

r 0 :

1/2(r 1+r 2)

R :

Reynolds number Ω 1 r 1 dρ/η 0

R c :

critical Reynolds number

T :

Taylor number r 1 Ω 21 d 3 ρ 2/η 20 *)

T c :

critical Taylor number

u 1, v 1, w 1 :

Fundamentals of the disturbance

u i , v i , w i , (i>1):

harmonics

\(\bar v\) :

mean velocity (not laminar velocity)

u :

−u 1/ar 1 Ω 1

v :

v 1/Rar 1 Ω 1

x :

(r−r 0)/d

β,γ :

material constants

η0:

viscosity

λ :

wave number αd

ρ :

density

Ω 1 :

angular velocity of inner cylinder

∼:

tilde denotes complex conjugate

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Authors and Affiliations

  1. Dept. of Mathematics, University of Reading, Reading, England

    C. F. Chan Man Fong

Authors
  1. C. F. Chan Man Fong
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Chan Man Fong, C.F. Nonlinear Taylor stability of viscoelastic fluids. Appl. Sci. Res. 23, 16–22 (1971). https://doi.org/10.1007/BF00413184

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  • DOI: https://doi.org/10.1007/BF00413184

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