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Rotational viscoelastic laminar boundary layer flow around a rotating disc

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Summary

The equations of motion for the laminar boundary layer flow over a rotating disc have been derived for a fluid which obeys aRivlin-Ericksen type of constitutive equation and whose material parameters are assumed to be arbitrary functions of the second invariant of the rate of deformation tensor. The analysis establishes the conditions under which a true similarity solution is possible. An inspectional analysis yields a relationship between the moment coefficient, a generalizedReynolds number and a modifiedWeissenberg number which incorporates a variable relaxation time with a process time characteristic of the boundary layer flow on the disc. Experimental data obtained are analysed in terms of the derived relationship and the agreement between the two, after the determination of the unknown constants, is found to be quite sound. A brief discussion follows which emphasizes the role of geometry, regime of flow and viscoelastic material parameters in giving a wide variety of flow phenomena.

Zusammenfassung

Die Bewegungsgleichungen für die laminare Grenzschichtströmung um eine rotierende Scheibe wurden für eineRivlin-Ericksen-Flüssigkeit abgeleitet. Die Materialparameter in dieser Zustandsgleichung wurden als beliebige Funktionen der zweiten Invarianten des Deformationsgeschwindigkeitstensors gesehen. Die Bedingungen wurden gegeben, unter denen eine echte Ähnlichkeits-Lösung existiert. Die Inspektionsanalyse wurde dann benutzt, eine Gleichung zwischen dem Widerstandskoeffizienten und derReynolds-Zahl abzuleiten, die auch eineWeissenberg-Zahl mit einer variablen Relaxationszeit und einer charakteristischen aus der Grenzschichtströmung abgeleiteten Prozeßzeit enthält. Die Versuchsdaten wurden mit Hilfe der Theorie analysiert, und eine gute Übereinstimmung wurde gefunden. Die Arbeit wird mit einer kurzen Diskussion beendet; in der Rolle der Geometrie des laminaren bzw. turbulenten Strömungsbereiches und der viskoelastischen Stoffparameter herausgestellt wird.

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Abbreviations

A :

material parameter\(\frac{{dyne}}{{cm^2 }}\) sec-b

A 1 :

Rivlin-Ericksen acceleration tensor of first order

A 2 :

Rivlin-Ericksen acceleration tensor of second order

b :

material parameter, dimensionless

C 1,C 2 :

the constants defined by eqs. [37] and [38] respectively

C M :

moment coefficient, dimensionless

C Minel :

Moment coefficient for inelastic fluids, dimensionless

C Mviscoel :

moment coefficient for viscoelastic fluids dimensionless

D :

rate of deformation tensor

\(\dot D\) :

material time derivative of rate of deformation tensor

F′ :

function ofζ, defined by eq. [15]

f(n) :

function ofn, defined by eq. [45]

G′:

function ofζ, defined by eq. [16]

H′:

function ofζ, defined by eq. [18]

K :

material parameter\(\frac{{dyne}}{{cm^2 }}\) (sec)n

M :

moment on a disc, dyne-cm

n :

material parameter, dimensionless

P :

material parameter\(\frac{{dyne}}{{cm^2 }}\) (sec)q

p :

isotropic pressure dyne/cm2

q :

material parameter dimensionless

r :

radial coordinate, cm

R :

radius of the disc, cm

Re0w :

Reynolds number

T :

stress tensor

T rr ,T θθ ,T zz ,T rθ ,T rz ,T θz :

components of stress tensor dyne/cm2

υ :

velocity vector

υ r ,υ 0,υ z :

components of velocity vector cm/sec

W :

vorticity tensor

Wi 1,Wi 2 :

Weissenberg numbers defined by eqs. [24] and [25] respectively

z :

axial coordinate, cm

β(n) :

function ofn defined by eq. [40]

\(\dot \gamma \) :

shear rate, sec−1

δ :

Kronecker delta in eq. [1]

δ′:

boundary layer thickness cm, (eq. [14])

Δ :

boundary layer thickness atr = R, cm (eq. [28])

ζ :

dimensionless variablez/δ, (eq. [14])

η :

dimensionless variabler/R, (eq. [14])

µ(\((\tilde II)\)):

scalar function, defined by eq. [11]

λ(\((\tilde II)\)):

scalar function, defined by eq. [13]

λ :

relaxation time of the fluid, sec

∇ :

nabla operator\(\frac{\partial }{{\partial r}} + \frac{1}{r}\frac{\partial }{{\partial \theta }} + \frac{\partial }{{\partial z}}\)

ρ :

density of the fluid gm/c. c.

ω(\((\tilde II)\)):

scalar function defined by eq. [12]

Ω :

rotational speed of the disc rad/sec

θ :

circumferential coordinate

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Authors and Affiliations

  1. Department of Chemical Engineering, University of Salford, M5 4WT, Salford, Lancashire, England

    D. D. Kale, R. A. Mashelkar & J. Ulbrecht

Authors
  1. D. D. Kale
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  2. R. A. Mashelkar
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  3. J. Ulbrecht
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Kale, D.D., Mashelkar, R.A. & Ulbrecht, J. Rotational viscoelastic laminar boundary layer flow around a rotating disc. Rheol Acta 14, 631–640 (1975). https://doi.org/10.1007/BF01520816

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