Summary
When a liquid drop is suspended in a viscous fluid undergoing shear flow, it deforms; if the deformation is small, the drop becomes an ellipsoid with its principal axes directed along the principal directions of strain of the fluid. In general, the lengths of the axes are all different; this is the case for hyperbolic flow, for which explicit theoretical formulas are given. Experimental observations of all three axes of deformed drops in hyperbolic flow agree with the theory.
The migration of a liquid drop in the non-uniform shear field between counter-rotating discs is calculated by finding in detail velocity fields that satisfy the creeping motion equations. If the drop shape is only slightly different from spherical, it is possible to find the velocity and pressure fields by a perturbation scheme in the small parameter characterizing the deformation, usingLamb's general solution in spherical harmonics (8). The harmonics required in the solution are found by first solving the problem of an infinitely viscous deformed drop, which is the same as a rigid body; the solution for drops of any viscosity is then determined by using the same harmonics but with different coefficients. A force is found to act on a fixed drop along the line joining the drop center to the axis of rotation of the discs. The velocity at which a free drop migrates along this line is then found by using the solution ofHadamard andRybczinski for a sedimenting liquid sphere. Experiments in a counter-rotating disc apparatus gave inconclusive results.
Zusammenfassung
Dieser Beitrag enthält eine Theorie des Verhaltens eines Flüssigkeitstropfens, der in einer viskosen Flüssigkeit suspendiert ist, die einer ungleichförmigen Scherströmung unterworfen wird und sich zwischen zwei parallelen Scheiben, die langsam um eine gemeinsame Achse gegeneinander rotieren, befindet. Es wird vorausgesagt, daß der Tropfen auf die Achse zuwandert, wenn seine Viskosität 13,9% der Viskosität der suspendierenden Flüssigkeit überschreitet; andernfalls entfernt er sich von der Achse mit einer Geschwindigkeit proportional seinem Verformungsparameter.
Eine Verallgemeinerung der Theorie der Verformung eines Tropfens zu einem Ellipsoid, die für willkürliche Scherfelder gilt, zeigt, daß seine drei Hauptachsen alle verschieden sein können. Dieses Ergebnis wird bestätigt durch Experimente beim hyperbolischen Fließen.
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Abbreviations
- A (m) n ,B (m) n ,C (m) n :
-
coefficients ofP (m) n inp (n) ,Φ (n) ,χ (n)
- b; b x ,b y ,b z :
-
radius of sphere; semiaxes of drop
- C :
-
a point
- D :
-
deformation parameter
- F;F :
-
force; surface function
- G :
-
velocity gradient
- h 1;h 2 :
-
y-coordinates of upper, lower disc
- i, j, k :
-
unit vectors alongx, y, z-axes
- I :
-
dyadic idemfactor
- k; K (λ) :
-
gradient in velocity gradient; viscosity ratio factor
- M :
-
1/(n+1) (2n+3)
- n;n :
-
unit normal vector; an integer
- p, P :
-
outer, inner pressure
- P n ,P (m) n :
-
Legendre function, associated Legendre function
- p (n) :
-
spherical harmonic
- r, r :
-
radial vector, its magnitude
- S, S :
-
drop surface, dilatation tensor
- t :
-
time
- u :
-
arbitrary velocity field
- v, V :
-
outer, inner velocity field
- w :
-
migration velocity vector
- x, y, z :
-
Cartesian coordinates
- γ :
-
interfacial tension
- ζ :
-
vorticity vector
- θ :
-
azimuthal angle
- η :
-
outer fluid viscosity
- λ :
-
viscosity ratio
- π, Π :
-
stress tensor
- ω, ω 0 :
-
radial distance, initial radial distance
- χ(n):
-
spherical harmonic
- Φ :
-
longitudinal angle
- Φ(n) :
-
spherical harmonic
- ω 1,ω 2 :
-
angular velocities of upper, lower disc
- 0:
-
referred to origin on axis of rotation; initial value
- 1, 2:
-
of upper, lower disc
- b, C, S :
-
value atb, C, S
- (n):
-
order
- r, θ, Φ:
-
Φ r-, θ-, orΦ-component
- 0, 1:
-
zero-order, first-order
- (∞):
-
undisturbed value
- ′, ″:
-
partial, complete
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This work was supported by the National Heart Institute of the United States Public Health Service (Grant-H-5911).
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Chaffey, C.E., Brenner, H. & Mason, S.G. Particle motions in sheared suspensions. Rheol Acta 4, 56–63 (1965). https://doi.org/10.1007/BF01968737
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DOI: https://doi.org/10.1007/BF01968737