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Hydrodynamics of a drop submerged in an unbounded arbitrary velocity field in the presence of surfactants

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Abstract

A solution is presented for the flow fields interior and exterior to a single spherical droplet submerged in an unbounded fluid, for the general case when the unperturbed velocity is Stokesian but otherwise arbitrary. Surface-active-agents are present in the system and are included in the analysis.

A general equation for the terminal settling velocity is derived which contains as special cases the familiar solutions of Levich and our previous solution.

Also derived is a general expression for the drag force. This expression contains as special cases the Laws of Faxen and our previous results.

The function describing the interface and its deviation from sphericity is derived. This may be used for determining more accurate flow fields in an iterative procedure.

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Abbreviations

a :

radius of a spherical bubble

A n , A m n , B n , B m n , C n , C m n :

constants defined by (18) and (19)

b :

distance from the point of maximum velocity to the center of the droplet

c :

the deviation of concentration of surfactants in the bulk of the continuous medium from the concentration of surfactants in the bulk near the interface

D b :

diffusivity of surfactants in the continuous medium

D s :

diffusivity of surfactants on the interface

g :

local gravity

j n :

the normal flux of surfactants on the interface

k :

unit vector in the z-direction

K :

constant defined by (22)

L m n :

constants defined by (39)

m :

integer

n :

integer

p :

pressure

p n :

solid harmonics of order n

P m n :

associated Legendre polynomials

r :

radius in spherical coordinates

t :

unit vector of a local coordinate system

R 0 :

radius to point of zero velocity in Poiseuillian distribution

R 1, R 2 :

radii of curvature of the droplet

u :

velocity vector interior to the droplet

U :

settling velocity

U 0 :

maximum velocity in Poiseuillian velocity distribution

v :

velocity vector exterior to the droplet

v ∞ :

the unperturbed velocity field relative to the droplet

v ′∞ :

the unperturbed velocity field relative to a stationary coordinate system

V :

velocity field defined by (11)

z :

cylindrical coordinate

α n , α m n , β n , β n n , γ n , γ m n :

constants defined by (19) and (20)

β :

defined by (26a)

Γ :

surface concentration of surfactants

Γ 0 :

equilibrium surface concentration of surfactants

Γ′ :

deviation concentration of surfactants from equilibrium

\(\Gamma '_n ,{\Gamma '_n}^m ,\hat {\Gamma '_n}^m \) :

constants defined by (25) and (19)

δ :

thickness of Nernest boundary layer of diffusion

ε :

overall absorption rate constant

θ :

coordinate in the spherical system (Fig. 1)

λ :

ratio of the viscosities interior to the droplet and outside it

λ n , λ′ n :

constants defined by (30) and (42), respectively

μ :

kinematic viscosity

ξ :

function describing the deviation of the interface from a sphere

π (r) :

the normal vector of the stress tensor acting on the interface, outside the droplet, based on v

Π (r) :

the normal vector of the stress tensor acting on the interface, based on V

ρ :

density

σ :

surface tension

τ (r) :

the normal vector of the stress tensor acting on the interface, interior to the droplet, based on u

φ :

coordinate in the spherical system (Fig. 1)

Φ n :

solid spherical harmonic of order n

χ n :

solid spherical harmonic of the order n

0:

denotes a function evaluated at Γ 0

e:

denotes a property of the field exterior to the droplet

i:

denotes a property of the field interior to the droplet

n:

denotes the normal component of a quantity at the interface

r :

denotes the radial component of a quantity at the interface

t:

denotes the tangential component of a quantity at the interface

∞:

denotes an unperturbed function

*:

indicates a function evaluated at the interface

References

  1. Hetsroni, G. and S. Haber, Rheo. Acta 9 (1970) 488.

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  2. Levich, V. G., Physiochemical Hydrodynamics, Prentice Hall Inc. N.J., 1962.

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  3. Gal-Or, B. and S. Waslo, Chem. Eng. Sci. 23 (1968) 1431.

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  6. Lamb, H., Hydrodynamics, Dover, New York, 1945.

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  7. Landau, L. D. and E. M. Lifshitz, Fluid Mechanics Addison- Wesley, Mass. 1959.

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Author information

Authors and Affiliations

  1. Facnlty of Mech. Eng., Technion — Israel Institute of Technology, Haifa, Israel

    S. Haber & G. Hetsroni

Authors
  1. S. Haber
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  2. G. Hetsroni
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Haber, S., Hetsroni, G. Hydrodynamics of a drop submerged in an unbounded arbitrary velocity field in the presence of surfactants. Appl. Sci. Res. 25, 215–233 (1972). https://doi.org/10.1007/BF00382297

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  • DOI: https://doi.org/10.1007/BF00382297

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