Abstract
The streaming motion past a spherical microcapsule is studied. The particle consists of a thin elastic membrane enclosing an incompressible fluid. Since the problem is highly nonlinear, a perturbation solution is sought in the limiting case where the deviation from sphericity is small. Obviously, the capsule remains nearly spherical when λ, the ratio of viscous forces to elastic (shape-restoring) membrane forces is small. In this limit, the rheology of the inside fluid is immaterial and the problem is essentially characterized by three parameters: λ, the Reynolds number Re (interia effect), and the Weissenberg number We (non-newtonian effect). The deformation is obtained explicitly under the restriction We<1, Re<1. It is shown that to leading order, the capsule deforms exactly into a spheroid which can be either oblate or prolate, depending mainly upon the elasticity number We/Re: for We/Re<0.57 the spheroid is oblate, while for We/Re>0.81 a prolate spheroid results. For 0.57<We/Re<0.81 additional details of the rheology of the membrane and of the suspending fluid are needed. The degree of the deformation is governed by the parameters λ Re. All parameters of the problem enter into the expression of the drag force. On a qualitative basis, these results are similar to those for droplets although major differences exist quantitatively.
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Abbreviations
- a :
-
sphere radius
- c, c′ :
-
density of outside and inside fluid, respectively
- D :
-
measure of deformation (5.4)
- E :
-
Young's modulus
- E1:
-
first elasticity number (5.7b)
- f :
-
rate of strain dyadic
- F = \(F_{\hat 2} \hat k\) :
-
drag force (4.10) \(\mathop {F_{ij} }\limits^{(k)}\): drag force corresponding to the field (\(\mathop {\upsilon _{ij} }\limits^{(k)} ,\mathop {P_{ij} }\limits^{(k)}\))
- h :
-
membrane thickness
- \(\hat k\) :
-
unit vector in z-direction
- K=λ Re:
-
parameter (5.7a)
- N 0 , N φ :
-
normal stress resultants of membrane
- \(\hat n\) :
-
outer unit vector of undeformed membrane
- \(\hat N\) :
-
outer unit vector of deformed membrane
- P 2(cos θ):
-
Legendre polynomial (P2(x)=1/2(3x2−1)
- P j i (cos θ):
-
associated Legendre functions (P2 = P 02 )
- p′ :
-
inside pressure
- p :
-
outside pressure, p→p ∞ far away; expanded as p = \(\mathop p\limits^{(0)}\) + γ \(\mathop p\limits^{(1)}\) + ... (4.1)
- \(\mathop p\limits^{(0)}\) :
-
outside pressure for undeformed particle; expansion in inner region r<r c : \(\mathop p\limits^{(0)}\) ≡ \(\mathop p\limits^{(i)}\) = p 0 + Rep 1 + ..., (3.7) with p 0 = p 00 + Rep 01 + ... (3.8c); expansion in outer region (use \(\mathop p\limits^{(0)}\) ≡ Rep (0)) r>r c : p (0) = \(\bar p\) 1+... (3.13)
- \(\mathop p\limits^{(1)}\) :
-
outside pressure for deformed particle (expanded like \(\mathop p\limits^{(0)}\))
- q = q r δ r + q θ δ θ :
-
load vector (2.4)
- r :
-
distance from particle center
- r c = 1/Re:
-
critical radius, separating inner and outer regions (3.11)
- Re=cυ ∞ a/η 0 :
-
Reynolds number of outside fluid
- s, s′ :
-
non-Newtonian contribution to the stress deviator for outside (inside) fluid (3.2)
- T, T′ :
-
stress tensor of outside (inside) fluid
- u = u r δ r + u θ δ θ :
-
displacement vector (2.10)
- υ, υ′ :
-
velocity field of outside (inside) fluid υ → υ ∞ = υ ∞ \(\hat k\) far away. Expanded as υ = \(\mathop \upsilon \limits^{(0)}\) + γ \(\mathop \upsilon \limits^{(1)}\) + ... (4.1)
- \(\mathop \upsilon \limits^{(0)}\) :
-
outside velocity field for undeformed particle; expansion in inner region (r<r c ): \(\mathop \upsilon \limits^{(0)}\) ≡ υ (i) = υ 0 + Reυ 1 + ..., (3.7) with υ 0 = υ 00 + Weυ 01 + ... (3.8a/8b). Expansion in outer region (r>r c ): \(\mathop \upsilon \limits^{(0)}\) ≡ υ (0) = \(\hat k\) + Re\(\bar \upsilon\) 1 + ..., (3.13)
- \(\mathop \upsilon \limits^{(1)}\) :
-
outside velocity for deformed particle (expanded like \(\mathop \upsilon \limits^{(0)}\))
- We = τυ ∞/a :
-
Weisenberg number of outside fluid (3.1)
- We′ = τ′υ ∞/a :
-
Weissenberg number of inside fluid (5.8a)
- X :
-
point on the undeformed membrane
- x :
-
the same point after deformation (2.12)
- α :
-
parameter appearing in the constitutive equation of a second order fluid, Eqn. (3.4). For viscometric flows −α/2 denotes the zero shear rate limit of the second normal stress coefficient relative to the first normal stress coefficient
- α′ :
-
corresponding parameter for inside fluid
- β :
-
parameter defined by (3.29)
- γ :
-
parameter describing the deformation (3.28)
- δ :
-
unit tensor \(\left( {\begin{array}{*{20}c} {100} \\ {010} \\ {001} \\ \end{array} } \right)\)
- δ r , δ θ :
-
unit vectors using spherical polar coordinates
- ∈ θ , ∈ φ :
-
the normal strains for the undeformed membrane (2.8)
- η 0, η′ :
-
viscosity of outside and inside fluid, respectively
- θ :
-
polar angle, measured relative to the direction of the undisturbed velocity υ ∞
- κ=η′/η 0 :
-
viscosity of inside fluid relative to the solvent viscosity (5.8a)
- λ = η 0 υ ∞/Eh/Eh :
-
ratio of viscous fluid stresses and elastic membrane stresses
- \(\overline \lambda \) :
-
ratio of viscous fluid forces and surface tension forces
- ν :
-
Poisson ratio (2.8)
- ρ = Rer :
-
(scaled) distance from sphere center used in outer region (3.11)
- σ :
-
surface tension
- τ :
-
characteristic time of suspending fluid (τ′ for inner fluid)
References
Barthès-Biesel D (1980) Motion of a spherical microcapsule freely suspended in a linear shear flow. J Fluid Mech 100:831–853
Brunn PO (1982) The time dependent deformation of a capsule in an arbitrary Stokes flow. To be published.
Chien S, King RG, Skalak R, Usami S and Copley AL (1975) Viscoelastic properties of human blood and red cell suspensions. Biorheol 12:341–346
Taylor TD and Acrivos A (1964) On the deformation and drag of a falling viscous drop at low Reynolds number. J Fluid Mech 18:466–476
Ajayi OO (1975) Slow motion of a bubble in a viscoelastic fluid. J Eng Math 9:273–280
Hassager O (1977) Bubble motion in structurally complex fluids. In: Chemical Engineering with Per Søltoff (eds Østergaard K and Fredenslund AA). Teknik Forlag, Copenhagen, pp 105–117
Wagner MG and Slattery JC (1971) Slow flow of a non-Newtonian fluid past a droplet. AIChE J 17:1198–1207
Skalak R, Tözeren A, Zarda R and Chien S (1973) Strain energy function of red blood cell membranes. Biophys J 13:245–264
Kraus M (1967) Thin elastic shells. Wiley, New York
Bird RB, Armstrong RC and Hassager O (1977) Dynamics of Polymeric Liquids. Vols I, II. John Wiley and Sons, New York
Giesekus H (1963) Die simultane Translations — und Rotations — bewegung einer Kugel in einer elastoviskosen Flüssigkeit. Rheol Acta 3:51–71
Happel J and Brenner H (1973) Low Reynolds Number Hydro-Dynamics 2nd ed Noordhof International Publishing, Leyden, The Netherlands
Milne-Thomson LM (1968) Theoretical Hydrodynamics. Macmillan Company.
Proudman I and Pearson JRA (1957) Expansion at small Reynolds number for the flow past a sphere and a circular cylinder, J Fluid Mech 2:237–263
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Brunn, P.O., Gilbert, J. & Jiji, D. Slow flow of a non-Newtonian fluid past a micro-capsule. Applied Scientific Research 40, 253–270 (1983). https://doi.org/10.1007/BF00388018
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DOI: https://doi.org/10.1007/BF00388018