THE DEFORMATION AND DRAG OF A MICROCAPSULE DUE TO INERTIA AND VISCOELASTICITY
P. Brunn, James H. Gilbert, Danny Jiji
May 1, 1983
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Journal
Annals of the New York Academy of Sciences
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Key Takeaway Key Takeaway: I f X denote the rheology of the interior fluid, and denote the membrane stresses.
Abstract
Microcapsules are (small) particles whose fluid content is enclosed by a membrane and which are usually suspended in some kind of solvent. Examples include red blood cells and encapsulated droplets. Previously, such particles were treated hydrodynamically as droplets, droplets with slip, or purely elastic particles. Only recently has the (two-dimensional) continuum character of the membrane been recognized and incorporated into a hydrodynamic analysis. The results obtained so far look promising. Assuming that the solvent is a Newtonian fluid of infinite extent, one can easily show that a spherical capsule does not deform in pure streaming motion when the flow field is governed by the Stokes equations. This result, which is independent of the actual membrane rheology, implies that deformation is a nonlinear effect. To study this effect in a viscoelastic fluid using a model that includes fluid inertia, we assume that a sufficiently stiff membrane would ensure that all possible deformations would be small. The advantages are two-fold. First, the rheology of the interior fluid becomes immaterial. Second, by concentrating on purely elastic membranes (considered as the two-dimensional limit of a three-dimensional isotropic elastic solid), we can employ the concept of linear elasticity as a first approximation. The velocity fields are continuous and fluid across the (deformed) membrane, and, provided that we neglect any acceleration within the membrane, membrane stresses are i n equilibrium with the viscoelastic fluid forces. This is a highly nonlinear problem. Consequently, we concentrate on a perturbation solution, starting from the limit of a rigid sphere (surface So). The resulting load (i.e., the relative force per unit area exerted by the fluid on the membrane) generates membrane stresses. Relating these to the strains via the constitutive equation and solving the strain-displacement relationship furnishes the new shape of the capsule. With this information, the drag force can be evaluated and the next-order approximation for the flow field can be computed. I f X denotes the ratio of viscous forces to elastic (shape-restoring) membrane forces, Re the Reynolds number, and We the Weissenberg number, then, for We 5 Re < X < I , the shape of the deformed surface is spheroidal,