Slow flow of a non-Newtonian fluid past a micro-capsule. (English) Zbl 0545.76001
This paper examines the slow flow of a non-Newtonian fluid past a spherical microcapsule consisting of an incompressible fluid enclosed by a thin elastic membrane. Of particular interest are the effects of fluid inertia and viscoelasticity on the deformation of the membrane. Since the problem is highly nonlinear, the authors assume that the ratio of viscous to elastic forces is small and seek a regular perturbation solution for the case when the deformation from sphericity is small. They also assume that the Weissenberg (We) and Reynolds (Re) numbers are also small. Non- Newtonian effects are included by writing the stress deviator in the form \(\underset \tilde{} T+p{\underset \tilde{} \delta}=2\underset \tilde{} f+We \underset \tilde{} s\) and expanding \(\underset \tilde{} s\), for small We, up to the level for a second order fluid.
Since inertial effects are small at distances r from the microcapsule less than \(r_ c(\equiv Re^{-1})\) but dominate when \(r>r_ c\), the authors seek separate series expansion solutions, up to first order in Re, for the velocity and pressure fields in both regions. Employing the known ”inertia free” fields in \(r<r_ c\) in terms of associated Legendre polynomials together with the general axisymmetric solution of the classical Oseen equation in \(r>r_ c\), the coefficients in the expansions are calculated by means of limit matching. The deformation of the membrane is then obtained by assuming that its constitutive equations are those of a linear, isotropic elastic material.
Neglecting terms that are second order in Re and We, it is found that the sphere deforms into an oblate spheroid if \((We/Re)<0.57\) and into a prolate spheroid if \((We/Re)>0.81\). For intermediate values more information is required about the Poisson ratio of the membrane and a material parameter in the expansion of \(\underset \tilde{} s\). The drag force is calculated and the variation of deformation with microcapsule size is examined. A comparison of qualitative results with those pertaining to the behaviour of surface tension droplets is given and it is shown that viscoelasticity tends to deform the sphere into a prolate spheroid while fluid inertia tends to deform it into an oblate spheroid.
Since inertial effects are small at distances r from the microcapsule less than \(r_ c(\equiv Re^{-1})\) but dominate when \(r>r_ c\), the authors seek separate series expansion solutions, up to first order in Re, for the velocity and pressure fields in both regions. Employing the known ”inertia free” fields in \(r<r_ c\) in terms of associated Legendre polynomials together with the general axisymmetric solution of the classical Oseen equation in \(r>r_ c\), the coefficients in the expansions are calculated by means of limit matching. The deformation of the membrane is then obtained by assuming that its constitutive equations are those of a linear, isotropic elastic material.
Neglecting terms that are second order in Re and We, it is found that the sphere deforms into an oblate spheroid if \((We/Re)<0.57\) and into a prolate spheroid if \((We/Re)>0.81\). For intermediate values more information is required about the Poisson ratio of the membrane and a material parameter in the expansion of \(\underset \tilde{} s\). The drag force is calculated and the variation of deformation with microcapsule size is examined. A comparison of qualitative results with those pertaining to the behaviour of surface tension droplets is given and it is shown that viscoelasticity tends to deform the sphere into a prolate spheroid while fluid inertia tends to deform it into an oblate spheroid.
Reviewer: P.J.Barratt
