Motion and shape of an axisymmetric viscoplastic drop slowly falling through a viscous fluid

Abstract

Slow sedimentation of a deformable drop of Bingham fluid in an unbounded Newtonian medium is studied using a variation of the integral equation method (Toose et al., J Eng Math 30:131–150, 1996, Int J Numer Methods Fluids 30:653–674, 1999). The Green function for the Stokes equation is used, and the non-Newtonian stress is treated as a source term. The computations are performed for a range of physical parameters of the system. It is demonstrated that initially deformed drop similar to Newtonian ones breaks up for high capillary number, Ca, and stabilizes to steady shapes at low Ca. Estimations of critical capillary number for specific initial deformations demonstrated its growth (increase in the stability of the drop) with the yield stress magnitude both for prolate and oblate initial shapes. Prolate initial shapes become more stable with the increase of the plastic viscosity. In contrast to this, for low yield stress, oblate shapes are destabilized with the growth of the plastic viscosity. This effect is similar to the effect of the viscosity of a Newtonian drop on its stability. However, at higher yield stress, the effect of plastic viscosity is reversed.

Appendix

The components of the kernels W and M in Eq. 12 are given by the following expressions.

$$ W_z^{zz} =r_y \Delta z\left( {3\Delta z^2E_{50} -E_{30} } \right), $$

$$ W_z^{zr} =r_y \left( {r_y E_{30} -r_x E_{31} -3\Delta z^2r_y E_{50} +3\Delta z^2r_x E_{51} } \right), $$

$$ W_z^{rz} =-r_y \left( {r_y E_{30} -r_x E_{31} +3\Delta z^2r_y E_{50} -3\Delta z^2r_x E_{51} } \right), $$

$$ W_z^{rr} =-r_y \Delta z\left( {E_{30} -3r_y^2 E_{50} +6r_x r_y E_{51} -3r_x^2 E_{52} } \right), $$

$$ W_r^{zz} =-r_y \left( {r_x E_{30} -r_y E_{31} -3\Delta z^2r_x E_{50} +3\Delta z^2r_y E_{51} } \right), $$

$$ \begin{array}{lll} W_r^{zr} &=&r_y \Delta z\left( {E_{31} -3r_x r_y E_{50} +3r_x ^2E_{51} +3r_y ^2E_{51}}\right. \\&&{\kern28pt}\left.{-\ 3r_x r_y E_{52} {\vphantom{3r_y ^2E_{51}}}} \right), \end{array} $$

$$ \begin{array}{lll} W_r^{rz} &=&-r_y \Delta z\left( {E_{31} +3r_x r_y E_{50} -3r_x ^2E_{51} -3r_y ^2E_{51}}\right.\\&&{\kern36pt}\left.{+\ 3r_x r_y E_{52} {\vphantom{-3r_y ^2E_{51}}}} \right), \end{array} $$

$$ \begin{array}{lll} W_r ^{rr}&=&-r_y \left( {r_x E_{30} -r_y E_{31} -3r_x r_y^2 E_{50} +3r_y^3 E_{51}}\right.\\ &&{\kern24pt}\left.{+\ 6r_y r_x^2 E_{51} -3r_x^3 E_{52}}\right. \\&&{\kern24pt}\left.{-\ 6r_x r_y^2 E_{52} +3r_x^2 r_y E_{53} } \right) \end{array} $$

$$ W_z^{\varphi \varphi } =-r_y \Delta z\left( {E_{30} -3r_x ^2E_{50} +3r_x ^2E_{52} } \right), $$

$$ \begin{array}{lll} W_r ^{\varphi \varphi }&=&-r_y \left( {r_x E_{30} -r_y E_{31} -3r_x^3 E_{50} +3r_y r_x^2 E_{51}}\right.\\ &&{\kern20pt}\left.{+\ 3r_x^3 E_{52} -3r_x^2 r_y E_{53} } \right), \end{array} $$

$$ M_z^z =r_y \left( {E_{10} +\Delta z^2E_{30} } \right), $$

$$ M_r^z =r_y \Delta z\left( {r_y E_{31} -r_x E_{30} } \right), $$

$$ M_z^r =r_y \Delta z\left( {r_y E_{30} -r_x E_{31} } \right), $$

$$ M_r^r =r_y \left( {E_{11} +r_y^2 E_{31} +r_x^2 E_{31} -r_x r_y E_{30} -r_x r_y E_{32} } \right), $$

Here Δz = z _x  − z _y ,

$$ \begin{array}{rll} E_{mn} \left( {\Delta z,r_x ,r_y } \right)&=&\frac{4k^m}{\left( {4r_x r_y } \right)^{m/2}}\int\limits_0^{\pi /2} {\frac{\left( {2\cos ^2\omega -1} \right)^n}{\left( {1-k^2\cos ^2\omega } \right)^{m/2}}} d\omega ,\\ k^2&=&\frac{4r_x r_y }{\Delta z^2+\left( {r_x +r_y } \right)^2}. \end{array} $$