A double emulsion droplet in a two-directional shear flow

This theoretical report explores a double emulsion droplet in a two-directional shear creeping flow. The droplet consists of a spherical inner drop (fluid 3) enclosed by a spherical shell (fluid 2), all immersed in external fluid 1. Six parameters govern: the capillary number ( Ca ), two viscosity ratios ( λ 21 , λ 31 ), the radii ratio ( κ ), the surface tensions ratio ( Ω ) and the flow-type ( G ). Using Lamb’s solution, an analytical solution for the spherical case ( Ca = 0, Ω = ∞) was found. We explored small deformations with: λ 21 = O (1), λ 31 = O (1), G = O (1), italic Ca much less than 1 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Ca</mml:mtext> </mml:mrow> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:math> $\textit{Ca} \ll 1$ and italic Ca slash normal upper Omega much less than 1 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Ca</mml:mtext> </mml:mrow> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:math> $ \textit{Ca}/\Omega \ll 1$ . If upper G greater than negative 1 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>G</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:math> $ G \gt -1$ , the outer surface is oriented at phi <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>ϕ</mml:mi> </mml:math> $\phi$ = +45°, while the inner surface at phi <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>ϕ</mml:mi> </mml:math> $\phi$ = −45°. The reverse occurs for upper G less than negative 1 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>G</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:math> $ G \lt -1$ . If G = −1, no deformation is possible. For a highly viscous shell ( lamda 21 much greater than 1 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mn>21</mml:mn> </mml:mrow>