12.3.1. Drag on contaminated drop in Stokes flow (\(\Gamma \sim \Gamma_{eq}\))
Let us consider a contaminated spherical drop moving at constant speed in the Stokes regime. The problem is similar to that for the Hadamard-Rybczinski solution [Had11]. Even with the presence of surfactant, the functional forms of the external and internal velocities have the same form as we obtained for the Hadamard-Rybczinski solution, so that,
(12.36)\[\begin{split}\begin{split}
&\mathbf{v}^{(e)} = \mathbf{u} - \alpha \frac{\mathbf{u} + (\mathbf{u} \cdot \mathbf{e}_{r}) \mathbf{e}_{r}}{r} + \beta \frac{ 3 (\mathbf{u} \cdot \mathbf{e}_{r}) \mathbf{e}_{r} - \mathbf{u}}{r^{3}} \\
&\mathbf{v}^{(i)}
= - A \mathbf{u} + B r^{2} \left( ( \mathbf{u} \cdot \mathbf{e}_{r} ) \mathbf{e}_{r} - 2 \mathbf{u} \right)
\end{split}\end{split}\]
We need to determine the four constants so as to satisfy the boundary conditions:
\begin{itemize}
\item The velocity normal to the interface is continuous and is zero because the sphere is motionless: \(\left. v_{r}^{(e)} \right|_{r=a} = \left. v_{r}^{(i)} \right|_{r=a} = 0\)
\item The velocity tangent to the interface is continuous: \(\left. v_{\theta}^{(e)} \right|_{r=a} = \left. v_{\theta}^{(i)} \right|_{r=a}\)
\item the difference in the tangential viscous stresses balances with the Marangoni stress (the surface tension gradient): \(\left. \tau_{r \theta}^{(e)} \right|_{r=a} - \left. \tau_{r \theta}^{(i)} \right|_{r=a} = - \frac{1}{a} \frac{\partial \sigma}{\partial \theta}\)
\end{itemize}
Only the third one is different from the conditions for the analysis in the Hadamard-Rybczinski solution.
In a coordinate system moving with the drop \(\mathbf{v} \cdot \mathbf{n} = 0\) at the interface, and Eq. (12.28) therefore reduces to
(12.37)\[ \nabla_{s} \cdot \Gamma \mathbf{v}_{s} = \nabla_{s} \cdot D_{s} \nabla_{s} \Gamma + \dot{S}_{\Gamma}\]
We assume that the drop interface is covered by surfactant at a concentration close to the equilibrium value, i.e.,
(12.38)\[ \Gamma = \Gamma_{eq} + \Gamma'\]
where \(\Gamma' / \Gamma_{eq} \ll 1\). Substituting this condition into the transport equation to linearize the problem gives
(12.39)\[ \Gamma_{eq} \nabla_{s} \cdot \mathbf{v}_{s} + \nabla_{s} \cdot \Gamma' \mathbf{v}_{s} = D_{s} \nabla_{s}^{2} \Gamma' + \dot{S}_{\Gamma}\]
where \(D_{s}\) is assumed to be constant. The interface velocity is retarded by the surfactant effect. For small \(\mathbf{v}_{s}\), the convective transport \(\nabla_{s} \cdot \Gamma' \mathbf{v}_{s}\) is negligible, so that,
(12.40)\[ \Gamma_{eq} \nabla_{s} \cdot \mathbf{v}_{s} = D_{s} \nabla_{s}^{2} \Gamma' + \dot{S}_{\Gamma}\]
With the spherical coordinates,
(12.41)\[ \frac{\Gamma_{eq}}{a \sin \theta} \frac{\partial v_{\theta} \sin \theta}{\partial \theta} = \frac{D_{s}}{a \sin \theta} \frac{\partial}{\partial \theta} \left( \frac{\partial \Gamma'}{\partial \theta} \sin \theta \right) + \dot{S}_{\Gamma}\]
The velocity profile is assumed to have the form of \(v_{\theta} \propto \sin \theta\) as in the Hadamard-Rybczinski solution. Writing \(v_{0}\) as the velocity at \(\theta = \pi / 2\), \(v_{\theta} = v_{0} \sin \theta\). Therefore,
(12.42)\[ \frac{\Gamma_{eq}}{a \sin \theta} \frac{\partial v_{\theta} \sin \theta}{\partial \theta}
= \frac{\Gamma_{eq}}{a \sin \theta} \frac{\partial v_{0} \sin^{2} \theta}{\partial \theta}
= \frac{2 \Gamma_{eq} v_{0}}{a} \cos \theta \]
By expanding the adsorption flux
(12.43)\[ \dot{S}_{\Gamma} = Q(C_{0}, \Gamma) - P(\Gamma)\]
we obtain
(12.44)\[ \dot{S}_{\Gamma} = Q(C_{0}, \Gamma_{eq}) - P(\Gamma_{eq}) + \left( \frac{\partial Q}{\partial \Gamma} - \frac{\partial P}{\partial \Gamma} \right) \Gamma'\]
For the equilibrium, \(Q(C_{0}, \Gamma_{eq}) = P(\Gamma_{eq})\). Hence,
(12.45)\[ \dot{S}_{\Gamma} = - \lambda \Gamma'\]
where
(12.46)\[ \lambda = \frac{\partial Q}{\partial \Gamma} - \frac{\partial P}{\partial \Gamma}\]
and for Eq. (12.10)
(12.47)\[ \lambda = k_{a} C_{0} + k_{d}\]
Since \(\lambda\) is positive, surfactant molecules come from the bulk liquid to the interface when \(\Gamma' < 0\). The linearized transport equation is thus given by
(12.48)\[ \frac{2 \Gamma_{eq} v_{0}}{a} \cos \theta = \frac{D_{s}}{a \sin \theta} \frac{\partial}{\partial \theta} \left( \frac{\partial \Gamma'}{\partial \theta} \sin \theta \right) - \lambda \Gamma'\]
The functional form \(\Gamma' = A_{\Gamma} \cos \theta\) satisfies the transport equation when
(12.49)\[ A_{\Gamma} = - \frac{2 \Gamma_{eq} v_{0}}{a \lambda + 2 D_{s}}\]
The diffusion coefficient is usually small (\(\sim 10^{-9}\)~m\(^{2}\)/s in many cases) and the diffusive flux is smaller than the advection flux. The diffusion is therefore neglected in the following discussion, and therefore,
(12.50)\[ \Gamma' = - \frac{2 \Gamma_{eq} v_{0}}{a \lambda} \cos \theta\]
The Marangoni stress can now be calculated using the expression of \(\Gamma'\), i.e.,
(12.51)\[ - \frac{1}{a} \frac{\partial \sigma}{\partial \theta}
= - \frac{1}{a} \frac{\partial \Gamma}{\partial \theta} \frac{\partial \sigma}{\partial \Gamma}
= - \frac{1}{a} \frac{\partial \Gamma'}{\partial \theta} \frac{\partial \sigma}{\partial \Gamma}
= - \frac{2 \Gamma_{eq}}{a^{2} \lambda} \frac{\partial \sigma}{\partial \Gamma} v_{0} \sin \theta
= \frac{3 \gamma}{a} v_{0} \sin \theta \]
where
(12.52)\[ \gamma = - \frac{2 \Gamma_{eq}}{3 a \lambda} \frac{\partial \sigma}{\partial \Gamma}\]
According to the velocity profile
(12.53)\[ \left. v_{\theta}^{(i)} \right|_{r=a}
= u \sin \theta \left( - A - 2 B a^{2} \right) \]
we may write
(12.54)\[ v_{0} = u ( - A - 2 B a^{2} )\]
The boundary condition for the tangential stresses is thus given by
(12.55)\[ \left. \tau_{r \theta}^{(e)} \right|_{r=a} = \left. \tau_{r \theta}^{(i)} \right|_{r=a} - \frac{1}{a} \frac{\partial \sigma}{\partial \theta}
~~\rightarrow~~
\beta \frac{6 \mu^{(e)} u}{a^{4}} \sin \theta = - 3 B \mu^{(i)} a u \sin \theta + \frac{3 \gamma}{a} ( - A - 2 B a^{2} ) u \sin \theta\]
The set of the boundary conditions to determine the constants are
(12.56)\[\begin{split}\begin{split}
&A + a^{2} B = 0 \\
&- 2 a^{2} \alpha + 2 \beta + a^{3} = 0 \\
&a^{3} A + 2 a^{5} B - a^{2} \alpha - \beta + a^{3} = 0 \\
&\left\{ \mu^{(i)} + 2 \gamma \right\} a^{5} B + \gamma a^{3} A + 2 \mu^{(e)} \beta = 0
\end{split}\end{split}\]
Solving the simultaneous equations gives
(12.57)\[\begin{split}\begin{split}
&\alpha = \frac{a}{4} \frac{2 \mu^{(e)} + 3 \mu^{(i)} + 3 \gamma}{\mu^{(e)} + \mu^{(i)} + \gamma} \\
&\beta = \frac{a^{3}}{4} \frac{\mu^{(i)} + \gamma}{\mu^{(e)} + \mu^{(i)} + \gamma} \\
&A = \frac{1}{2} \frac{\mu^{(e)}}{\mu^{(e)} + \mu^{(i)} + \gamma} \\
&B = - \frac{1}{2 a^{2}} \frac{\mu^{(e)}}{\mu^{(e)} + \mu^{(i)} + \gamma}
\end{split}\end{split}\]
By substituting \(A\) and \(B\) into Eq. (12.53) we obtain
(12.58)\[ \left. v_{\theta}^{(e)} \right|_{r=a}
=
\left. v_{\theta}^{(i)} \right|_{r=a}
= \left( \frac{\mu^{(e)}}{\mu^{(e)} + \mu^{(i)} + \gamma} \right) \frac{u}{2} \sin \theta \]
This results shows that the surface tension gradient (\(\gamma\)) causes retardation of the interface motion and the interface becomes immobile when \(\gamma\) is large.
The drag force is given by
(12.59)\[ F_{D} = 8 \pi \mu^{(e)} u \alpha = \left( \frac{2 \mu^{(e)} + 3 \mu^{(i)} + 3 \gamma}{\mu^{(e)} + \mu^{(i)} + \gamma} \right) 2 \pi \mu^{(e)} u a\]
The drag coefficient is
(12.60)\[ C_{D} = \frac{8}{Re} \left( \frac{2 \mu^{(e)} + 3 \mu^{(i)} + 3 \gamma}{\mu^{(e)} + \mu^{(i)} + \gamma} \right)\]
\(\gamma\) is positive for \(\partial \sigma / \partial \Gamma < 1\). When
\(\gamma \gg \mu^{(e)} + \mu^{(i)}\), the Stokes solution recovers:
(12.61)\[ C_{D} \rightarrow \frac{24}{Re}\]
On the other hand, when \(\gamma \ll \mu^{(e)} + \mu^{(i)}\), the drag correlation becomes the Hadamard-Rybczinski solution
(12.62)\[ C_{D} \rightarrow \frac{8}{Re} \left( \frac{2 \mu^{(e)} + 3 \mu^{(i)}}{\mu^{(e)} + \mu^{(i)}} \right)\]