Estimating the effective viscosity of bubble suspensions in oscillatory shear flows by means of ultrasonic spinning rheometry

Abstract

We have proposed a novel methodology using ultrasonic velocity profiling to estimate the effective viscosity of bubble suspensions that are accompanied by non-equilibrium bubble deformations in periodic shear flows. The methodology was termed “ultrasonic spinning rheometry” and validated on measurement of the effective viscosity of particle suspensions that has a semi-empirical formula giving good estimation of the actual viscosity. The results indicated that the proposed technique is valid for particle volume fractions below 3.0 %. Applying this to bubble suspensions suggested that the effective value of temporal variations in the capillary number, $\hbox{Ca}_{\rm rms}$, is an important indicator to distinguish regimes in estimating the effective viscosity: Unsteady flows having larger $\hbox{Ca}_{\rm rms}$ number than the critical capillary number for the deformation of bubbles are categorized into Regime 2 that includes both highly unsteady conditions and large steady deformation of bubbles.

Appendix

Variable separation as $u_{\theta }(r, t) = F(t)G(r)$ modified Eq. (10) into two ordinal differential equations,

$$ F^{\prime } + \gamma F = 0, $$

(15)

$$ G^{{\prime \prime }} + \frac{1}{r}G^{\prime } + \left( {\beta ^{2} - \frac{1}{{r^{2} }}} \right)G = 0,\quad \beta ^{2} = \frac{\gamma }{\nu }, $$

(16)

with connecting parameter $\gamma$. By variable transformation of $\zeta = \beta r$, Eq. (16) can be assumed as Bessel differential equation with $n = 1$,

$$ \frac{{\hbox{d}}^2 G}{{\hbox{d}} \zeta ^2} + \frac{1}{\zeta } \frac{{\hbox{d}}G}{{\hbox{d}}\zeta } + \left( 1-\frac{n^2}{\zeta ^2} \right) G = 0.$$

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The general solution of this equation is given as a linear combination of Bessel function of first kind $J_n$ and second kind $Y_n$. To satisfy the boundary conditions, $u_{\theta }(r=R, t) = U\sin \omega t = {\rm Im}[U\exp (i\omega t)]$ and $u_{\theta }(r=0, t) = 0$, the solution should be

$$G(r) = C J_1(\beta r), \quad J_1(\beta r) = \sum _{m=0}^{\infty } \frac{(-1)^m}{m!(m+1)!} \left( \frac{\beta r}{2} \right) ^{2m+1},$$

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where $C$ is arbitrary constant. Assuming $F(t) = B\exp (i\omega t)$ as the solution of Eq. (15) with arbitrary constant $B$ gives $\gamma = -i\omega$. Then, $\beta$ is determined as

$$ \beta ^{2} = \frac{\gamma }{\nu } = - i\frac{\omega }{\nu },\quad \beta \pm ( - 1 + i)k,\quad k = \sqrt {\frac{\omega }{{2\nu }}} . $$

We adopt $\beta = (-1+i)k$ to represent the real phenomenon. Finally, $u_{\theta }(r, t)$ is given as the imaginary part of the combined solution between $F(t)$ and $G(r)$ as

$$\begin{aligned} u_{\theta }(r, t)&= \frac{U}{\varPhi _R^2 + \varPsi _R^2} \left[ (\varPhi \varPhi _R + \varPsi \varPsi _R)\sin \omega t \right. \nonumber \\&\quad \left. + (\varPhi _R \varPsi - \varPhi \varPsi _R) \cos \omega t \right] . \end{aligned}$$

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Here, the $\varPhi (r)$, $\varPsi (r)$, $\varPhi _R$, $\varPsi _R$ are, respectively, real and imaginary part of $J_1(r)$ and $J_1(r=R)$, and given in series as

$$ \varPhi (r) = \sum\limits_{{m = 0}}^{\infty } {\phi _{m} } (r),\quad \varPhi _{R} = \sum\limits_{{m = 0}}^{\infty } {\phi _{m} } (r = R), $$

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$$\varPsi (r) = \sum\limits_{{m = 0}}^{\infty } {\varPsi_{m} } (r),\;\;\varPsi $$

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where

$$\begin{aligned} \phi _m(r)&= \frac{2^m}{m!(m+1)!} \left( \frac{kr}{2} \right) ^{2m+1}f_m, \nonumber \\&\ \quad f_m = {\left\{ \begin{array}{ll} (-1)^{(m + 2)/2}&{}: m = \text{even}\,\text{number}\\ (-1)^{(m + 1)/2}&{}: m = \text{odd}\,\text{number}\\ \end{array}\right. }, \end{aligned}$$

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$$ \begin{aligned} \psi _{m} (r) & = \frac{{2^{m} }}{{m!(m + 1)!}}\left( {\frac{{kr}}{2}} \right)^{{2m + 1}} g_{m} , \\ g_{m} & {\text{ = }}\left\{ {\begin{array}{*{20}l} {( - 1)^{{m/2}} } \hfill & {:m = {\text{even}}\,{\text{umber}}} \hfill \\ {( - 1)^{{(m + 1)/2}} } \hfill & {:m = {\text{odd}}\,{\text{number}}} \hfill \\ \end{array} } \right. \\ \end{aligned} $$

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Phase delay of the oscillation propagating in the fluid layer, $\varTheta (r)$, in the form of

$$ u_{\theta }(r, t) = UA(r) \sin [\omega t + \varTheta (r)] $$

is given as

$$ \varTheta (r) = \tan ^{-1} \frac{\varPhi _R \varPsi (r) - \varPhi (r) \varPsi _R}{\varPhi (r) \varPhi _R + \varPsi (r) \varPsi _R}. $$

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