Prediction of the shape and pressure drop of Taylor bubbles in circular tubes

Abstract

A model based on a combination of lubrication theory and capillary effects (ignoring inertial effects and gravity) is developed to predict the shape and pressure drop of long bubbles flowing in circular tubes in pressure-driven flows. An analytical solution for the thickness of the wetting film left on the tube wall as a function of the Capillary number (the ratio between viscous effects and surface tension) was derived by Klaseboer et al. (Phys Fluids 26:032107, 2014), which considerably extends the original result of Bretherton (J Fluid Mech 10:166–188, 1961) and confirms the empirical law of Aussillous and Quéré (Phys Fluids 12:2367–2371, 2000). It is based on a crucial condition that requires that the bubble must fit inside the tube. An extension of this formulation allows for an analytical expression of the pressure drop across the bubble by applying the tube fit condition for the front and the back of the bubble and a force balance. The complete shape of the bubble can then be obtained numerically by applying boundary conditions at the tube centre. The interesting physics occurring at the back of the bubble is also investigated. A theoretical condition for the minimal length of such a bubble is given. Comparisons with experimental and numerical data for the shape of the bubble, pressure drop and curvature at the front and rear of the bubble for small to intermediate Capillary numbers give excellent agreement.

Appendix: Numerical implementation of the governing equations

Numerically we solve Eq. 13 in non-dimensional form. Using the scalings of Eq. 12, the boundary conditions of Eq. 40 are written as

$$\begin{aligned} \left. \begin{array}{lll} \eta (0) &{}= R/b \\ \eta (\xi _{m}) &{}= R/b \\ \frac{d^2\eta (0)}{\hbox {d}\xi ^2}&{}= A_{\text {B}} \end{array}\right\} \end{aligned}$$

(47)

where $A_{\text {B}}$ is given by Eq. 39.

The domain size [$0,\xi _{m}$] is divided in n equally spaced points from $i = 1$ to n. Eq. 13 has a third-order derivative and is discretised as for $i=3$ to $n-2$.

$$\begin{aligned} \frac{\eta _{i+2}-2\eta _{i+1}+2\eta _{i-1}+\eta _{i-2}}{2\hbox {d}\xi ^3}= \frac{\eta _i-1}{\eta _{*i}^3} \end{aligned}$$

(48)

where we linearise the systems by taking $\eta _*$ from the previous iteration. This equation is solved for $i=3$ to $n-2$. For $i=1$ and $i=n$ we take $\eta _i=R/b$. Special consideration is needed for the points $i=2$ and $i=n-1$. For $i=n-1$ we take backward difference scheme. Discretisation of the boundary condition allows the calculation of $i=2$ as follows

$$\begin{aligned} \frac{\eta _{2}-2\eta _{1}+\eta _{0}}{\hbox {d}\xi ^2}=A_{\text {B}} \end{aligned}$$

(49)

which provide the missing $\eta _{0}$ point. For the initial condition, we take $\eta =R/b$ at the boundaries and $\eta =1$ for the internal points. We build the matrix and solve the system implicitly using MATLAB. After each iteration we update $\eta _*$ until convergence is achieved.