Bubble dynamics in a microfluidic chamber under low-frequency actuation

Abstract

This paper presents a study of bubble dynamics in a resonator-shaped microfluidic chamber, including the bubble generation, radial oscillation and translation. The microfluidic chamber is incorporated with a piezoelectric actuator as part of the chamber wall, and DI-water is pumped throughout the chamber. Our experiments show that under the actuation at 1 kHz, the bubbles are generated, starting near the center of the chamber, growing up in size and even moving upstream against the main flow. Such type of bubble generation is different from those conventional bubble generations in microchannels by introducing external gas or laser radiation, and the bubble dynamics presented in this study is different from existing studies focusing on high-frequency actuations. To interpret our experimental findings, both numerical simulations and analytical modeling are conducted for studying the bubble dynamics in the low-frequency actuation case. The results show that the bubble generation and oscillations are due to the high-amplitude pressure fluctuations inside the chamber, corresponding to the actuations. Specifically, under the low-frequency actuation, the primary Bjerknes force becomes insignificant and the bubble translation against the main flow is attributed to the chamber sidewall attracting effect on the bubbles.

Appendices

Appendix 1: numerical simulation using Ansys Fluent

A three-dimensional numerical simulation has been conducted by using the commercial software Ansys Fluent, to study pressure distribution in the microfluidic chamber. The computation domain for the numerical simulation is identical to that used in experiments, and the dimensions and meshes are shown in Fig. 13. There are 25 layers of meshes along the channel height, and the meshing is specifically refined inside the chamber and at the Y junction. Under the assumptions of incompressible, Newtonian fluid, the governing equations of continuity and momentum are expressed as

$$\nabla \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V} = 0$$

(8a)

$$\frac{{\partial \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V} }}{\partial t} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V} \cdot \nabla \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V} = - \frac{\nabla p}{\rho } + \upsilon \nabla^{2} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V}$$

(8b)

where $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V}$ is the velocity vector, p the pressure, ρ the density and υ the kinetic viscosity.

Fig. 13

a Geometry and dimensions of the numerical simulation domain. b Meshing scheme in x–y plane (not to scale). c Meshing scheme in the height direction (not to scale)

The bottom wall with PZT disk is set as a moving boundary, and the top wall is stationary. The vibration of the PZT disk is described by the displacement ζ(x, y, t) (Timshenko and Krieger 1959)

$$\zeta (x,y,t) = \left\{ {\begin{array}{*{20}l} {A_{0} \sin (2\pi ft)\left( {1 - \left( {\frac{{\sqrt {x^{2} + y^{2} } }}{{R_{\text{chamb}} }}} \right)^{2} } \right)^{2} ,} &\quad {{\text{within}}\;{\text{disk}}} \\ {0,} &\quad {{\text{outside}}\;{\text{disk}}} \\ \end{array} } \right.$$

(9)

where the amplitude of vibration A ₀ is 0.22 μm, the frequency imposed to the PZT disk f is 1.0 kHz, and R _chamb is radius of the chamber, which is 8 mm in our experiment. All the boundary conditions are listed in Table 1. Based on a mesh independence study, it is found that the meshing scheme with 877,296 cells in total is sufficient for our simulations.

Table 1 Boundary conditions used in the numerical simulations

Appendix 2: the analytical model and detailed calculations

The model for a bubble between two intersected walls is illustrated in Fig. 14a, together with the coordinate system. The image system which satisfies the boundary condition at the rigid walls is illustrated in Fig. 14b. There are 2n − 1 image bubbles in total, where n = 180/2α (n is an integer). R and x denote the time-dependent radius and displacement, respectively, the variables with dots corresponding to the time derivatives, and d _i is the distance between the real bubble and the ith image bubble. Local coordinates, (r, θ) and (r _i , θ _i ), are located at the center of the real bubble and ith image bubbles, respectively. The velocity potential in response to the actuation in the model is defined as

$$\Phi = \dot{R}\Phi^{r} + \dot{x}\Phi^{x}$$

(10)

where R and x with dots are the corresponding time derivatives for the bubble’s radius and displacement, respectively, Φ^r is the unit potential for bubble radial motion and Φ^x is the unit potential for bubble translational motion.

Fig. 14

The model and image system for a bubble located nearby two inclined walls

The velocity potential from the bubble radial pulsation of unit strength is obtained by the superposition of contributions from all bubbles, including the real one and its images, i.e.

$$\Phi^{r} = \phi^{r} + \sum\limits_{i = 1}^{2n - 1} {\phi_{i}^{r} } = - \frac{{R^{2} }}{r} + \sum\limits_{i = 1}^{2n - 1} {\left( { - \frac{{R^{2} }}{{r_{i} }}} \right)}$$

(11)

where $\phi^{r}$ is the velocity potential corresponding to the real bubble, and $\phi_{i}^{r}$ is the velocity potential responding to the ith image bubble. The velocity potential from the bubble translation of unit strength is obtained by the superposition of contributions from all bubbles, including the real one and images, i.e.

$$\Phi^{x} = \phi^{x} + \sum\limits_{i = 1}^{2n - 1} {\phi_{i}^{x} } = - \frac{{R^{3} \cos \xi }}{{2r^{2} }} + \sum\limits_{i = 1}^{2n - 1} {\left( { - \frac{{R^{3} \cos \xi_{i} }}{{2r_{i}^{2} }}} \right)}$$

(12)

where $\phi^{x}$ is the velocity potential for the real bubble, and $\phi_{i}^{x}$ is the velocity potential for the ith image. ξ and ξ _i , are the coordinate angles denoted in Fig. 14b. The kinetic energy T, associated with the motions (including pulsation and translation) of all bubbles, is written as

$$T = \frac{\rho }{2}\left[ {\dot{R}^{2} \iint_{b} {\Phi^{r} {\text{d}}S + 2\dot{R}\dot{x}}\iint_{b} {\Phi^{x} {\text{d}}S + \dot{x}^{2} }\iint_{b} {\Phi^{x} \cos \xi {\text{d}}S}} \right]$$

(13)

The first integral in Eq. 13 can be calculated by substituting the potential Eq. 11 into the integral and evaluating it over the real bubble surface. The integration, denoted as I ₁, gives

$$\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}\nolimits_{b} {\Phi^{r} {\text{d}}S} = 4\pi R^{3} + \frac{{4\pi R^{4} }}{{D_{1} }} = I_{1}$$

(14)

where $\frac{1}{{D_{1} }} = \sum\nolimits_{i = 1}^{n - 1} {\frac{2}{{d_{i} }}} + \frac{1}{{d_{n} }}$.

By substituting the velocity potential Eq. 12 into the second and third integrals in Eq. 13, the two integrals, denoted as I ₂ and I ₃, can then be obtained by evaluating integrals over the real bubble surface,

$$\iint_{b} {\Phi^{x} {\text{d}}S} = - \frac{{2\pi R^{5} }}{{D_{2} }} = I_{2}$$

(15)

$$\iint_{b} {\Phi^{x} \cos \xi {\text{d}}S} = \frac{2}{3}\pi R^{3} + \frac{{2\pi R^{6} }}{{D_{3} }} = I_{3}$$

(16)

In Eq. 15 and 16, $\frac{1}{{D_{2} }} = \sum\nolimits_{i = 1}^{n - 1} {\frac{2\sin i\alpha }{{d_{i}^{2} }} + } \frac{1}{{d_{n}^{2} }}$ and $\frac{1}{{D_{3} }} = \sum\nolimits_{i = 1}^{n - 1} {\frac{{\sin^{2} i\alpha + 1}}{{d_{i}^{3} }}} + \frac{1}{{d_{n}^{3} }}$.

The kinetic energy T can therefore be given as

$$T = \frac{\rho }{2}[\dot{R}^{2} I_{1} + 2\dot{R}\dot{x}I_{2} + \dot{x}^{2} I_{3} ]$$

(17)

The potential energy U of the liquid is calculated by

$$U = - P_{1} V_{\text{b}} - xF_{\text{ex}}$$

(18)

where V _b is the time-dependent volume of the bubble, P ₁ is the pressure at the surface of the real bubble and the drag force on the bubble, F _ex, is calculated by $F_{\text{ex}} = - 12\pi \mu_{\text{eff}} R\dot{x}$ (Doinikov 2002). The Lagrangian L can be obtained by the kinetic energy T and potential energy U,

$$L = T - U = \frac{\rho }{2}\left[ {\dot{R}^{2} I_{1} + 2\dot{R}\dot{x}I_{2} + \dot{x}^{2} I_{3} } \right] + P_{1} V_{\text{b}} + xF_{\text{ex}}$$

(19)

By substituting the Eq. 19 into the Lagrangian equation

$$\frac{d}{{{\text{d}}t}}\left( {\frac{\partial L}{{\partial \dot{q}_{i} }}} \right) - \frac{\partial L}{{\partial q{}_{i}}} = 0$$

(20)

and carrying out the differentiations with respect to the generalized coordinates R and x, the governing equations Eqs. 7a and 7b for the bubble translation are derived. In the calculations, the scattered pressure at the surface of the real bubble P ₁ is defined by $P_{1} = P_{\text{g}} + P_{\text{v}} - \frac{2\sigma }{R} - \frac{{4\mu_{\text{eff}} \dot{R}}}{R} - P_{\infty }$, where P _g is the partial pressure of gas inside the bubble, P _v the vapor pressure, σ the surface tension, and μ _eff the effect viscosity. The pressure actuations applied at infinity is, $P_{\infty } = P_{0} - P_{\text{a}} \sin (2\pi ft)$, where P ₀ is the ambient pressure, P _a is the pressure fluctuation amplitude, and f is the actuation frequency. P _a is set as 0.93 P ₀ (Doinikov 2002). The effective viscosity μ _eff consists of the thermal damping μ _t and fluid viscous damping μ (Prosperetti 1977). All the parameters in the model are listed in Table 2.

Table 2 The parameters used in the analytical model