Transport control within a microtube

Investigation of the entrainment of gases induced by a surface wave propagating along the wall of a microtube is conducted by using a relaxed model with slip velocity boundary conditions. Flow patterns tuned by critical reflux values $a_0$, Knudsen numbers, Reynolds numbers, and the wave number are demonstrated. Results show that once the cross section of the microtube is narrowed down (slip velocity increasing) there are earlier backward flows and the flow pattern is much more complicated. Our results should be useful for the design of micro total analytical systems.

Figure 1

Schematic longitudinal diagram of the deformable motion of the microtube’s wall inside one cross-section with respect to the tube axis (the orientation is counted from, say, the $x$ axis which is normal to the $z$ axis). The tube is axisymmetric.

Figure 2

Dependence of the mean velocity profile $U\left(r\right)$ on $a_0$ for the wave number $\alpha =0.8$, the Reynolds number $\mathrm{Re}=50$, $\mathrm{Kn}=0$. $a_0$ increases from $-25$ to 15. $a_{0_{\mathit{cr}}}=7.3422$ Kn is the Knudsen number (see the text for the definition). $a_0$ is related to the second-order pressure gradient. All quantities shown here and in subsequent figures are dimensionless.

Figure 3

Dependence of the mean velocity profile $U\left(r\right)$ on $a_0$ for the wave number $\alpha =0.8$, the Reynolds number $\mathrm{Re}=50$, $\mathrm{Kn}=0.1$. $a_0$ increases from $-25$ to 15. $a_{0_{\mathit{cr}}}=5.2303$. As $a_0=a_{0_{\mathit{cr}}}$, $U(r=0)=0$, i.e., a backward flow starts to form.

Figure 4

(Color online) Three-dimensional view of the mean velocity profile $U\left(r\right)$ for $\mathrm{Re}=50$, $\alpha =0.8$, $\mathrm{Kn}=0$, $a_0=-22$.

Figure 5

(Color online) Three-dimensional view of the mean velocity profile $U\left(r\right)$ for $\mathrm{Re}=50$, $\alpha =0.8$, $\mathrm{Kn}=0$, $a_0=-28$.