Stability of the flow in a soft tube deformed due to an applied pressure gradient

A linear stability analysis is carried out for the flow through a tube with a soft wall in order to resolve the discrepancy of a factor of 10 for the transition Reynolds number between theoretical predictions in a cylindrical tube and the experiments of Verma and Kumaran [J. Fluid Mech. 705, 322 (2012)]. Here the effect of tube deformation (due to the applied pressure difference) on the mean velocity profile and pressure gradient is incorporated in the stability analysis. The tube geometry and dimensions are reconstructed from experimental images, where it is found that there is an expansion and then a contraction of the tube in the streamwise direction. The mean velocity profiles at different downstream locations and the pressure gradient, determined using computational fluid dynamics, are found to be substantially modified by the tube deformation. The velocity profiles are then used in a linear stability analysis, where the growth rates of perturbations are calculated for the flow through a tube with the wall modeled as a neo-Hookean elastic solid. The linear stability analysis is carried out for the mean velocity profiles at different downstream locations using the parallel flow approximation. The analysis indicates that the flow first becomes unstable in the downstream converging section of the tube where the flow profile is more pluglike when compared to the parabolic flow in a cylindrical tube. The flow is stable in the upstream diverging section where the deformation is maximum. The prediction for the transition Reynolds number is in good agreement with experiments, indicating that the downstream tube convergence and the consequent modification in the mean velocity profile and pressure gradient could reduce the transition Reynolds number by an order of magnitude.

Figure 1

Undeformed tube of diameter $1200\mu \mathrm{m}$ (a), schematic of undeformed tube (b) and deformed tube (c), and close-up views of deformed tube where the soft section is made of PDMS with shear modulus 18 kPa, at a Reynolds number of 950 (d). The locations are $L_1=2$ mm upstream, and $L_2=1$ cm, $L_3=3.5$ cm, $L_4=6$ cm, and $L_5=9.5$ cm downstream of the entrance to the soft section. The scale bar in the figures is 1 mm in length. Panels (a) and (d) are reproduced from Verma and Kumaran [3].

Figure 2

The side view (a) and the magnified side view at different streamwise locations (b) and the cross-sectional view (c) at different streamwise locations of the tube reconstructed from the experimental images in Fig. 1 with soft walls made with shear modulus 18 kPa at Reynolds number 950. The locations are $L_1=2$ mm upstream and $L_2=1$ cm, $L_3=3.5$ cm, $L_4=6$ cm, and $L_5=9.5$ cm downstream of the entrance to the soft section. The scale bar in the figures is 1 mm.

Figure 3

Schematic diagram, not to scale, showing the coordinate system, the fluid and gel, local tube diameter $A$, and the outer diameter $HA$ for the soft wall. A schematic of the parabolic profile at the entrance to the test section is shown on the left, and the qualitative variation in the velocity in the diverging and converging sections, plotted later in Fig. 4, are also shown.

Figure 4

The mean velocity ${\overline{v}}_z$ (m/s) as a function of the radius $r$ $\left(\mu \mathrm{m}\right)$ (a) and the scaled mean velocity $[{\overline{v}}_z/(\pi Q/A^2)]$ as a function of the scaled radius $(r/A)$ (b) for the flow in a tube of undeformed diameter $1200\mu \mathrm{m}$ in which the developing section is fabricated with shear modulus 18 kPa at a Reynolds number (based on mean diameter) of 1025 at distances $x=0$ cm $(\circ )$, 1 cm $\left(▵\right)$, 3.5 cm $\left(\nabla \right)$, 6 cm $(◃)$, 8.8 cm $(▹)$, and at 9 cm $(\diamond )$ from the joint between the developing and test sections. Here $A$ is the deformed radius at the location $z$.

Figure 5

The pressure as a function of axial distance $z$ for a tube in which the developing section is made of PDMS gel with shear modulus 18 kPa for Reynolds numbers 126 $(\circ )$, 521 $\left(▵\right)$, 716 $\left(\nabla \right)$, 888 $(◃)$, 1025 $(▹)$, and 1240 $(\diamond )$. The joint between the developing and test sections is at $z=0$, and the vertical line shows the location of the pressure transducer.

Figure 6

The pressure difference between the pressure transducer and the tube outlet for tubes in which the developing section is made of soft gel with shear modulus 18 kPa (a), shear modulus 25 kPa (b), and shear modulus 35 kPa (c), and a hard tube with shear modulus 0.5 MPa (d). The dashed lines show the experimental results, while the results from flow simulations are shown by the solid lines.

Figure 7

The imaginary part for the wave speed $c_i$ for the mode with the largest $c_i$ as a function of the wave number $k$ for the flow through a tube with wall made of gel with shear modulus 18 kPa at Reynolds numbers 521 (a), 716 (b), and 888 (c) at different downstream locations $x$ from the inlet of the soft section, $x=0(\circ ),x=1$ cm, $\left(▵\right),x=3.5$ cm $\left(\nabla \right)$, $x=6$ cm $(◃)$, and $x=8.8$ cm $(▹)$ and at the outlet $x=9.5$ cm $(\diamond )$.

Figure 8

The maximum of the imaginary part of the wave speed $c_i$ (a) and the scaled wave number of the most unstable mode $k_{\max }$ (b) as a function of downstream distance $x$ from the entrance to the soft section for gels in which the soft wall is made with shear modulus 18 kPa for Reynolds numbers 521 $(\circ )$, 716 $\left(▵\right)$, 888 $\left(\nabla \right)$, 1025 $(◃)$, and 1139 $(▹)$.

Figure 9

The maximum of the imaginary part of the wave speed $c_i$ (a) and the scaled wave number of the most unstable mode $k_{\max }$ (b) as a function of downstream distance $x$ from the entrance to the soft section for gels in which the soft wall is made with shear modulus 25 kPa for Reynolds numbers 978 $(\circ )$, 1077 $\left(▵\right)$, 1217 $\left(\nabla \right)$, 1414 $(◃)$, and 1582 $(▹)$.

Figure 10

The maximum of the imaginary part of the wave speed $c_i$ (a) and the scaled wave number of the most unstable mode $k_{\max }$ (b) as a function of downstream distance $x$ from the entrance to the soft section for gels in which the soft wall is made with shear modulus 35 kPa for Reynolds numbers 1182 $(\circ )$, 1305 $\left(▵\right)$, 1439 $\left(\nabla \right)$, 1533 $(◃)$, and 1679 $(▹)$.

Figure 11

The slope of the wall as a function of downstream distance $z$ for tubes in which the soft wall is made with shear modulus 35 kPa for an average Reynolds number of 521 $(\circ )$, 716 $\left(▵\right)$, 888 $\left(\nabla \right)$, 1025 $(◃)$, and 1139 $(▹)$.

Figure 12

The scaled streamwise fluid velocity fluctuations ${\Phi }_v={\tilde{v}}_z/{\left.{\tilde{v}}_z\right|}_{r=A}$ (a) and the scaled streamwise displacement fluctuations in the solid ${\Phi }_u={\tilde{w}}_Z/{\left.{\tilde{w}}_Z\right|}_{r=A}$ (b) as a function of $r/A$ at two downstream locations 8.8 cm from the entrance of the soft section (solid line) and 9.5 cm from the entrance of the soft section (dashed line) for tube with soft section made of gel with shear modulus 18 kPa at $\mathrm{Re}=888$ $(\circ )$, tube with soft section made of gel with shear modulus 25 kPa and $\mathrm{Re}=1217$ $\left(▵\right)$, and for a tube with soft section made of gel with shear modulus 35 kPa and $\mathrm{Re}=1419$ $\left(\nabla \right)$.

Figure 13

The velocity gradient at the wall from the simulations for the base flow (solid symbols) and the velocity gradient for a parabolic flow with the same flow rate (open symbols) as a function of downstream distance $z$ for tubes in which the soft wall is made with shear modulus 35 kPa for an average Reynolds number of 521 $(\circ )$, 716 $\left(▵\right)$, 888 $\left(\nabla \right)$, 1025 $(◃)$, and 1139 $(▹)$. The lines are drawn to provide visual guidance.

Figure 14

The actual gradient from the simulations for the base flow (solid lines) and the pressure difference between the entrance of the soft section $(z=0$ cm) and the outlet $(z=14.5$ cm) divided by the tube length (9.5 cm) as a function of downstream distance $z$ (dashed lines) for tubes in which the soft wall is made with shear modulus 35 kPa for an average Reynolds number of 521 $(\circ )$, 716 $\left(▵\right)$, 888 $\left(\nabla \right)$, 1025 $(◃)$, 1139 $(▹)$.