# Mechanics of Fluid-Conveying Microtubes: Coupled Buckling and Post-Buckling

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fluid-Structure Interaction Model of the Microtube

_{a}). As seen in the figure, a Cartesian coordinate frame with axes x and z is employed to describe the geometrical properties of the microsystem.

_{u}= N

_{w}= 10 (see Appendix A for the convergence analysis). To the best of our knowledge, the above Equations (i.e., Equations (8) and (9)) with this high degree of freedom have not been presented for the post-buckling of fluid-conveying microtubes. It is worth noticing that in Equations (8) and (9), the first term of each Equation incorporates the effects of the fluid flow on the coupled buckling and post-buckling of microtubes. A continuation method is employed to solve this high-dimensional discretised model; the Floquet theory for the stability analysis is used. The continuation method is a numerical technique for periodic vibration problems as well as nonlinear buckling analyses. Near the divergence state, there are multiple solutions for the nonlinear differential equations. The continuation method is employed since it is capable of obtaining both stable and unstable solutions. In this method, there are two main loops: (1) internal loop, and (2) external loop. It is worth mentioning that in the external loop, a predictor is applied to the system of Equations. For more information about this numerical technique, the reader is referred to chapter 4 (pages 169–197) of the book written by Seydel [48] about bifurcation analysis.

## 3. Results and Discussion

_{p}= 1220 kg/m

^{3}, and ρ

_{f}= 1000 kg/m

^{3}, respectively. Moreover, in this paper, the length scale parameter, the inner diameter, the outer diameter and the length-to-diameter ratio of the microtube are assumed as l = 17.6 µm, D

_{i}= 25 µm, D = 50 µm, L/D = 100, respectively. It should be noted that the small-scale parameter is a material constant obtained from the results of experimental measurements. In Appendix B, the experiment setup for obtaining the small-scale parameter is explained. For instance, for epoxy microtubes, it is 17.6 μm while a value of 53.7 μm is obtained for polypropylene microtubes [24,49].

^{5}. It is seen in Figure 2 that both transverse and axial displacements are zero until a certain dimensionless velocity of the flowing fluid (u

_{f}= 7.8845). At this critical point, a buckling instability occurs through a branch point bifurcation, causing lateral and longitudinal displacements to increase suddenly. After this point, there are two possible stable solutions (solid line) and one unstable solution (dashed line) for the transverse displacement. For the case of the axial displacement, there is only one stable solution, as well as one unstable solution after the occurrence of buckling.

^{5}and S = 100. It is assumed that the microscale tube containing flowing fluid is not embedded in an elastic foundation, i.e. both the linear and nonlinear spring coefficients are set to zero $({K}_{1}={K}_{2}=0)$. As seen, the critical fluid velocity related to the buckling of the microsystem increases with increasing the axial pretension. In other words, the presence of a pretension strengthens the microtube conveying fluid against buckling. Moreover, it is found that the lateral deflection of the system decreases when larger axial pretensions are applied. Finally, to show the accuracy and reliability of the present results, the bifurcation diagrams of a clamped-clamped pipe conveying fluid obtained by the present model are compared with those obtained in [57] (see Figure 9). An excellent agreement is found between the present results and those previously published in the literature.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Convergence Analysis

## Appendix B. Experiment Setup for Obtaining the Small-Scale Parameter

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**Figure 1.**(

**a**) A fluid-conveying microtube resting on a nonlinear elastic foundation; (

**b**) the free-body diagram of an ultrasmall element of the microtube; (

**c**) the free-body diagram of an element of the fluid.

**Figure 2.**Bifurcation diagrams of the microtube conveying fluid; (

**a**) the transverse displacement at x

_{d}= 0.5; (

**b**) the longitudinal displacement at x

_{d}= 0.125.

**Figure 3.**Comparison of the bifurcation diagrams of the microtube obtained via the classical ($\overline{\mu}$ = 0.0) and MCS ($\overline{\mu}$ = 0.5746) theories: (

**a**) the transverse displacement at x

_{d}= 0.5; (

**b**) the longitudinal displacement at x

_{d}= 0.125.

**Figure 4.**Bifurcation diagrams of the microtube showing the effect of the linear spring coefficient on the transverse displacement at the centre.

**Figure 5.**Bifurcation diagrams of the microtube showing the effect of the nonlinear spring coefficient on the transverse displacement at the centre.

**Figure 6.**Bifurcation diagrams of the microtube showing the effect of the slenderness ratio on the transverse displacement at centre; U is the dimensional flow velocity.

**Figure 8.**Bifurcation diagrams of the microtube showing the effect of the axial pretension on the transverse displacement at centre.

**Figure 9.**Bifurcation diagrams of a clamped-clamped pipe conveying fluid, showing the centre transverse displacement; solid and dashed lines show the result of the present study while symbols show the results obtained in [57].

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**MDPI and ACS Style**

Farajpour, A.; Farokhi, H.; Ghayesh, M.H.
Mechanics of Fluid-Conveying Microtubes: Coupled Buckling and Post-Buckling. *Vibration* **2019**, *2*, 102-115.
https://doi.org/10.3390/vibration2010007

**AMA Style**

Farajpour A, Farokhi H, Ghayesh MH.
Mechanics of Fluid-Conveying Microtubes: Coupled Buckling and Post-Buckling. *Vibration*. 2019; 2(1):102-115.
https://doi.org/10.3390/vibration2010007

**Chicago/Turabian Style**

Farajpour, Ali, Hamed Farokhi, and Mergen H. Ghayesh.
2019. "Mechanics of Fluid-Conveying Microtubes: Coupled Buckling and Post-Buckling" *Vibration* 2, no. 1: 102-115.
https://doi.org/10.3390/vibration2010007