# An Analytical Solution for Unsteady Laminar Flow in Tubes with a Tapered Wall Thickness

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Flow through a circular tube with varying wall thickness and, consequently, varying wave propagation velocity. (Wall thickness and diameter are exaggerated.)

**Figure 2.**The wave speed profile assumed in Equation (11) results in a wave speed that approximates both the case where the wave speed varies linearly and the case where the tube’s wall thickness varies linearly, as long as the variation in wave speed is small relative to its absolute value. (This example was calculated for E = 200 GPa and K = 1 GPa).

**Figure 3.**Outlet pressure response to a unit step in inlet pressure, for a closed exit, showing effect of taper ratio.

**Figure 4.**Outlet flow response to a unit step in inlet pressure, for an open exit, showing effect of taper ratio.

**Figure 5.**Inlet flow response to a unit step in inlet pressure, for a closed exit, showing effect of taper ratio.

**Figure 6.**Outlet flow response to a unit step in inlet flow, for an open exit, showing effect of taper ratio.

**Figure 7.**Outlet pressure response to a unit step in inlet pressure for a closed exit, showing the effect of dissipation number (for $\frac{{c}_{B}}{{c}_{A}}=0.5$).

**Figure 8.**Outlet flow response to a unit step in inlet pressure for an open exit, showing the effect of dissipation number (for $\frac{{c}_{B}}{{c}_{A}}=0.5$).

**Figure 9.**Inlet flow response to a unit step in inlet pressure for a closed exit, showing the effect of dissipation number (for $\frac{{c}_{B}}{{c}_{A}}=0.5$).

**Figure 10.**Outlet flow response to a unit step in inlet flow for an open exit, showing the effect of dissipation number (for $\frac{{c}_{B}}{{c}_{A}}=0.5$).

**Figure 11.**Computation time required to calculate a sample solution, as the time resolution is varied. For the MOC case, ${N}_{x}$ corresponds to the number of spatial grid points. For the frequency-domain case, the frequency points were selected to give the same time resolution as the MOC case.

**Figure 12.**Normalized error for the same data as Figure 11.

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**MDPI and ACS Style**

Wiens, T.; Etminan, E.
An Analytical Solution for Unsteady Laminar Flow in Tubes with a Tapered Wall Thickness. *Fluids* **2021**, *6*, 170.
https://doi.org/10.3390/fluids6050170

**AMA Style**

Wiens T, Etminan E.
An Analytical Solution for Unsteady Laminar Flow in Tubes with a Tapered Wall Thickness. *Fluids*. 2021; 6(5):170.
https://doi.org/10.3390/fluids6050170

**Chicago/Turabian Style**

Wiens, Travis, and Elnaz Etminan.
2021. "An Analytical Solution for Unsteady Laminar Flow in Tubes with a Tapered Wall Thickness" *Fluids* 6, no. 5: 170.
https://doi.org/10.3390/fluids6050170