## Shape Optimization of Peristaltic Pumping

## Introduction

Peristalsis is present in the esophagus, intestines, and other parts of the human body [5, 1, 2] and is able to move material through the body by sending traveling waves of contraction along the esophageal and intestinal walls. Peristaltic pumping falls into the category of a positive displacement pump because the channel/tube walls deform in order to move the interior fluid. Peristalsis is common in other biological contexts, such as blood flow in small vessels [11] and the transport of urine from the kidney to the bladder.

In industry, peristalsis is used in mechanical pumps to move very viscous or non-newtonian fluids through flexible deformable tubes. One common method is to have ‘rollers’ move over and locally compress the tube to induce a net motion of the fluid. Knowing the optimal shape of the deformation would be useful for designing the mechanical parts of the pump, like the rollers, so as to take advantage of efficient pumping profiles. See my paper [15] for more details.

## Shape Optimization

We developed a variational method for optimizing peristaltic pumping in a two dimensional periodic channel whose upper and lower walls move to pump fluid. No prior assumption is made on the wall motion, except that it is a traveling wave in shape. Hence, we consider an infinite dimensional optimization problem. To the best of our knowledge, optimizing the general wave shape in peristaltic pumping has never been done. Our optimization algorithm consists of finding the shape of the traveling wave such that the input fluid power is minimized subject to a given amount of mass flux and given channel volume.

Most wave-forms assumed in the literature have relatively mild displacements [6, 12, 3, 13, 4, 8] or are sinusoidal [9, 10, 7, 14]. Moreover, our optimization method can be generalized to include more complicated fluid models, such as generalized Newtonian and viscoelastic flow models.

## Fluid Problem

The problem we solve is to find the shape of the top and bottom walls (see Figure 1) such that the input fluid power is minimized subject to constraints on the mass flux and the enclosed area (2-D volume). See my paper [15] for more details.

## Optimized Shapes

Here are some movies showing the shape evolution (from our optimization method) that eventually lead to optimal shapes. Each frame of the movie shows the current shape with the steady-state flow field illustrated by streamlines. Everything is plotted with respect to the wave frame of the traveling wave. Periodic boundary conditions are imposed on the left and right ends of the channel. The Reynolds number is **Re = 500**.

--- Asymmetry Allowed

In these movies, both top and bottom walls are independent but each moves with the same wave speed.

In these movies, the bottom wall corresponds to a line of symmetry. Only the top-half of the domain is shown.

## References

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[3] P. Hariharan, V. Seshadri, and R. K. Banerjee. Peristaltic transport of non-newtonian fluid in a diverging tube with different wave forms. Mathematical and Computer Modelling, 48(7-8):998 – 1017, 2008.

[4] M. H. Haroun. Effect of wall compliance on peristaltic transport of a newtonian fluid in an asymmetric channel. Mathematical Problems in Engineering, 2006:19, 2006.

[5] F. Kiil. The Function of the Ureter and Renal Pelvis: pressure recordings and radiographic studies of the normal and diseased upper urinary tract of man. W. B. Saunders Co., Philadelphia, 1957.

[6] N. Liron. On peristaltic flow and its efficiency. Bulletin of Mathematical Biology, 38(6):573 – 596, 1976.

[7] M. Mishra and A. R. Rao. Peristaltic transport of a newtonian fluid in an asymmetric channel. Zeitschrift Angewandte Mathematik und Physik, 54:532–550, 2003.

[8] J. C. Misra and S. K. Pandey. Peristaltic transport in a tapered tube. Mathematical and Computer Modelling, 22(8):137 – 151, 1995.

[9] C. Pozrikidis. A study of peristaltic flow. Journal of Fluid Mechanics, 180:515–527, 1987.

[10] A. M. Provost and W. H. Schwarz. A theoretical study of viscous effects in peristaltic pumping. Journal of Fluid Mechanics, 279:177–195, 1994.

[11] L. M. Srivastava and V. P. Srivastava. Peristaltic transport of blood: Casson model–ii. Journal of Biomechanics, 17(11):821–829, 1984.

[12] S. Takabatake, K. Ayukawa, and A. Mori. Peristaltic pumping in circular cylindrical tubes: a numerical study of fluid transport and its efficiency. Journal of Fluid Mechanics, 193(-1):267–283, 1988.

[13] D. Tang and S. Rankin. Numerical and asymptotic solutions for peristaltic motion of nonlinear viscous flows with elastic free boundaries. SIAM Journal on Scientific Computing, 14(6):1300–1319, 1993.

[14] J. Teran, L. Fauci, and M. Shelley. Peristaltic pumping and irreversibility of a stokesian viscoelastic fluid. Physics of Fluids, 20(7), 2008.

[15] S. W. Walker and M. J. Shelley. Shape optimization of peristaltic pumping. accepted to Journal of Computational Physics, 2009.