Numerical simulations of flows in an elastic cylinder with two chambers

Sunmi Lee1, and Eunok Jung1

1 Department of Mathematics, Konkuk University, Seoul, Korea.

We present a mathematical model of valveless pumping in a tube with two elastic chambers, which are motivated by theLiebau’s two-tank model. The tube consists of partially soft and partially (almost) rigid. We employ a two-dimensionalmodel using the immersed boundary method. In this new model, we have showed the important features of valveless pumpingas the previous experiments and mathematical models have been discovered. For instance, we have observed that the directionand the amount of a net flow are sensitively dependent on the pumping frequency.

Valveless pumping is closely related to the cardiovascular system and Liebau suggested that asymmetric periodic compressionof the heart causes the blood circulation. He did a physical experiment using elastic tubes with different length and materialsand observed the net flow of those systems. Liebau made two models: first, a closed loop with different length or materialsand secondly, two open tanks are connected with an elastic tube. There has been more work of computational, analytical andexperimental research on the valveless pumping phenomenon in the first model [3], [4] than the second model [1], [2].

We develop new mathematical models of valveless pumping: a tube with two elastic chambers which is analogous to theLiebau’s two-tank model. We concerns with a new mathematical model, a two-dimensional IB model. In the two-dimensionalIB model, the boundary should be immersed in a fluid, which resulting in modifying the two-tank model to the two-chambermodel that closes the system with an elastic chamber at each end (see Figure 1). Numerical results are presented. There aremany parameters that affect flows of valveless pumping such as the frequency, the compression duration, and the location ofperiodic pumping, and material or geometrical properties of a system. Two most important parameters that affect a net floware examined: the frequency and the compression duration of periodic pumping. These two factors influence on the amountand the direction of a net flow. Especially the pumping frequency is the most critical factor we are greatly interested. Meanflows at two cross sections are computed as functions of the frequency in Figure 2.

Fig. 1 A fluid filled in the entire box and an elastic cylinder with two chambers are immersed in the fluid (dark blue part is (almost) rigidand light blue is soft) and periodic pumping is given at the second sixth of the elastic tube (arrows).

PAMM · Proc. Appl. Math. Mech. 7, 2100075–2100076 (2007) / DOI 10.1002/pamm.200701018

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

0 1 2 3 4 5 6 7 8 9 10−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

duration = 0.5, kc1 =200 and kc2 =40

aver

age

net fl

ow fo

r 15

sec

net flow versus frequency

RightLeft

Fig. 2 Averaged net flow versus frequency: averaged net flow is computed at cross sections of two dark blue parts (right and left) for30 seconds and frequency is varied from 0.1s

−1 to 10s−1. Positive net flow represents the flow from left to right and negative net flow

represents the flow from right to left.

References[1] Alfio Borzi and Georg Propst, “ Numerical investigation of the Liebau phenomenon”, Z. angew. Math. Phys., 54, 1050–1072 (2003).[2] Georg Propst, “Pumping effects in models of periodically forced flow configurations”, Physica D, 217, 193–201 (2006).[3] E. Jung, “2-D Simulations of valveless pumping using Immersed Boundary Method”, Ph.D. thesis, Courant Institute of Mathematical

Sciences, New York University, 1998.[4] Ottesen J., “Valveless pumping in a fluid-filled closed elastic tube-system: one-dimensional theory with experimental validation”,

Journal of mathematical biology, 46(4), 309–332 (2003).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ICIAM07 Contributed Papers 2100076