%%%%%%%%%%%%%%%%   Introduction/Ellipsometry %%%%%%%%%%%%%%%%%%%

\BigHeading{Introduction}

Pumping blood is the main function of the heart. 
Is it possible to pump blood without valves? This report is intended 
to show by numerical simulations that there can be a net flow 
generated by a valveless mechanism in a circulatory system.
Simulations of valveless pumping are motivated by the physical experiment
Kilner. He introduced net flow in one direction depending on the location 
of periodic forcing in two different types of physical experiments; 
one with chambers and the other with just tubing.
We have examined flows driven by pumping without valves in Kilner's second 
model, a loop of tubing of which part is almost rigid and the other part
is flexible. We find that there is a phenomenon of changing direction of 
flow in the loop even when the periodic forcing is given in the fixed 
location. In particular, the crucial parameter that determines the 
direction is the driving frequency of the periodic forcing. 

As reviewed by Moser \& Noordergraaf, there has been a few earlier 
investigators of valveless pumping. Harvey W(1628), Weber EH(1834), 
Donders FC(1856) contributed to give theory of valveless pumping and claim 
that the heart is not only essential element to pump blood through the 
cardiovascular system. Ozanam M(1881) and Liebau G(1954) made various types 
of physical experiments to explain mechanism of valveless pumping.
Moser \& Noordergraaf(1998) try to identify the responsible
mechanism and conditions under which this mechanism operates.
They claim that average flow is to be generated by providing energy, 
the circuit which has a compliant reservoir, and difference of the 
impedance in the two pathways. Although there has been considerable
interest in valveless pumping, not so much attention has been directed 
to the change in direction of pumping that may occur with change in 
parameters.

One of applications of valveless pumping is cardiopulmonary resuscitation(CPR).
During CPR, it has been reported that the valves in our heart do not function.
Instead, they remain open throughout the pumping cycle. Despite the observed 
lack of valve function, some patients with cardiac arrest are
successfully resuscitated by the external chest-compression CPR. 
If the magnitude and even the direction of flow in valveless pumping 
are indeed frequency dependent, as our results seen to indicate, it is of 
obvious important to know what frequency of chest compression will
produce the most effective CPR. Another biological example of valveless
pumping may occur in the human embryological aspects is the first 
circulatory embryo at the end of the third week. The valves have not been 
made at this stage yet. Nevertheless, there is a net flow in the circulatory
system. An industrial application of valveless pumping is 
microelectromechanical systems(MEMS) devices where there is a need to produce 
fluid motion without inserting anything. The fluid motion we have examined 
during our numerical simulations can be driven by either a traveling wave 
or a standing wave by giving a periodic forcing. Each case might be applied 
in MEMS. 

It is not easy to give a clear explanation in theory for valveless 
pumping. This report uses numerical simulations as an ``experimental''
tool to study this mysterious phenomenon. 
\BigHeading{Immersed Boundary Method} 
Consider a viscous incompressible fluid which is contained in a periodic
rectangular domain containing an immersed massless elastic boundary. 
The fluid equations are in Eulerian form and the immersed boundary is in 
Lagrangian form. The equations of motion of the elastic immersed boundary
and fluid system are as follows
\vspace{0.5cm}
\Equations{
\begin{eqnarray}
 \rho (\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla)\mathbf{u})+\nabla p=\mu\nabla^2\mathbf{u}+\mathbf{F}. \\
 \nabla\cdot\mathbf{u}(\mathbf{x},t)=0.
\end{eqnarray}
\begin{equation}
 \mathbf{F}(\mathbf{x},t)=\int\mathbf{f}(s,t)\,\delta(\mathbf{x}-\mathbf{X}(s,t))\,ds.
\end{equation}
\begin{equation}
 \mathbf{f}(s,t)=\kappa_{1}(\mathbf{X}(s,t)-\mathbf{X_{target}}(s,t))+
\kappa_{2}(\frac{\partial^2 \mathbf{X}(s,t)}{\partial s^2}).
\end{equation}
\begin{eqnarray}
 \frac{\partial\mathbf{X}(s,t)}{\partial
t}(s,t)&=&\mathbf{u}\left(\mathbf{X}(s,t)\right)\\
 &=&\int\mathbf{u}(\mathbf{x},t)\,\delta(\mathbf{x}-\mathbf{X}(s,t))\,d\mathbf{x}.
\end{eqnarray}
}
\vspace{0.3cm}
Here, $\mathbf{u}$ is the fluid velocity, $p$ is the fluid pressure, and $\mathbf{F}$ is the force density generated by the boundary in the fluid. $\mathbf{X}(s,t)$ and $\mathbf{f}(s,t)$ are the elastic boundary configuration and force density arising from the elastic stresses, respectively.

{\bf Applications of the Immersed Boundary Method} : Any problem involving 
elastic structure interacting with a viscous incompressible 
fluid which is in regions of complicated geometry. This approach has been 
applied to problems of blood flow in the heart, the design of prosthetic 
cardiac valves, wave propagation in the cochlea, 
aquatic animal locomotion, and flow in collapsible tubes.

{\bf Philosophy} : The elastic material is treated as a part of fluid 
in which singular forces are applied.

{\bf Strength} : It can handle the complicated and time dependent geometry 
of the elastic immersed boundary which interacts with fluid while using a fixed 
regular lattice.
%\Summary{
%\[
%\hspace{1.in} \mathbf{X}(s,t) \longrightarrow{} \mathbf{f}(s,t) 
%\longrightarrow{\text{\delta fn}} \mathbf{F}(\mathbf{x},t) \\
%\hspace{1.5in} \uparrow{} \hspace{4.in} \downarrow{\text{N-S eq.}\\
%\longleftarrow{\text{no-slip}} \mathbf{U}(s,t) \longleftarrow{\text{\delta fn}} \mathbf{u}(\mathbf{x},t) 
%\ifPoster{}\else{\newpage}\fi
%\]
%\vspace{2in}
%{\bf Summary of numerical method}
%\Summary{\[
%\mathbf{X}(s,t) \hspace{0.1cm} \longrightarrow{} \hspace{0.1cm}
% \mathbf{f}(s,t) \hspace{0.1cm} 
%\longrightarrow{} \hspace{0.1cm} \mathbf{F}(\mathbf{x},t)
% \longrightarrow{} \hspace{0.1cm} \mathbf{u}(\mathbf{x},t) 
%\hspace{0.1cm} \longrightarrow{} \hspace{0.1cm} \mathbf{U}(s,t) 
%\hspace{0.1cm} \longrightarrow{} \mathbf{X}(s,t) 
%\]}
%}
%\vspace{1cm}
%\Summary{
%For given the velocity $\mathbf{u}$ and boundary position $\mathbf{X}$ at 
%time level n we are looking for update $\mathbf{u}$ and $\mathbf{X}$.
%\begin{itemize}
%\item  Find the force on the immersed elastic boundary from 
%the given configuration. : equation(4)
%\item Spread the boundary force into the nearby lattice points
%for fluid using $\delta$ function. : equation(3)
%\item Solve the Navier-Stokes equations on the rectangular lattices
%to get update $\mathbf{u}$ and $p$ using the periodic boundary condition. 
%: equation(1),(2)
%\item Find the update velocity and positions on the immersed boundary 
%points using the interpolate equation and no-slip condition 
%: equation(5),(6)
%\end{itemize}
%}
\vspace{3in}
%%%%%%%%%%%%%%%%   VIP-MAI Considerations    %%%%%%%%%%%%%%%%%%%
\ifPoster{}\else{\newpage}\fi
\BigHeading{Introduction of the 2-D model of the valveless pumping}
%\vspace{0.5cm}
%\SubHeading{Initial position} 
\centerline{\epsfxsize \PlotWidth
  \epsffile{realb_inital.eps}}
We consider the incompressible viscous flow with constant density $\rho$ 
and viscosity $\mu$ in a periodic rectangular box which contains the immersed 
elastic boundary. The immersed elastic boundary is consisted of two parts 
which are partially almost rigid(blue color) and partially flexible(red color).
There are fluid markers inside of the loop of tubing. 
%{\bf Giving the external force as time goes by}
%\centerline{\epsfxsize \PlotWidth
%  \epsffile{realb_target.eps}}
%  \title{The motion of target positions over one cycle}
%\vspace{.5cm}
\vspace{0.5cm}
The physical boundary is controlled by the periodic motion
of the target positions which are not visible.
The two dimensional flow is driven by periodic external force which is 
giving by vertical oscillations of the left one third of the lower and 
upper flexible boundary during most of our simulations.
\vspace{.5cm}
%%%%%%%%%%%%%%%%   VIP-MAI Considerations    %%%%%%%%%%%%%%%%%%%
\BigHeading{Discussion}
\vspace{0.5cm}
\SubHeading{The direction of flow inside a loop depend on the driving 
frequency of the periodic forcing.} 
\vspace{.3cm}
\centerline{\epsfxsize \PlotWidth
  \epsffile{Plot_real_amp04.eps}}
  \title{Average flow vs Period}
%  \caption{Average Flow vs Period : The two different amplitudes 
%	of target positions are compared.}
\vspace{.5cm}
The time average flow around a loop as a function of period are investigated 
in two different cases. As we mentioned before the target positions control
the motion of flow, so two different amplitudes for target positions,
amplitude=0.6(cm) in case I and amplitude=0.4(cm) in case II, are chosen 
to compare. The positive values and negative values denote the direction 
of the flow inside a loop with clockwise direction and counter-clockwise 
direction, respectively. Each case is investigated under the same 
circumference except period. In each of the numerical experiments,
the code was run until a periodic steady state was reached. 
\vspace{.5cm}
\begin{center}
  \begin{tabular}{ || p{4in} | r  ||}\hline
   \hspace{.7in}Parameters&\hspace{.3in}                 \\ \hline \hline 
   \hspace{.7in}Density   &\hspace{.3in}  1($gm/cm^{3}$) \\ \hline
   \hspace{.7in}Viscosity &\hspace{.3in}  0.01(gm/cm s)  \\ \hline
   \hspace{.7in}Diameter of loop  &\hspace{.3in}  1.2(cm) \\\hline
   \hspace{.7in}Grid      &\hspace{.3in}  256 x 128      \\ \hline
   \hspace{.7in}Physical boundary &\hspace{.3in}  4654     \\ \hline
   \hspace{.7in}Time      &\hspace{.3in}  150(s)         \\ \hline
   \end{tabular}
  \end{center}
\vspace{.5cm}
\Results{
\begin{itemize}
  \item It is clear to see there are at least 6 turning points in case I,
	and at least 2 turning points in case II. One of the interesting
	phenomenon is that there exist symmetric motion of flow, almost zero 
	flow, in finite frequencies even though the axisymmetric force is given.
  \item The motion of flow in case I is much more active than one in case II.
	It is hard to get the positive value of mean flow, which denote the 
	direction of flow inside a loop with clockwise direction, in case II.
	The traveling wave is easier to get than the standing wave. We need 
	certain strength of force to get a flow with clockwise direction.
  \item The time average flow tends to almost zero flow for very short and 
	long period in both of cases.
  \item The maximum flux with right direction is very important in the 
	application of CPR. Therefore, We investigate here the following 
	special cases, maximum mean flow with clockwise direction at 
	period(0.325), and maximum mean flow with counter-clockwise direction
	at period(0.21) in case I. 
\end{itemize}  
}
\vspace{3in}
\SubHeading{The time average flow around a loop as a function of amplitude of 
	the target positions} 
\vspace{.3cm}
\centerline{\epsfxsize \PlotWidth
  \epsffile{Plot_amplitude.eps}}
\vspace{.5cm}
\Results{
\begin{itemize}
  \item The average flow as a function of amplitude of 7 different periods, 
	T = 0.1, 0.3, 0.375, 0.4, 0.525, 1.1 and 1.7, are compared.
  \item The direction of flow inside a loop also depends on the amplitude.
	It is clear to see in Figure that there is a point of changing 
	direction in the cases of T=0.3, 0.375, 0.4 and 1.1. 
  \item To get a clockwise direction the certain value of amplitude are 
	necessarily needed.
\end{itemize}
}
\vspace{.5cm}
\SubHeading{Check our numerical method}
\vspace{.3cm}
{\bf Numerical convergence} 
\vspace{.3cm}
 \begin{center}
  \begin{tabular}{ || p{4.in} | r ||}\hline
   \hspace{.7in}$L_2$ difference ratio        &          \\ \hline \hline 
   \hspace{.7in}$\|$$u_{64}$-$u_{128}$$\|_2$/$\|$$u_{128}$-$u_{256}$$\|_2 $   & 1.910726 \\ \hline
   \hspace{.7in}$\|$$u_{128}$-$u_{256}$$\|_2$/$\|$$u_{256}$-$u_{512}$$\|_2 $  & 1.995359 \\ \hline \hline
   \hspace{.7in}$\|$$v_{64}$-$v_{128}$$\|_2$/$\|$$v_{128}$-$v_{256}$$\|_2 $   & 1.662576 \\ \hline
   \hspace{.7in}$\|$$v_{128}$-$v_{256}$$\|_2$/$\|$$v_{256}$-$v_{512}$$\|_2 $  & 1.869106 \\ \hline
  \end{tabular}
 \end{center}
\vspace{.3cm}
The table displays the $L_{2}$ difference ratios of velocities for four cases. 
The lattices 32x64, 64x128, 128x256, and 256x512 are discussed. These results 
are based on time=150(second), number of cycles=96. We have almost first order accuracy.
\vspace{2.cm}
\centerline{\epsfxsize \ThirdPlotWidth
  \epsffile{thesis_period_real.eps}
  \hfill
  \epsfxsize \ThirdPlotWidth
  \epsffile{Plot_area_real.eps}
  \hfill
  \epsfxsize \ThirdPlotWidth
  \epsffile{thesis_period_hreal.eps}}
\Results{
\begin{itemize}
  \item In the leftmost figure, the time average flow on the horizontal(blue) 
and vertical(red) cross sections of the loop are compared. The two 
graph are pretty close. 
  \item In the center figure, the area inside a loop are calculated on various
periods. The difference from the area of the initial position is small 
enough to show the conservation of volume in the incompressible flow.
  \item In the rightmost figure, The time average flow as a function 
of period are compared at $t=75$(sec) with blue color and 
$t=150$(sec) with red color. This figure shows us that the time 
which we chose during most of simulations, $t=150$(sec), is 
enough time to get the steady state motion.	
\end{itemize}
}
\vspace{1cm}
%323-677-6865
%%%%%%%%%%%%%%%%   VIP-MAI Considerations    %%%%%%%%%%%%%%%%%%%
\ifPoster{}\else{\newpage}\fi
\BigHeading{The Special Cases : Maximum Flux}
\vspace{0.5cm}
The maximum flux with right direction is crucially important 
in the application of CPR.
\vspace{.5cm}
\SubHeading{Maximum average flux with clockwise direction :
\\ \hspace{3in}Period = 0.325(s)} 
\vspace{.5cm}
\centerline{\epsfxsize \PlotWidth
  \epsffile{awm_maxl_pos.eps}}
\medskip
\vspace{.5cm}
\centerline{\epsfxsize \HalfPlotWidth
  \epsffile {max_flux1_angle.eps} \hfill
  \epsfxsize \HalfPlotWidth
  \epsffile{max_flux1_flowmeter1.eps}}
\medskip
\vspace{.5cm}
\SubHeading{Maximum average flux with counter-clockwise direction : 
\\\hspace{3in}Period = 0.21(s)} 
\vspace{.3cm}
\centerline{\epsfxsize \PlotWidth
  \epsffile{awm_minl_pos.eps}}
\medskip
\vspace{.5cm}
\centerline{\epsfxsize \HalfPlotWidth
  \epsffile{min_flux1_angle.eps} \hfill
  \epsfxsize \HalfPlotWidth
  \epsffile{min_flux1_flowmeter1.eps}}
\medskip
\vspace{.5cm}
%\SubHeading{Almost zero net flow} 
%\vspace{.3cm}
%
%\centerline{\epsfxsize \PlotWidth
%  \epsffile{awm_zerol_pos.eps}}
%  \caption{Parameters : period=1.34(s), time=25(s), amplitude=0.6(cm)}
%   \title{Figure(7)}
%\medskip
%\vspace{.5cm}

%\centerline{\epsfxsize \HalfPlotWidth
%  \epsffile{awm_zero_angle.eps} \hfill
%  \caption{Parameters : period=1.34(s), time=25(s), amplitude=0.6(cm)}
%  \epsfxsize \HalfPlotWidth
%  \epsffile{zero_flux_flowmeter1.eps}}
%  \caption{Parameters : period=1.34(s), time=150(s), amplitude=0.6(cm)}
%
%\medskip
%\vspace{.5cm}
\Results{
\begin{itemize}
  \item The dimensionless variable, Reynolds number, is defined by
  	Average-flux/viscosity. The Reynolds numbers of these
	two cases are around 60 for maximum flow with clockwise direction,
	161 for maximum flow with counter-clockwise direction.
  \item The figures of the positions and angles of fluid markers inside of 
	a loop show us that there exist net flows. 
	Flows driven by a standing wave, which is investigated in CPR, is shown 
	in upper figure.
	Flows driven by a traveling wave propagating along the flexible
	boundary is investigated in lower figure. The only different parameter
	of these two numerical simulations is the driving frequency(1/period).
  \item Flowmeters show us that each cases is 
	investigated in the periodic steady-state motion.
\end{itemize}
}
\vspace{.5cm}
\BigHeading{Conclusions}
\Conclusions{
\begin{itemize}
  \item From numerical simulations it became evident that the frequency of the 
	driving force is a critical parameter in changing the direction of flow. 
	Especially, the cases of maximum flow provide good applications in CPR.	
  \item The bigger amplitude, the easier to get a fluid motion by a standing
	wave.
  \item When we compare the cases of the giving driving force at the end of the 1/3
	left and right hand side of the flexible boundary, we got the following 
	results : 
 \begin{center}
  \begin{tabular}{ || p{2.in} | r | r ||}\hline
   \ Period &   Average Flux(Left) & Average Flux(Right)      \\ \hline \hline 
   \ T = 0.325 &   0.6969 &   -0.6812 \\ \hline
   \ T = 0.21  &  -1.5411 &    1.4889 \\ \hline 
   \ T = 1.34  &  -0.0194 &    0.0271 \\ \hline
  \end{tabular}
 \end{center}
	where the positive and negative values denote the clockwise and counter-
	clockwise direction, respectively.
	In some our simulations we have the same phenomenon as physical
	experiments in Dr. Kilner's.
\end{itemize}
}	
\begin{thebibliography}{99}

\bibitem{1} {\sc C. P. Peskin and D. M. Mcqueen} (1995),
{\em A general method for the computer simulation of biological
systems interacting with fluids}, Symposia of the Society for Experimental
Biology, vol. 49. Cambrige, UK: The Company of Biologists Limited. 
{310, 314, 337}

\bibitem{2}{\sc M. Moser, J. W. Huang, G. S. Schwarz, T. Kenner, and 
A. Noordergraaf} (1998), {\em Impedance defined flow generalization of William 
Harvey's concept of the circulation - 370 years later}, International Journal
of Cardiovascular Medicine and Science, vol. 1. Nos 3/4, pp 205-211

\end{thebibliography}