MICROCHANNEL IMPEDANCE FOR QUASI NEWTONIAN

FLUIDS WITH SPATIAL MODULATED VISCOSITY

Tatiana Tabares Medina

Logic and Computation Group

Physics Engineering Program

School of Sciences and Humanities

EAFIT University

Medellin, Colombia

ttabares@eafit.edu.co

ABSTRACT

Using the Navier-Stokes equation and assuming a viscosity radially modulated for a quasi-Newtonian fluid, we obtain

the impedance of a fluid through microchannels and their corresponding electrical analogs. To solve the Navier-Stokes

equation will use the Laplace transform, the Bromwich integral, the residue theorem and Bessel functions. This will give

a formula for the impedance in terms of Bessel functions and from these equations to be constructed equivalent electrical

circuits. These solutions correspond to the case of quasi-Newtonian fluid it is to say a fluid that does not stagnate in the

channel wall as is the case if the fluid is Newtonian. The formulas obtained may have applications in the general theory

of microfluidics and microscopic systems design for drug delivery.

Keywords: Bessel function, Bromwich integral, quasi-Newtonian fluid, Navier Stokes equation, microchannel, Laplace

transform, impedance, residue theorem

INTRODUCTION

Navier-Stokes equations are a set of differential equations that describe the behavior of moving fluids, involving the

relation between velocity, pressure, temperature and density, the Navier Stokes equations also include the effects of

viscosity.[1]

The use of the Navier-Stokes equations promotes the analysis of motion of fluids trough a specific vessel. This paper will

use the equations mentioned in order to study the behavior of quasi-Newtonian fluids through microchannels, when the

flow in a system is known is possible identifies the impedance of fluid. This type of analysis allows doing a relation

between the behavior of a fluid and the behavior electric in a circuit.

This paper has the objective know the microchannels impedance for quasi-Newtonian fluids, these fluids make possible

to generate a mathematical model including a variable viscosity trough the a interpretation of Navier-Stokes and have

been considered in several paper and investigations, because it is a case that allows analyzing different boundary

conditions, initial and small changes of some variables.

One of the central topics of the paper is the study the impedance of the fluid mentioned above, so is important to

consider that impedance in a circuit is roughly opposition to the passage of current, the impedance can be analyzed in un

fluid and the expression that represent this one is`[2]

P(s) = Q(s)*Is (1)

Where P(s) is the pressure in terms of the Laplace transform, Q (s) is the flow that flowing through a cross-section and Is

is the impedance.

The above relation stipulates that knowing the system flow and pressure is possible to know what the opposition to the

passage of fluid through the system that is being studied, this relationship turns out to be very useful for example in the

Smart Biomedical and Physiological Sensor Technology X, edited by Brian M. Cullum, Eric S. McLamore, Proc. of SPIE Vol. 8719, 87190F · © 2013 SPIE · CCC code: 1605-7422/13/$18 · doi: 10.1117/12.2014627

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'

design of medical systems whose objective is the drug distribution or any instrument is of particular interest where the

distribution of fluids through channels.

In order to obtain the flow velocity and impedance of a fluid, will discuss the initial and boundary conditions which is

exposed to the fluid in the microchannels, it is important to implement these conditions in the Laplace domain for this

study is the fluid velocity in this domain and then made use of the residue theorem and the Bromwich Integral to obtain

this speed in the time domain, is also considered fluid flow and will be analyzed for the same time know the efficiency

corresponding to the time it takes for the fluid to reach a steady movement starting from rest, knowing the flow rate is

possible to find a relation for the impedance and study the corresponding analogous to a circuit.

PROBLEM

The measure of the impedance has been extensively treated in medicine and biology because it provides information

about the content of volume in different organs in the body, determinations features of body tissues and the way that

blood flows trough of arteries important to understand the control and regulation of cardiovascular functions.[3]

In order to develop a model that allows observing results for impedance fluids in microchannels and the impedance for

circuits analogs, we will use the Navier-Stokes equation for a fluid with variable viscosity flowing in a very long

cylindrical micro-channel with radius a.

(2)

Where (r) is the variable viscosity. In the particular case when (r) = r the Navier-Stokes equation takes the form

(3)

With the boundary condition

u(a,t)=0 (4)

In this paper we solve the following two problems:

Problem 1: Starting the flow: Initially the liquid is at complete rest for t <0 and there is not pressure gradient.

Then suddenly at t = 0 a constant pressure gradient is applied and an accelerating flow starts to run with the initial

condition

u(r,0)=0 (5)

Problem 2: Stopping the flow by internal viscous forces: Initially, for time t<0, a fully developed steady-state flow

with velocity profile v(r) was present, driven by a constant pressure gradient. Then at t = 0, the pressure gradient is

suddenly removed such that for t>0 the pressure gradient vanishes; and an decelerating flow starts to run until the

complete rest is reached. In this case the initial condition is

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u(r, 0) = v(r) (6)

METHOD

The method by which the different equations were obtained will be show in the figure (1). This figure show the

processing for obtain a solution of the Navier-Stokes equation which describe the motion of quasi-Newtonian fluids

trough the microchannels establishing the initial conditions and boundary condition, this equation will be solution for

two situations where it is starting the flow with a gradient pressure and a variant viscosity and the other case is when the

flow is stopping and the fluid has an initial motion but in a time higher than zero and its pressure gradient is removed.

Represent the behavior of quasi-newtonian fluid trough of microchannels using the Navier-Stokes equation

Setting the initial condition and boundary condition for a accelerating flow

Solving the Navier-Stokes equation in Laplace domain

using the residue theorem and Bronwich integral.

Finding the flow in time domain and determination the effective

time when the flow develop stationary movement.

Define the specific conditions in order to visualize the behavior of fluid in the time. Findindg of

impedance fluid.

Setting the corresponding analogs of pressure, flow and

impedance in a circuit.

Figure 1. Mothod for obtainig the impedance for a quasi-newtonian fluid using the Navier-Stokes equation [5,6,7,8].

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According the initial part of figure 1, the mathematical model starts with the Navier-Stokes equation which was

discussed in the previous item as well the initial conditions and boundary conditions, the other conditions in the figure 1

will be discussed in the results of this paper.

RESULTS

Results for Problem 1: Starting the flow

Using the boundary and initial conditions in the Navier-Stokes equation for microchannels is possible to obtain a

solution in terms of the Laplace transform, then the fluid velocity U(r) is given by

(7)

Where U(r) = L {u(r, t)}, P (z, s) = L {p (z, t)} and U(r) is expressed in terms of Bessel functions. Is important to

consider that

(8)

Inserting this in U (r) is obtained

(9)

In order to find the inverse Laplace transform for U (r) and express it in terms of time, applies the residue theorem and

integral Bromwich.

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(10)

The volumetric flow rate is the measure of volume of fluid which passes through a surface per unit time, in this case the

flow is

(11)

Also is possible define the flow as

(12)

Inserting the velocity U(r) in the flow equation gives

(13)

Flow is evaluated at infinity to obtain a stationary flow

(14)

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(15)

Considering this is possible to obtain the effective time teff

(16)

The effective time is the time that takes for the fluid to go from rest to develop a stationary movement.

Is possible to write q(s) as

(17)

Solving for K(s) in the last equation

(18)

K (s) can be extended through series around s to facilitate obtaining the inverse Laplace transform of q (s):

(19)

In this case K(s) corresponds to impedance in fluid, so is possible to see that the impedance is related with flow and

pressure and you can obtain a solution to this using expansion through series, the equation (19) show that the impedance

only depends of radius and density.

Results for Problem 2: Stopping the flow

Using the Navier Stokes equations as in the previous case in a fluid with a flow steady of velocity v(r) and considering

that the pressure gradient is removed, the Navier Stokes equation that describe this behavior is

(20)

for t = 0 is known that v (r) is

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(21)

The solution of the Navier Stokes equation in terms of the Bessel function is

(22)

Where V(r) = L {v(r, t)}. Using the residue theorem and integral Bromwich is possible to find the inverse Laplace

transform in order to obtain V(r) in terms of time t.

(23)

In a particular case where ρ=1, derivate of the pressure=-1, µ=1, is possible to obtain

(24)

If in the last equation v(r,t) is expanding for the first 20 zeros of BesselJ function is possible to obtain a graph that

describes the behavior of the velocity for different times with a variation of the radius

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40)

o.s

0.4

0.3

0.2

0.1

r

The last graph shows the behavior of the velocity for different values of the time varies between 0 and 10 seconds. Is

possible to observe that the velocity decreases more and more as time elapses and the same form show that boundary and

initial conditions are met.

The flow of fluid which passes through a surface is

(25)

As defined according to the problem is known that the flow at infinity is

(26)

The time required to decelerate the fluid and arrives to rest is

(27)

In a particular case where ρ=1, derivate of the pressure=-1, µ=1, is possible to obtain

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QR)

0.s

0.4

03

02

0.1

0 02 0.4 0.6 0.8 1 12 7.4 16 18

(28)

If the last equation is expanding for 20 terms of the summation, is possible obtain a graph where the flow and its time

related.

The last graph shows as the flow decrease a long of the time until zero. According to this graph and the previous is

possible to determine the behavior and development of a fluid after defining some parameters.

In order to study the behavior of fluid in the electronic circuit is necessary to identify elements with the same function in

each system. In this case the pressure gradient in the fluid corresponds to potential in a circuit electric, the flow

corresponds to current.

The electric impedance is represented as the sums between a real part and an imaginary part

(29)

Where Z is the impedance, R is the real part and X is the imaginary part that can be produce by existence of inductors or

existence of capacitors.

The impedance can be expressed as a generalization of Ohm law.

(30)

This last equation corresponding to the relation between the results obtain in this paper in the equation (19) and its

relations with a circuit electric.

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CONCLUSIONS

The use of Navier-Stokes equations allowed studying the flow, velocity and impedance of a quasi-Newtonian fluid for

two cases where the initial conditions varies and in the same way the results of equations. The study of different

conditions enables obtaining some graphs which describe the behavior of flow and velocity respect to variables as the

variation in radius and time.

The results obtain in this paper facilitate identify a behavior in a circuit with similar characteristics to those studied and

finding possible applications for similar model mathematics.

ACKNOWLEDGMENT

The present research work was developed under the collaboration of Dr. Felix Londoño EAFIT Research Director, Dr.

Mauricio Arroyave Director of Engineering Physics Program Dr. Andrés Sicard Ramirez, director of Logic and

Computation Group and Prof. Juan Fernando Giraldo Ospina, thanks to this latter for his continues accompaniment and

for making possible the realization of this study.

REFERENCES

[1]Gatski, Thomas B., Chester E. Grosch, y Milton E. Rose. «The numerical solution of the Navier-

Stokes equations for 3-dimensional, unsteady, incompressible flows by compact schemes». Journal

of Computational Physics 82, n.o 2 (1989): 298–329.

[2]Cox, Robert H. «Wave propagation through a Newtonian fluid contained within a thick-walled.

Viscoelastic tube: The influence of wall compressibility». Journal of biomechanics 3, n.o 3 (1970):

317–335.

[3]Salazar Muñoz, Yolocuauhtli. «Caracterización de tejidos cardíacos mediante métodos

mínimamente invasivos y no invasivos basados en espectroscopia de impedancia eléctrica» (2004).

http://www.tdx.cat/handle/10803/6187.

[4]Olufsen, Mette S., y Ali Nadim. «On deriving lumped models for blood flow and pressure in the

systemic arteries». Math Biosci Eng 1, n.o 1 (2004): 61–80.

[5] Bromwich integral, http://en.wikipedia.org/wiki/Inverse_Laplace_transform

[6] Residue Theorem, http://en.wikipedia.org/wiki/Residue_theorem

[7] Bessel functions, http://en.wikipedia.org/wiki/Bessel_function

[8] Maple, www.maplesoft.com

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