Numerical simulations and analysis of a micropump actuated by traveling plane waves

A. Fatih Tabak, Serhat Yeşilyurt*

Sabanci University, Tuzla, Istanbul, 34956, Turkey

ABSTRACT

Traveling plane-wave deformations on a solid thin film immersed in a fluid can create viscous propulsion in the direction opposite to wave propagation. Here, we present modeling and analysis of a valveless dynamic micropump that incorporates a traveling plane-wave actuator. Our numerical model incorporates direct coupling between solid deforming boundaries and the fluid by means of a deforming mesh according to Arbitrary Lagrangian Eulerian implementation, and 2D unsteady Stokes equations to solve for the flow realized by the traveling-wave actuator in the channel. A commercial finite-element package, COMSOL, is used for simulations. Analysis is carried out to identify the effects of operating conditions such as wavelength, frequency and amplitude of the waves. For specified dimensions of the pump, maximum time-averaged flow rate, exit pressure, rate-of-work done on the fluid and efficiency of the micropump are calculated for different operating conditions.

Keywords: Micropump, deforming mesh, ALE, traveling-wave actuator, pump efficiency

1. INTRODUCTION Sustaining controllable flows and pressure loads with micropumps remains a challenge in many areas such as drug delivery systems, biotechnology applications, microelectronic cooling, sensor applications and fuel delivery systems for energy applications [1-8]. From one aspect, micropump actuation mechanisms can be categorized into three main groups as magnetic, electric or mechanical [9]. Typically, common mechanical micropumps employ a reciprocating positive displacement mechanism, and utilize structural components such as pistons, membranes and valves in order to control the flow [10,11]. Due to limited size of the stroke volume, piezoelectric drivers are utilized, and large voltages at high frequencies are applied to maintain high flow rates [12-14].

A micropump that uses the propulsion mechanism of microorganisms such as bacteria and spermatozoa [15,16] can be a viable option for the actuation of micropumps. A spermatozoon utilizes sudden stress induced beating motions to propel itself opposite to the propagation direction of the waves [17]. Sir Taylor analyzed the interaction between an inextensible infinite sheet and surrounding fluid, concluding that the average-speed of the organism, U, is proportional to the wave speed, c, and the square of the amplitude of the deformation waves, B2, divided by the square of the wavelength, λ2, i.e. U ~ cB2/λ2 [18]. The rate of work done on the fluid due to the traveling-waves on an infinitely long sheet in an infinite medium is presented by the analysis of Childress [19], where, based on Taylor’s analysis it is shown that the average rate of work done on the fluid, Π, is proportional to the flow rate multiplied by the viscosity, i.e. Π ~ µω2B2/λ. Analysis of the propulsion due to traveling wave deformations that take place on an infinitely long sheet placed between parallel plates, is carried out by Katz [20]; for the case in which the separation between the channel walls, H, is much smaller than the wavelength, H << λ, the speed of the organism is obtained as proportional to the wave speed divided by square of the channel height to amplitude ratio added to a constant, i.e. U ~ c/((H/B)2 + constant). According to the Katz’s asymptotic analysis, the organism reaches its maximum speed when the wave amplitude equals the half of the channel height [20], which clearly indicates the displacement pump limit in the case the sheet is fixed and works as a displacement pump.

In this work we propose a pumping mechanism, which uses traveling-wave deformations on a finite-length thin inextensible film as the actuation mechanism. The finite-length film is placed inside a channel as shown in figure 1. Performance of a typical pump, which is characterized by the flow rate, exit pressure, rate of work done on the fluid and its mechanical efficiency is demonstrated as a function of its operating conditions, which are the wavelength, amplitude and the drive frequency. The unsteady flow created by the motion of the thin film inside the channel is modeled by two-

* syesilyurt@sabanciuniv.edu; phone +90 216 483 9579; fax +90 216 483 9550; sabanciuniv.edu

dimensional time-dependent Stokes equations on the domain that deforms according to the motion of the film; an arbitrary Lagrangian Eulerian (ALE) method with the Winslow smoothing [21] is used to model the fluid-structure interaction.

Figure 1: Wave propagation on a thin solid film placed in micro channel filled with incompressible viscous fluid.

2. METHODOLOGY Vertical motion of the inextensible film due to traveling-wave deformations is given by a sinusoidal wave-form as a function of time, t, lateral position on the film, xf, angular frequency, ω=2πf, wave number, k=2π/λ and an amplitude function B(xf,t) as follows:

( ) ( ) ( ), , sin ωf f f fy x t B x t t kx= − . (1)

The function B(xf,t) limits the wave-deformations of the film (see figure 1) by a parabolic envelop, which ensures fixed ends, with a maximum extend, B0, in the middle, which is given by:

( ) 01, 4 1 min ,f f

ff f

x xB x t B t

f

⎛ ⎞⎛ ⎞ ⎛ ⎞= −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠l l

, (2)

where ( )min ,1t f introduces an initial ramp that lasts exactly one period, and ensures a start from the rest initial condition. Since Reynolds number for such a flow is small due to small length and velocity scales, typically much less than one, the flow is governed by Stokes equations in a time-dependent deforming domain, Ω(t):

( )2ρ µ in m P tt

∂⎛ ⎞+ ⋅∇ = −∇ + ∇ Ω⎜ ⎟∂⎝ ⎠

U u U U , (3)

subject to conservation of mass:

( )0 in t∇ ⋅ = ΩU . (4)

In (3) and (4), U=[u,v]′ is the velocity vector, P is the pressure, ρ is fluid’s density, and µ is the viscosity of the fluid. The time-dependent domain, Ω(t), corresponds to the volume containing the fluid at time t, and bounded by boundaries of the micropump which are the channel’s walls, inlet and outlet, and the location of the surface of the thin film at time t. The um in the left-hand-side of (4) is an artificial velocity of the domain, Ω(t), due to its time-dependent deformation. No slip conditions are assigned at channel walls:

( ) ( )( ) ( )

,0, , , 0

,0, , , 0.

u x t u x H t

v x t v x H t

= =

= = (5)

y x x=x0

xf

x=x0+λf

yf

x=L

y = H

y = 0

y = H/2

Limiting parabolic envelop

Horizontal velocity is set to be zero at all times on the surface of the film:

( ), , 0f fu x y t = . (6)

After the initial ramp, film’s vertical velocity is given by the time-derivative of (1):

( ) ( ) ( ), , ω , cos ωv x y t B x t t kxf f f= − (7)

Channel inlet pressure is set to zero, and the outlet pressure is specified by an exit pressure that corresponds to the pressure load on the pump:

[ ] [ ] ( )0, , , ,

0; out inx y t x L y tP P P P P

= =− ⋅ = − ⋅ = ∆ ≡ −I n I n , (8)

where n is the outward normal of the surface. The initial condition for (3) is the flow at rest, i.e. the velocity components are all equal to zero at t = 0:

( ) ( ), ,0 , ,0 0u x y v x y= = (9)

Arbitrary Lagrangian Eulerian (ALE) with Winslow smoothing [21-23] method is used to calculate the velocity of the deforming mesh from the Laplace equation:

( )2 0 in m t∇ Ωu = , (10)

subject to the boundary conditions: ( ), , 0m f fu x y t = (11)

( ), , fm f f

dyv x y t

dt= (12)

( )( )

( )( )

0, , , , 00, , , , 0

m m

m m

u y t u L y tv y t v L y t⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦

(13)

( )( )

( )( ) .

, 0, , , 0,0, , , 0

m m

m m

u x t u x H tv x t v x H t⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦

(14)

Once um is obtained, Stokes equations given by (3) are solved subject to the incompressibility condition in (4), boundary conditions in (5-8), and initial conditions in (9) with the commercial software, COMSOL [22].

The instantaneous flow rate per unit depth for a particular set of parameters, namely B0, f, λ, can be computed by integrating the x-velocity in the channel over inlet or the exit:

( ) ( ) ( ) ( ), ,0

H

in out in outy

Q t Q t t dy=

= = ⋅∫ u nm (15)

where nin and nout correspond to inlet and outlet surface normals. Since flow is perpendicular to the outlet and inlet surfaces, ‘+’ and ‘–’ signs defines outlet and inlet flows respectively. To be sure if there is a violation of conservation of mass given by (4), flow rates are checked and observed to be well below the numerical tolerance used in simulations, which keeps the relative error below 10-3.

Time-averaged flow rate can be computed by integrating the instantaneous flow rate in (15) over a suitable time interval, preferably starting after the initial ramp is completed and for long enough duration to capture at least one entire wave displacement on the film, i.e.

( )1

0

, , ,1 0

1t

av av in out in outt

Q Q Q t dtt t

= =− ∫ . (16)

The rate of work exerted on the fluid by the thin film is computed by integrating the y-component of the stress induced by the fluid on the film and the vertical velocity of film over entire surface of the film:

( ) ( ) ( )Film surface

, , , ,y f f f ft x y t v x y t dAΠ = Σ∫ , (17)

where Π(t) is the rate of work done on the fluid and Σy is the y-component of the stress exerted by the film on the fluid, which is defined as [24]:

( ), , 2µ µy f f y xv u vx y t P n ny y x

⎛ ⎞ ⎛ ⎞∂ ∂ ∂Σ = − + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ (18)

where, µ is the viscosity of the fluid, nx and ny are x- and y-components of the corresponding surface normal. The time-averaged power, Πav, can be computed by the same method as the time-averaged flow rate calculated in (16).

Hydraulic efficiency of the pump, η, based on the imposed pressure load at the exit, ∆P, time-averaged flow rate given by (16), and time time-averaged power based on (17) is defined as [25]:

η 100 av

av

P Q∆=

Π (19)

The hydraulic efficiency given by (19), in effect, is the net portion of the rate of mechanical work done on the fluid and converted to flow against the pressure load imposed at the exit and zero inlet pressure.

3. RESULTS Numerical results that are presented here are obtained for pump dimensions that correspond to a typical micropump, of H = 625 µm, W =2.5 mm and ℓf =1.25mm. For the base-case operating conditions with water, we have: f = 2 Hz, λ = ℓf = 1.25 mm and B0 = 14.5 µm. Approximately 30000 linear equations are solved for each time-step for 5 dimensionless time units that correspond to 5 full periods for the base case with COMSOL. Simulation outputs converge to a steady-periodic state within two periods. To calculate time-averaged quantities, last three periods are used. A typical simulation takes about 6 hours on a single processor of a dual 2.4 GHz 32-bit Xeon workstation with 1GB of RAM running on SUSE Linux operating system.

In figure 2, the pressure distribution and the velocity vectors of the flow are shown for the base operating conditions, i.e. f = 2 Hz, λ = ℓf = 1.25 mm, B0 = 14.5 µm, and ∆P = 0. When the wavelength is large compared to film’s length, deformations on the film create a pressure distribution that extends to channel walls. The pressure variations induced by the deformation of the film force recirculations that are much stronger than the average flow rate, which are indicated by the relative size of the velocity vectors in figure 2. Detailed analysis of the flow regime in the micropump is presented elsewhere [26].

In figure 3, the pressure distribution and the streamlines of the flow are shown for f = 2 Hz, λ = ℓf/10 = 125µm, B0 = 14.5 µm and ∆P = 0. In this case, pressure variations due to traveling-wave deformations do not extend to channel walls, and steady pressure distribution and steady streamlines away from the film are observed. Local variations in the pressure result in smaller recirculations near the film. As the wave propagates downstream, local recirculations move with the wave and drag the neighboring fluid along. There are two resident vortices near the walls that make the flow resemble to the one due to a converging diverging nozzle. Pressure towards the end of the film is somewhat larger than the pressure at the front of the film, which is less than the inlet pressure of zero. Pressure at the end of the film is larger than the exit pressure, which is also zero for this case.

Figure 2: Color shaded pressure distribution and velocity’s arrow plot for λ = λf = 1.25 mm, B0

= 14.5 µm with f = 2 Hz. Note that dimensions in the figure are normalized with 2.5×10-4 m, and pressure with 2.5×10-4 Pa.

Figure 3: Color shaded pressure distribution and streamlines for λ = λf /10 = 125 µm, B0

= 14.5 µm with f = 2 Hz. Note that dimensions in the figure are normalized with 2.5×10-4 m, and pressure with 2.5×10-4 Pa.

In figure 4, the pressure load and the efficiency are plotted against the time-averaged flow rate for the base case, for which λ = λf = 1.25 mm, B0

= 14.5 µm and f = 2 Hz. The performance is that of a typical dynamic pump without inertial effects as the pressure difference due to higher exit pressure varies linearly with the flow rate, and, hence, the efficiency of the pump is given by a perfect parabola. The maximum time-averaged flow rate is obtained for zero pressure load at Qmax = 0.42 µl/min, and the maximum pressure load for the zero time-averaged flow rate is 0.99 mPa. The pump operates at constant fluid power (rate of work done on the fluid by the film) of 2.60 pW with small variations (<10-2 pW) as the pressure load changes. The energy of the flow is distributed between the kinetic energy of the flow and the pressure according to the pressure load as expected. The maximum efficiency is obtained at the average flow rate that equals the half of the Qmax as 0.067 %; based on the linearity between the pressure load and the average flow rate. To calculate the efficiency, we also have:

max maxmax

max4P Qη ∆

=Π

(21)

where maxΠ is the time-averaged maximum power exerted on fluid in Ω(t) for specified wavelength, λ, frequency, f, and amplitude, B0. The power, maxΠ , remains nearly constant as the flow rate or the applied pressure varies. Note that, had the power varied with the flow rate and the pressure load for a fixed operating condition, linear dependence between the pressure and the average flow rate (see figure 4) would no longer have held.

Figure 4: Inlet-to-outlet pressure increase, ∆P, and efficiency, η, vs. time-averaged flow rate, Qav, for the base case, λ = 125

µm, B0 = 14.5 µm, and f = 2Hz; results indicated with symbols are from numerical simulations, and dashed lines

indicate the linearity of ∆P with Qav.

In figures 5, 6 and 7, variations of the maximum time-averaged flow rate for zero ∆P in the channel (in 5a, 6a and 7a), the maximum pressure difference for zero time-averaged flow rate (in 5b, 6b and 7b), the rate-of-work done on the fluid (in 5c, 6c and 7c), and maximum pump efficiency (in 5d, 6d and 7d) are plotted against the wavelength (in figure 5), frequency (in figure 6) and the amplitude (figure 7). In figures 5, 6 and 7, the reference case is set to λ = 1.25 mm, B0

= 14.5 µm and f = 2Hz.

The flow regime in the pump changes with the wavelength as depicted in figures 2 and 3. For shorter wavelengths than half of the film’s length, the maximum flow rate and maximum pressure load decrease with increasing wavelength (figures 5a and 5b). For the same range of wavelengths, the power exerted on the fluid tends to decrease with a faster rate than the one for Qmax and ∆Pmax (figure 5c). Hence the efficiency tends to decrease with a rate similar to that of Qmax and ∆Pmax. Unfortunately, it is harder to quantify the rates especially for Qmax and ∆Pmax, as there are not enough data points especially for short wavelengths due to numerical instability of the solutions as the resolution of each wave requires increasing number of mesh nodes on the film, which, in turn, poses a difficulty in deforming mesh calculations. However for the power, one can argue that Πmax is proportional to 1/λ.

For larger wavelengths than about the half of the film’s length, in a relatively small intermediate regime Qmax and ∆Pmax increase with the wavelength for up to about twice the length of the film (see figures 5a and 5b). Further increases in the wavelength do not yield larger flow rates or pressure increases in the channel due to the shape function given by (2), which results in a simple vertical motion of the film without an effective wave propagation and, hence, pumping. In that window, the power exerted on the fluid (figure 5c) increases with the wavelength with a faster rate than that of Qmax and ∆Pmax. In combination of responses of the flow rate, pressure and the power, the efficiency of the pump drops with the increasing wavelength (figure 5d).

In figure 6, the frequency is varied while the wavelength and the amplitude are kept at their reference values. Hence, the flow regime corresponds to the case for large wavelengths as shown in figure 2. Increasing the frequency results in increased flow rate, pressure and the power, but efficiency remains unaffected from the frequency variations. From figures 6a and 6b, one can easily inspect that the Qmax and ∆Pmax vary linearly with the frequency, i.e. Qmax ~ f, and ∆Pmax ~ f. From figure 6c, it is observed that the power exerted on the fluid varies with the square of the frequency, i.e. Πmax ~f 2. Therefore the efficiency remains constant at its reference value (.067 %) according to (21).

Lastly, in figure 7, we observe that the flow rate (7a), pressure (7b), power (7c), and, hence, the efficiency increase with the square of the amplitude, i.e. 2 2 2 2

max 0 max 0 max 0 0~ , ~ , ~ , and η ~ .Q B P B B B∆ Π

According to asymptotic results presented by Taylor [18] in his analysis for traveling waves on an infinite-sheet in an infinite medium, the average velocity of the swimming sheet scales with the product of the wave-speed and the square of the amplitude, i.e. 2

0~U B c , which agrees well with our observations for constant wavelength (and constant channel cross-section) as the flow rate is observed to be proportional to the product of the amplitude-squared and the frequency, i.e. ( ) 2

max max 0~ λQ U WH B f= (see figures 6a and 7a). Similarly according to the analysis of the effect of the traveling waves on an infinite-sheet between parallel plates by Katz [20], the time-averaged velocity of the sheet is given by:

( )20~ / constantU c H B + , which also agrees well with our results for constant wavelength and the channel height.

Clearly, the effect of the wavelength differs from the asymptotic analysis for a finite-length film in finite-channel.

An energy analysis of the Taylor’s infinite sheet is carried out by Childress [19]; according to his results time-averaged rate-of-work done on the fluid is given by 2 2

0~ µω λB∞Π , which agrees well with our results for constant wavelength, as the power is found to be proportional to the frequency in figure 6c and the square of the amplitude in 7c, i.e.

2 2max 0~ f BΠ . Moreover, our results exhibit an inverse relationship between the power and the wavelength capturing

Childress’ asymptotic result for infinite-sheet in infinite-medium in full for small wavelengths compared to the film’s length and the channel height (see figure 5c).

4. CONCLUSION An effective pumping mechanism based on the propulsion of microorganisms is demonstrated using numerical simulations of the interaction between the solid deforming film and the fluid inside a channel. The mechanism incorporates traveling-wave deformations on a thin finite-length film as the source of pumping that can provide flow and pressure at the outlet of the channel. The effect of operation parameters such as wavelength, frequency and amplitude is quantified by means of a number of numerical simulations. In simulations, the flow between parallel plates due to traveling deformation waves on the solid film is modeled with Stokes equations and solved on a deforming mesh due to the motion of the boundary with the ALE method.

Numerical simulation results for amplitude and frequency dependence of the flow rate and the rate of work agree qualitatively well with the asymptotic behavior of the infinite-sheet placed in an infinite-medium given in [18,20]. According to simulations: (1) As amplitude increases, the flow rate, exit pressure and the efficiency of the micropump increases with the square of the amplitude; (2) The flow rate and the exit pressure vary linearly with the frequency while the rate of work done on the fluid varies with the square of the frequency, thus, resulting in efficiency of the micropump to remain constant with the increasing frequency. The effect of the wavelength varies with respect to the flow regime, which depends on the relative size of the wavelength with respect to channel’s dimensions. For small wavelengths, a somewhat steady flow in the channel results in due to drag effect of the train of waves on the fluid which results in a steady pressure distribution away from the film. For large wavelengths, on the other hand, a different regime that relies on the propagation of recirculations is observed by traveling-wave deformations on the film due to the pressure interaction between the film and the channel walls. For small wavelengths, the flow rate, exit pressure and power tend to decrease with increasing wavelength; the power decrease is proportional to 1/λ but the behavior is inconclusive for the flow rate and pressure. For large wavelengths, a somewhat stable window appears insofar as the flow rate, exit pressure and the rate-of-work done on the fluid increases with wavelength resulting in somewhat improvement of the efficiency of the micropump.

Finally, based on the numerical simulation results, the micropump must be operated at small wavelengths compared to the film’s length, and as high amplitude as the power requirements of the flow can be met. Even though, the maximum flow rate and the exit pressure of the pump increases with the frequency as well, efficiency of the pump does not improve with the frequency adjustments.

ACKNOWLEDGMENTS

We kindly acknowledge the partial support for this work from the Sabanci University Internal Grant Program (contract number IACF06-00418).

Figure 5: Time-averaged flow rate for zero pressure load (a), pressure rise for zero flow rate (b), pressure (c) and efficiency

(d) vs the wavelength.

(a) (b)

(c) (d)

Figure 6: Time-averaged flow rate for zero pressure load (a), pressure rise for zero flow rate (b), pressure (c) and efficiency

(d) vs the frequency.

(c)

(a) (b)

(d)

Figure 7: Time-averaged flow rate for zero pressure load (a), pressure rise for zero flow rate (b), pressure (c) and efficiency

(d) vs the amplitude.

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