Analysis of Tesla-type valveless micropumps

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POLITECNICO DI MILANO

Scuola di Ingegneria Industriale e dell’Informazione

Corso di Laurea Magistrale in Ingegneria Aeronautica

Dipartimento di Scienze e Tecnologie Aerospaziali

Analysis of Tesla-type valveless

micropumps

Relatore: Prof. Nicola Parolini

Tesi di Laurea Magistrale di:

Davide Losapio

Matricola: 852441

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Ringraziamenti

Per primi voglio ringraziare i miei genitori e mio fratello. Senza i loro sacrifici, il loro continuo supporto e incoraggiamento nei momenti più difficili di questo lungo percorso universitario (senza contare poi il continuo e indispensabile rifornimento di barattolini, mozzarelle, taralli e tante altre amenità, senza il quale la permanenza a Milano sarebbe stata un inferno), non sarei oggi qui ad indossare una corona di alloro.

Un sentito ringraziamento va anche, e soprattutto, al mio relatore, il Prof. Nicola Parolini, per essere stato sempre disponibile ad aiutarmi e darmi consigli per lo sviluppo del lavoro di ricerca e per la stesura di questa tesi.

Grazie a tutti gli amici di Andria, in particolare a Sara, Arianna, Floriana ed Enrico, ormai amici di una vita su cui posso sempre contare, nonostante la distanza fisica che ci separa.

Ringrazio Mattia per i bei momenti trascorsi in questi ultimi anni di università e anche per la sua generosità quando si trattava di offrire gnocchi e cotolette a pranzo. Grazie ad Alberto, con cui ho trascorso una vita insieme e condiviso momenti strambi, spastici, al limite del surreale e, soprattutto, unici e autentici. Pensava di essersi fi-nalmente liberato di me con la fine della laurea magistrale, e invece... (Mi dispiace). Grazie ad Anna, una delle persone più sincere che conosca, e con cui non ho timore a confidarmi su qualsiasi cosa.

Ringrazio tutti gli altri miei compagni di università, e in particolare Valentina e Fed-erico per aver reso piacevoli questi anni di permanenza al Politecnico (soldaTina non dimentica).

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spero torni prepotente anche nel prossimo.

In particolare voglio ringraziare le "Petti" Clarissa e Debora per i momenti da Pomerig-gio Cinque passati insieme e non solo, che sono stati tra i più divertenti e autentici dell’ultimo anno passato a Milano, e che sono sicuro continueranno ancora.

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Abstract

The Tesla-type valveless micropump is analyzed using both numerical simulations and approximate solution methods. The first are based on the resolution of the 2D incompressible Navier–Stokes equations with the Finite Volume Method on un-structured grids, using the CFD open-source software OpenFOAM. The definition of geometries and meshes is done with the software Gmsh.

The main components of the micropump are first studied: the Tesla valves. The diodicity is the key parameter investigated: it describes the ability of the valve in passing flow in the forward direction while inhibiting flow in the reverse direction. Steady-state simulations are performed in order to characterize the diodicity of the T45A valve, presented in the work of Bardell [1], in the low-Reynolds number regime. Then, transient simulations are considered to investigate the dynamic behaviour of the same valve under oscillatory flow conditions. Due to the poor performances of the T45A valve, an optimized valve is considered to maximize diodicity, which is the result of optimization algorithms by Gamboa et al. [2].

2D Transient simulations of a valveless micropump incorporating optimum valves are then performed in the case of imposed harmonic motion of the membrane. Hence, the fluid-membrane coupling is neglected. Dynamic mesh handling is involved to take into account the movement of the membrane. The micropump performance is in-vestigated in terms of net flow produced and real efficiency. The results from the simulations are then compared with those obtained through lumped-system models developed for the micropump, inspired to the work of Tsui et al. [3, 4], still assuming harmonic motion of the membrane. A reasonable agreement in flow rate predictions and other performance quantities is found. Nonlinear fluid-membrane coupling mod-els are also developed, partially following the work of Pan et al. [5, 6], since in real applications nonlinear effects predominate in the vibration response due to the cou-pling between fluid and membrane. The net flow through the pump is characterized, and an optimal frequency, at which the system must be excited in order to obtain the maximum net flux, is determined.

Keywords

Valveless micropumps; Tesla valves; OpenFOAM; Computational Fluid Dynamics; Lumped-system analysis; Fluid-structure coupling.

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Sommario

Una micropompa alternativa del tipo Tesla è analizzata usando simulazioni nu-meriche e metodi di risoluzione approssimati. Le prime sono basate sulla risoluzione delle equazioni di Navier–Stokes incomprimibili in 2D con il Metodo dei Volumi Finiti su griglie non strutturate, usando il software OpenFOAM.

In una prima fase si è proceduto ad analizzare singolarmente la componente principale, ovvero la valvola di Tesla. La diodicità è il parametro chiave da studiare: esso descrive la capacità della valvola di far scorrere del fluido nella direzione favorev-ole e di impedirne il movimento nel senso contrario. Sono state svolte simulazioni stazionarie in modo da caratterizzare la diodicità della valvola T45A, presentata nel lavoro di Bardell [1], per bassi numeri di Reynolds. In seguito, sono state svolte sim-ulazioni instazionarie in modo da studiare il comportamento della valvola quando è sottoposta a condizioni di flusso oscillatorie. A causa delle limitate prestazioni della valvola T45A, si è deciso di utilizzare una valvola ottimizzata che garantisca diodicità superiori, che è il risultato del lavoro svolto da Gamboa et al. [2].

Sono poi state svolte simulazioni instazionarie 2D di una micropompa che include le valvole ottime nel caso in cui il movimento della membrana sia supposto armon-ico. In particolare, l’accoppiamento fluido-membrana è in questo caso trascurato. A causa del movimento della membrana, è stato anche tenuto conto del movimento della griglia. Sono poi analizzati il flusso netto prodotto e l’efficienza della microp-ompa. I risultati ottenuti dalle simulazioni sono stati poi confrontati con quelli di modelli a parametri concentrati del sistema, ispirati al lavoro di Tsui et al. [3, 4]. Si sono osservate concordanze in termini di flussi e di quantità prestazionali. Sono stati anche sviluppati modelli non lineari che accoppiano fluido e membrana, che seguono in parte il lavoro svolto da Pan et al. [5, 6], dato che nella realtà non si può prescindere da questo accoppiamento. Viene poi caratterizzato il flusso netto che scorre nella pompa e determinata una frequenza ottimale a cui eccitare il sistema in modo da garantire il massimo flusso netto possibile.

Parole chiave

Micropompe alternative; Valvole di Tesla; OpenFOAM; Fluidodinamica computazionale; Modelli a parametri concentrati; Accoppiamento fluido-struttura.

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List of Figures

1.1 A valveless micropump with Tesla-type NMP valves. Behind it, an electronic computer chip gives an idea about the spatial scale of the

system. Figure by Gamboa et al. [2]. . . 2

1.2 Illustration of the Tesla valvular conduit from Tesla’s patent [14]. . . 3

3.1 Control Volume. . . 22

3.2 Vectors d and S on a non-orthogonal mesh. . . 27

3.3 Vectors ∆ and k in the "orthogonal correction" approach. . . 29

3.4 Mesh deformation problem. . . 36

4.1 Sketch of the T45A valve by Nobakht et al. [15]. . . 47

4.2 Generic Tesla-type valve parametrization proposed by Bardell [1]. . . 49

4.3 T45A valve 2D geometry. . . 50

4.4 T45A valve - Physical surfaces. . . 52

4.5 T45A valve - Coarse mesh. . . 54

4.6 T45A valve - Volume flow rate response to an applied pressure differ-ence (at inlet). Backward flow case, ∆P = 0.1 atm, Medium grid. . . 61

4.7 T45A valve - Velocity magnitude field (units are m/s). Forward flow case. Medium grid. . . 63

4.8 T45A valve - X component velocity field (units are m/s). Forward flow case. Medium grid. . . 63

4.9 T45A valve - Y component velocity field (units are m/s). Forward flow case. Medium grid. . . 63

4.10 T45A valve - Velocity magnitude field (units are m/s). Backward flow case. Medium grid. . . 64

4.11 T45A valve - X component velocity field (units are m/s). Backward flow case. Medium grid. . . 64

4.12 T45A valve - Y component velocity field (units are m/s). Backward flow case. Medium grid. . . 64

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Medium grid. . . 65 4.15 T45A valve - Pressure drop (in atm units) versus Reynolds number for

forward and backward flow. Cross symbols are numerical data; curves are the fitted power-law relations, equation (4.3.4). Medium grid. . . 69 4.16 T45A valve - Log of pressure drop (in Pa units) versus log of 2D volume

flow rate for forward and backward flow. Cross symbols are numerical data; lines are the logarithm of the fitted power-law relations, equation (4.3.3). Medium grid. . . 69 4.17 T45A valve - Diodicity versus Reynolds number. The curve is the

ratio of the fitted power-law relations for the backward and forward flow directions, equation (4.3.5). Medium grid. . . 70 4.18 PIN versus time. The time is normalized by the period of oscillation

T, the pressure by the peak value Pmax. One period of oscillation is

depicted. . . 73 4.19 Outlet Volume flow rate response to an applied harmonic pressure

difference. Pmax = 0.5 atm, Medium grid case. . . 75

4.20 QOU T versus time superimposed on PIN versus time. Pmax = 0.5 atm,

Medium grid case. . . 76 4.21 QOU T versus time superimposed on QIN versus time. Pmax = 0.5 atm,

Medium grid case. . . 76 4.22 hQOU Tiversus time. Pmax = 0.5 atm, Medium grid case. . . 77

4.23 New parametrization of a generic Tesla-type valve proposed by Gam-boa, Morris and Forster [2, 17]. . . 79 4.24 Optimum valve 2D geometry. . . 81 4.25 Comparison between the T45A and optimum valves 2D geometries. . 81 4.26 Optimum valve - Physical surfaces. . . 82 4.27 Optimum valve - Coarse mesh. . . 83 4.28 Optimum valve - Pressure difference responses (in atm units) to

im-posed volume flow rates. Pressure differences are computed in a way to yield positive values. Re = 1500. . . 87 4.29 Optimum valve - Pressure drop (in atm units) versus Reynolds number

for forward and backward flow. Numerical data (circle symbols) are joined with straight lines. . . 89 4.30 Optimum valve - Diodicity versus Reynolds number. Numerical data

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4.31 Optimum valve - Log of pressure drop (in Pa units) versus log of 2D volume flow rate for forward and backward flow. Circle symbols are numerical data; lines are the logarithm of the fitted power-law

relations, equation (4.3.3). . . 90

4.32 T45A valve - Pressure difference responses (in atm units) to imposed volume flow rates. Pressure differences are computed in a way to yield positive values. Re = 1500. . . 91

4.33 Pressure drop (in atm units) versus Reynolds number for forward and backward flow. Numerical data (cross and circle symbols) are joined with straight lines. . . 93

4.34 Diodicity versus Reynolds number. Numerical data (cross and circle symbols) are joined with straight lines. . . 93

5.1 2D geometry of the micropump with optimum Tesla valves. L = 20wv, H = 3wv. . . 97

5.2 2D geometry of the micropump with optimum Tesla valves. L = 50wv, H = 2wv. . . 98

5.3 Micropump - Physical surfaces. . . 99

5.4 Micropump - Coarse mesh. . . 100

5.5 Micropump - Right valve. Coarse mesh. . . 101

5.6 Micropump - Left valve. Coarse mesh. . . 101

5.7 Volume flow rate response to an imposed oscillating velocity of the membrane. Positive flow rates are exiting from the domain, and vicev-ersa for negative flow rates. . . 108

5.8 Outlet/inlet flow rate ratio β in the period [6T + T/4, 7T + T/4]. . . 109

5.9 hQOU Tiversus time. . . 110

5.10 Valveless micropump electrical equivalent circuit. Steady model. . . . 113

5.11 Valveless micropump electrical equivalent circuit. Unsteady model. . 117

5.12 Volume flow rate responses to an imposed oscillating velocity of the membrane for the lumped-system models and the numerical simula-tion. Positive flow rates are exiting from the domain and viceversa for negative flow rates. . . 121

5.13 Outlet/inlet flow rate ratio β in the period [6T + T/4, 7T + T/4] for the lumped-system models and the numerical simulation. . . 123

5.14 hQOU T(t)i versus time for the lumped-system models and the numer-ical simulation. Tw = 2T . . . 124

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values of excitation frequencies. . . 142

5.18 Membrane vibration response versus coupling parameter β in terms of amplitude η0 and phase lag θ. . . 144

5.19 Membrane vibration response versus time in terms of η for different values of the coupling parameter β. . . 144

5.20 Net flux versus excitation frequency for different values of the loading amplitude F . . . 145

5.21 Sketch of a Tesla-type valveless micropump. Unsteady model. . . 146

5.22 θ versus ω/ω1 for different values of λ and F . . . 162

5.23 Π versus ω/ω1 for different values of λ and F . . . 163

5.24 Q00 net versus ω/ω1 for different values of λ and F . . . 165

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List of Tables

4.1 T45A valve - independent design variables values. . . 50 4.2 T45A valve - Grids used. . . 54 4.3 Forward flow - recorded 2D volume flow rates

"

m2

s

#

. . . 60 4.4 Backward flow - recorded 2D volume flow rates

"

m2

s

#

. . . 60 4.5 Forward flow - Relative errors on the 2D volume flow rates between

successively refined grids. . . 66 4.6 Backward flow - Relative errors on the 2D volume flow rates between

successively refined grids. . . 66 4.7 T45A valve - Parameters β and n of the power-law fit of ∆P versus

volume flow rate following equation (4.3.2). Medium grid case. The units of β are " P a · s m2 # . . . 67 4.8 T45A valve - Correlation coefficients R2 between the simulation data

and the calculated ones from the power-law functions, equation (4.3.2). Medium grid case. . . 67 4.9 Computed 2D net volume flow rates

"

m2

s

#

. . . 77 4.10 Optimum valve - Independent design variables values. Lengths are

normalized by the channel width wv. . . 80

4.11 Optimum valve - Parameters β and n of the power-law fit of ∆P versus volume flow rate following equation (4.3.2). The units of β are

"

P a · s m2

#

. 88

4.12 Optimum valve - Correlation coefficient R2 between the simulation

data and the calculated ones from the power-law functions, equation (4.3.2). . . 88

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after having performed steady-state simulations. The units of I are " P a · s2 m2 # . . . 120 5.3 Net flow rates and real efficiencies values for the lumped models and

the numerical simulation. . . 124

5.4 Optimum valve - Correlation coefficients R2 between the simulation

data and the calculated ones from the quadratic function, equation (5.4.4). . . 141 5.5 Input parameters for the fluid-membrane coupling steady model. L is

normalized by the channel width wv. . . 141

5.6 Derived parameters for the fluid-membrane coupling steady model. . 141 5.7 Fixed input parameters for the fluid-membrane coupling unsteady

model. . . 161 5.8 Variable input parameters for the fluid-membrane coupling unsteady

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Chapter 1 Introduction

1.1 Overview

MEMS (Micro-Electro-Mechanical Systems) devices consist of microfluidic systems whose purpose is that of moving very small quantities of fluid (of the order of micro-liters or nanomicro-liters) within them. The design and fabrication of microfluidic systems is an active research field since at least two decades and many of them have been in-cluded in microsystems for medical, biotechnical, environmental and microelectronics cooling applications.

A micropump is one of the most important devices in this field and most of the research works are devoted to this particular topic. Micropumps require the presence of microvalves in order to direct properly the flow, and they can be classified in active and passive microvalves, where mechanical and non-mechanical moving parts can be employed in both microvalve types [7, 8].

Of particular relevance is the so-called "valveless micropump", which is of recipro-cating type, and in a typical configuration it consists of a cylindrical chamber where the bottom surface is fixed, while the upper one is a flexible diaphragm that oscillates due to a periodic voltage excitation by a piezoelectric actuator, positioned on top of the diaphragm, and this causes a periodic volume change of the chamber. Moreover, no-moving-parts (NMP) microvalves, whose behaviour relies only on fluidic phenom-ena, are connected to the inlet and outlet of the chamber in order to direct the flow in the desired direction. The single NMP microvalve is designed so that when the flow occurs in the forward direction the hydraulic resistance is lower with respect to the case in which the flow takes place in the backward direction. Typically, NMP valves are used because they increase the pump reliability, simplify the fabrication process and are cheap. Moreover they allow the transport of fluid containing particles with

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Figure 1.1: A valveless micropump with Tesla-type NMP valves. Behind it, an

electronic computer chip gives an idea about the spatial scale of the system. Figure by Gamboa et al. [2].

.

size of the order of the valves’ channels width. An example of such micropump is shown in Fig. 1.1.

The working principle of the valveless micropump is as follows: there exists a suction phase, in which the diaphragm is moving upwards, leading to an increase of fluid volume inside the chamber and, due to the low pressure within it, the flow is directed into the chamber. The flow rate through the inlet is greater than that through the outlet. Contrarily, a pump phase exists when the diaphragm is moving downwards, leading to a decrease of fluid volume inside the chamber and, due to the high pressure within it, the flow is directed outside the chamber. The flow rate through the outlet is greater than that through the inlet. Consequently there will be a net transport of fluid from the inlet to the outlet of the micropump.

The main types of microvalves used in valveless micropumps are nozzle/diffusers and Tesla microvalves. The present thesis is focused on the study of the second

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1.1. OVERVIEW valve type, since many research works agree on the fact that the best performances in terms of flow-rectification efficiency are achieved by using Tesla valves [9, 10, 11, 12, 13]. Moreover, most of the works related to the study of valveless micropumps incorporate nozzle/diffuser valves rather than Tesla valves. This is another valid reason to explain the adoption of Tesla valves over the nozzle/diffusers.

The first Tesla valve was invented by the homonym inventor [14], and it is dis-played below.

Figure 1.2: Illustration of the Tesla valvular conduit from Tesla’s patent [14].

If the valve is filled with fluid (water, for example) and if a pressure difference exists between the inlet (marked as 5 in the figure) and the outlet (marked as 4 in the figure) there will be fluid motion from one side to the other. In particular, if the inlet pressure is higher with respect to the outlet one the flow will go from right to left following the path indicated by 7 (the forward direction). In the opposite case the flow will go from left to right crossing both the main channel and the curved ones (the backward direction). In the latter case the fluid travels more distance with respect to the former one and so the fluid is subjected to higher work done by the friction forces. Moreover vorticity generates at the end of each curved channel, where the flows rejoins. For these reasons the resistance that the fluid encounters when the flow is going from left to right is higher than that encountered when the flow is in the opposite direction.

The Tesla valve was invented for macroscale applications, where the channel widths are of the order of the centimeters and the pressure gradients are of the order of the atmosphere. As a consequence, this type of valve is known to perform well with fully-turbulent high-Reynolds-number flow, in the sense that the flow rates achieved by operating the valve in the forward direction are way higher than those obtained operating it in the backward direction. The particular valve shape invented by Tesla has then been adopted even for microscale applications, where the channel widths are of the order of hundreds of microns and still the pressure gradients are of the order of the atmosphere, such as in MEMS and, in this particular case, in valveless micropumps. Consequently laminar low-Reynolds-number flows are expected and many research works deal with this subject in order to develop an understanding of

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the fluidic mechanism of the Tesla type valve in this flow regime through numerical simulations [1, 15, 16]. The results can then be utilized in order to design valves with improved performances with the aid of shape parametrization, computational fluid dynamics and cost function minimization, discarding the so-called "build & test" method [2, 17].

The ability to pass flow in the forward direction while inhibiting flow in the reverse direction can be determined quantitatively through the diodicity of the valve, which is defined as: Di ≡ ∆Pbackward ∆Pf orward ! Q , (1.1.1)

where ∆Pbackward is the pressure difference between the outlet and the inlet of the

valve when a constant volume flow rate Q is imposed on the outlet, while ∆Pf orward is

the pressure difference between the inlet and the outlet of the valve when a constant volume flow rate Q is imposed on the inlet. The ratio of these two quantities defines the diodicity of the valve. By definition Di ≥ 1, otherwise the specifications of forward and backward directions would be interchanged. Moreover, a valve with Di = 1 would not produce net flow at all. So, the higher the diodicity of the valve, the higher the capability of inhibiting the backward flow in favour of the forward one. The pressure drops can be also decomposed in their independent and depedent parts on flow direction as follows:

Di = ∆Pindependent+ ∆Pdependent,backward

∆Pindependent+ ∆Pdependent,f orward

!

Q

, (1.1.2)

which shows that the diodicity is maximized when ∆Pindependent is minimized.

Diod-icity is also a function of Reynolds number (hence of flow rate) for a particular valve shape, and the functional relation varies significantly with it [10]. Typically 1 < Di < 2 for NMP microvalves, which is a relatively low value, but values higher than 2 can be obtained with optimized valves.

For what concerns the study of valveless micropumps employing Tesla valves, instead, linear models based on first principles have been developed, which take into account the coupling between the fluid and the membrane [18, 19], with no experimental information needed to predict the resonant behaviour. This model made it possible to determine optimal valve size. However, the literature is scarce in providing numerical simulation studies of this particular system, which would give more accurate results with respect to the cited linear models. Valveless micropumps employing nozzle/diffuser valves are instead frequently studied: Nguyen and Huang [20] reported a numerical simulation of pulse-width-modulated micropumps with

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1.2. AIMS nozzle/diffuser elements. The membrane vibration was modeled by either a moving wall or moving velocities as the boundary condition. The fluid-structure interaction is here neglected. Yang et al. [21] evaluated the performance of micropumps with two chambers arranged in parallel or series combination. A study including the coupling between the electrical, mechanical and fluid systems in valveless micropumps was performed by Fan et al. [22] using FEM and computational fluid dynamics. The behaviour of the membrane at different frequencies, that affect the pumping rate, was investigated. The fluid-structure interaction and the electro-mechanical coupling were also included in the studies of Yao et al. [23] and Jeong and Kim [24]. Nonlinear models based on the lumped-system method, that focuses only on time variation of the quantities, have been developed for the valveless micropump with nozzle/diffuser valves, where part of them assumes sinsuoidal motion of the membrane (the fluid-membrane coupling is neglected) [3, 4, 25] while others consider the fluid-fluid-membrane coupling [5, 6, 26]. In the study of Tsui and Lu [3], the flow in a valveless pump was analyzed using a static lumped-system analysis and CFD simulations. It was shown that the resulting inlet and outlet flow rate variations and pumping efficiencies obtained from the lumped model are close to the results obtained through CFD and experimental methods. Additionally, in the study of Pan et al. [5], the movement of the membrane was modeled by a PDE taken from the clamped-thin-plate theory. The equations were solved in an approximate manner by the small perturbation method and the Galerkin method. However, the flow unsteadiness was not taken into account. Afterwards, the inertial terms were then taken into account by Pan et al. [6] and it was shown that these terms were not negligible, compared with the viscous terms, in the dynamic coupling analysis. Moreover, the linear model presented in [18] can also be used when nozzle/diffuser valves are adopted instead of Tesla valves.

1.2 Aims

The present thesis aims at extending the analysis of Tesla-type valveless micropumps. In particular, the problem will be investigated through the following steps:

• perform 2D steady state simulations of a Tesla type valve in order to charac-terize its diodicity through the help of linear regression analysis and reveal the low-Reynolds-number diodicity mechanism of this type of valve;

• perform 2D transient simulations of the same Tesla valve used before in order to accurately predict the transient flow-rate response to an imposed harmonically-varying pressure between the inlet and the outlet of the valve and verify that

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a net flow going from the inlet to the outlet develops;

• perform 2D steady state simulations of an optimized Tesla type valve in order to characterize its diodicity and compare the results obtained with those of the valve considered before;

• perform 2D transient simulations of a valveless micropump incorporating the

optimized Tesla type valves considered before, where the velocity of the mem-brane is imposed and it is assumed to vary harmonically both in space and time. Hence, the fluid-membrane coupling is neglected and dynamic mesh handling is involved in these simulations. The performance of the micropump in terms of net flow produced, efficiency, and other useful parameters is investigated;

• develop a lumped-system model for the valveless micropump incorporating

Tesla valves assuming that the motion of the membrane is imposed and as-sumed to vary harmonically both in space and time (the fluid-membrane cou-pling is neglected). The model will be first illustrated in its steady version, where inertial effects of the flow are neglected and then it will be expanded by including an inertial term in order to capture properly the real behaviour of the fluid motion inside the micropump: this will be referred to as "unsteady model". From these models the transient flow rate response at the inlet and outlet of the micropump will be determined, so as the net flow rate that will develop and other useful performance quantities. In this way it will be easy to adjust geometrical parameters, frequency and amplitude of the membrane motion in order to meet specific requirements in terms of flow rates, for exam-ple. The results coming from the steady/unsteady models will be compared between them and with those obtained through the 2D numerical simulations; • develop an approximate model for the valveless micropump incorporating Tesla valves which includes the fluid-membrane coupling. The model will be first il-lustrated in its steady version, where inertial effects of the flow are neglected and then it will be expanded by including an inertial flow term in order to capture properly the real behaviour of the membrane displacement and of the fluid motion inside the micropump: this will be referred to as "unsteady model". To this end a simple 1D structural model for the membrane displacement and the incompressible version of the Navier–Stokes equations will be employed, where the pressure will be the variable that couples the equations that gov-ern the fluid and structural domains. The net flow rate that develops in the pump is determined, so as the optimal working frequency for the micropump and other quantities. Differences in the results obtained with the steady and unsteady models will be highlighted. Moreover, the results of steady and

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un-1.3. THESIS OUTLINE steady models will be analyzed by varying the values of different parameters, such as geometrical quantities, the amplitude of the piezoelectric force and the excitation frequency.

1.3 Thesis outline

The thesis is structured in the following way:

• Chapter 1 presents an overview on the valveless micropump and on the typical valve configurations used for this particular microfluidic system, provides a background on previous studies of NMP valves (Tesla-type valves, in particular) and valveless micropumps and lists the thesis objectives and structure;

• Chapter 2 presents the assumptions and the equations that govern the be-haviour of the fluid flow inside NMP valves and valveless micropumps. The integral version of the equations will be derived for both static and moving domain cases, while the differential version of the equations will be derived for the static domain case. A dimensionless form of the governing equations is also presented;

• Chapter 3 presents the numerical discretization and the solution algorithms adopted in order to solve the equations presented in the previous chapter. The moving mesh problem is also considered for the case of moving fluid domains, as it happens in valveless micropumps. OpenFOAM, the simulation tool that will be used to simulate the flow inside the valves and in the valveless microump, is also shortly described;

• Chapter 4 presents the definition of the 2D geometry and mesh of two different Tesla valves: the T45A [1] and one with high-diodicity capabilities, termed as "optimum valve" [2, 17]. The numerical methodology employed, the setting and resolution of the problem in OpenFOAM and the numerical procedures used in order to determine the diodicity of the valves over a low-Reynolds-number range when constant flow rates or pressure differences are imposed are also presented. The transient flow rate response of the T45A valve is accurately predicted when a harmonic pressure difference is imposed between the inlet and outlet of the valve. It is also verified that a net flow develops from the inlet to the outlet of the valve. The results obtained in terms of pressure drops versus imposed flow rates for the forward and backward directions and of diodicity versus Reynolds number of the two valves will be compared.

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• Chapter 5 presents the definition of the 2D geometry and mesh of a valve-less micropump incorporating the optimum Tesla valves and the setting and resolution of the problem in OpenFOAM when the motion of the membrane is assumed as harmonic. A lumped-system model for this system will be devel-oped in both its steady and unsteady versions. From the simulations and the models, the transient flow rate response at the inlet and outlet of the microp-ump and other useful performance quantities, such as the micropmicrop-ump efficiency, will be determined, and it will be verified that a net flow rate develops in the right direction. The results coming from the steady/unsteady models will be compared between them and with those obtained through the 2D numerical simulations, considered as benchmark solutions.

An approximate model for the same system that includes the fluid-membrane coupling is also developed in its steady and unsteady versions. The net flow rate that develops in the pump is determined, so as the optimal working frequency for the micropump and other quantities. Differences in the results obtained with the steady and unsteady models will be highlighted. Moreover, the re-sults of steady and unsteady models will be analyzed by varying the values of different parameters, such as geometrical quantities, the amplitude of the piezoelectric force and the excitation frequency.

• Chapter 6 summarizes the work done in the thesis, draws the conclusions about the addressed subject, and presents suggestions for possible future works.

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Chapter 2 The governing equations

In this chapter the assumptions and the equations that govern the behaviour of the fluid flow inside NMP valves and valveless micropumps. The integral version of the equations will be derived for both static and shape changing domain cases, while the differential version of the equations will be derived for the static domain case. A dimensionless form of the governing equations is also presented.

2.1 Assumptions

Even if NMP microvalves channel widths are rather small (the order of hundreds of microns), the continuum hypothesis still holds, so that the fluid can be treated as continuous. This means that it is possible to take fluid properties such as density, pressure, velocity, temperature as well defined over volume elements which are small in comparison to the length scale of the valve, but large in comparison to molecular length scale. In this way the fluid properties can be mathematically treated as continuous functions of space and time. The continuum hypothesis is no longer valid if Kn > 0.01, where Kn is the Knudsen number, defined as the ratio of a characteristic dimension of molecular structure to the characteristic length scale.

Since it is also assumed that the working fluid is water, a suitable characteristic dimension of molecular structure could be equal to several intermolecular spacings (for water, the intermolecular spacing lm is of the order of 10−10 m). Taking the

characteristic length scale of the valve equal to its channel widths (≈ 10−4 m) it is

clear that the resulting Kn is far less than 0.01, so that the continuum hypothesis holds.

Considering isothermal conditions and the fact that the typical velocity scales involved are such that the Mach number is low (Ma < 0.3) it can also be stated

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that the flow is incompressible and the viscosity is constant (since viscosity is mainly temperature dependent for water).

Throughout this work it is also assumed that 2D domains will be employed in order to lower the computational effort of CFD simulations and that low-Reynolds-number flow will be investigated such that the flow regime is always laminar. It will be later described in more details how to distinguish a laminar flow from a turbulent one when dealing with NMP microvalves, and what are the consequences of having one type of flow regime or the other inside the valves in terms of pressure drops and flow rates.

2.2 Mass conservation

The first governing equation for the fluid flows that we consider is the one that expresses the mass conservation.

Let CM be a control mass defining a certain portion of fluid that moves in space with a velocity given by the vector field u = u(x, t). The mass of fluid present in the CM at a given time t is:

m ≡

Z

ΩCM

ρ(x, t)dΩ,

where ΩCM denotes the volume occupied by the CM and ρ(x, t) denotes the density

of the fluid at the point x at the instant t.

The mass conservation principle applied on the CM states that m must be con-stant, hence independent from time:

dm dt = 0 ⇒ d dt Z ΩCM ρ(x, t)dΩ = 0. (2.2.1)

Often it is more convenient switching to a control volume approach, where only the flow within a certain spatial region, called control volume (CV ), is considered, so that the left hand side of the previous equation can be written as [27]:

d dt Z Ω ρdΩ + I ∂Ω ρ(u − ub) · ndS = 0, (2.2.2)

where Ω is the volume occupied by the CV , ∂Ω is the surface enclosing the CV , n is the unit vector normal to ∂Ω pointing outwards, u is the fluid velocity on ∂Ω, ub is

the CV surface velocity. This equation states that the rate of change of mass in the

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2.2. MASS CONSERVATION to the CV boundary is null. For a fixed CV , ub = 0, and the first derivative of the

previous equation becomes a partial derivative:

Z Ω ∂ρ ∂tdΩ + I ∂Ω ρu · ndS = 0. (2.2.3)

Equations (2.2.2) and (2.2.3) express the integral form of mass conservation for a moving CV and a fixed one, respectively.

Applying the Gauss’ divergence theorem to the surface integral term of equation (2.2.3) we obtain: Z Ω " ∂ρ ∂t + ∇ · (ρu) # dΩ = 0,

and exploiting the arbitrariness of the volume Ω, the previous integral relation can be written as a differential equation:

∂ρ

∂t + ∇ · (ρu) = 0. (2.2.4)

This is the differential or local form of mass conservation, known also as continuity

equation.

Since we are assuming that the flow is incompressible it can be stated that:

Z ΩCM dΩ = const → d dt Z ΩCM dΩ = 0.

This assumption on fluid motion is called incomprimibility and it states that the value of the volume of the CM must be constant in time. The left hand side of the last equation can be written also as:

d dt Z Ω dΩ + I ∂Ω(u − ub) · ndS = 0, (2.2.5)

as we’ve done previously when passing from equation (2.2.1) to equation (2.2.2). For a fixed CV , ub = 0, and the first derivative of the previous equation becomes a

partial derivative: Z Ω ∂1 ∂tdΩ + I ∂Ωu · ndS = I ∂Ωu · ndS = 0, (2.2.6)

Equations (2.2.5) and (2.2.6) express the incomprimibility for a moving CV and a fixed one, respectively.

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Applying the Gauss’ divergence theorem to equation (2.2.6) we obtain:

Z

Ω

∇ ·udΩ = 0,

and exploiting the arbitrariness of the volume Ω, the previous integral relation can be written as:

∇ ·u = 0. (2.2.7)

This is the differential version of the incomprimibility condition and it states that the velocity field must be solenoidal. Equation (2.2.7) is also called incomprimibility

constraint. Using the incomprimibility constraint in the continuity equation after

developing the divergence term results in:

∂ρ

∂t + u · ∇ρ = 0,

which is a transport equation for ρ. Assuming that the initial density is uniform, namely ρ(x, 0) = ρ, the solution to the previous equation is ρ(x, t) = ρ, ∀t > 0 and ∀x, and so the density is constant. For this reason, the incomprimibility constraint is equivalent to the continuity equation when considering a fixed CV and a constant initial density.

The integral form of mass conservation for a fixed CV , then, becomes simply:

I

∂Ωu · ndS = 0,

which means that the volumetric flux through ∂Ω must be null.

If we instead consider a moving CV , it can be shown that in order to ensure mass conservation the so-called space conservation law must be enforced, otherwise artificial mass sources could appear when the equations are discretized [27]:

d dt Z Ω dΩ − I ∂Ωub ·ndS = 0. (2.2.8)

Inserting this last equation in the incomprimibility condition for a moving CV , equa-tion (2.2.5), results in getting equaequa-tion (2.2.6). So, the incomprimibility condiequa-tion for a moving CV is equal to that for a fixed CV , and, reasoning as already done before, the density is constant when considering a uniform initial density. Then, from equation (2.2.2) and equation (2.2.8), it follows that the integral form of mass conservation for a moving CV can also be expressed as:

I

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2.3. MOMENTUM BALANCE and that the differential form of mass conservation for a moving CV coincides with the incomprimibility constraint. Hence, the mass conservation is stated in the same way both in NMP valves, that involve a fixed CV , and valveless micropumps, that instead involve a moving CV .

2.3 Momentum balance

Now we consider another governing equation: the so-called momentum balance. It corresponds to the application of Newton’s II law on an arbitrary fluid volume, which states that in every inertial frame of reference the rate of change of momentum of a fluid volume is equal to the sum of the forces applied on it.

Let’s take in consideration the same control mass defined before. The fluid mo-mentum present in CM at a given time t is:

q ≡Z ΩCM ρ(x, t)u(x, t)dΩ = Z ΩCM ρudΩ.

Hence the application of Newton’s II law on the CM can be written as:

dq dt = F → d dt Z ΩCM ρudΩ = F,

where F is the vector of applied forces on the CM. Switching to a control volume approach, as done for mass conservation, allows to write the the previous equation as: d dt Z Ω ρudΩ + I ∂Ω ρu(u − ub) · ndS = f, (2.3.1)

where f is the vector of applied forces on the considered CV . This equation states that the rate of change of momentum in the CV plus the momentum flux through the CV boundary due to the fluid motion relative to the CV boundary is equal to the sum of the applied forces on the CV . For a fixed CV the previous equation becomes: Z Ω ∂(ρu) ∂t dΩ + I ∂Ω ρuu · ndS = f. (2.3.2)

f can be split in two different contributions: f = f∂Ω+ fΩ,

where f∂Ω denotes the surface forces (due to pressure, viscous stresses, etc.), while fΩ

denotes the body forces (gravity, electromagnetic forces, etc.). fΩ can be written as:

fΩ =

Z

Ω

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where b represents the body forces per unit mass. f∂Ω is instead defined as:

f∂Ω =

Z

∂ΩT · ndS, (2.3.4)

where T is the surface stress tensor, which takes into account the contributions due to pressure and viscous stresses. Hence it can be shown that T can be written as:

T = −pI + σ, (2.3.5)

where p is the static pressure, I the unit tensor and σ the viscous stress tensor. Inserting equation (2.3.5) into equation (2.3.4) and then using what obtained and equation (2.3.3) into equations (2.3.1) and (2.3.2) we get for a moving and fixed CV , respectively: d dt Z Ω ρudΩ + I ∂Ω ρu(u − ub) · ndS = − Z ∂Ω pndS + Z ∂Ω σ ·ndS + Z Ω ρbdΩ. Z Ω ∂(ρu) ∂t dΩ + I ∂Ω ρuu · ndS = − Z ∂Ω pndS + Z ∂Ω σ ·ndS + Z Ω ρbdΩ. (2.3.6)

These two equations represent the integral forms of momentum balance for a moving and a fixed CV , respectively.

Using the gradient theorem for the first term on the right hand side of equation (2.3.6), the Gauss’ divergence theorem on the remaining surface integral terms of the same equation, and exploiting the arbitrariness of the volume Ω we get:

∂(ρu)

∂t + ∇ · (ρuu) = −∇p + ∇ · σ + ρb, (2.3.7)

that is the conservative version of the differential form of momentum balance for a fixed CV . A non conservative version of the last relation can be obtained by developing the time derivative and the divergence term on the left hand side of equation (2.3.7). This results in getting:

ρ∂u ∂t + ρ(u · ∇)u + u " ∂ρ ∂t + ∇ · (ρu) # = −∇p + ∇ · σ + ρb.

Using mass conservation in the form of equation (2.2.4), the third term on the left hand side of the equation is null and we obtain:

ρ∂u

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2.3. MOMENTUM BALANCE that is the non conservative version of the differential form of momentum balance for a fixed CV .

Now we need to make an adequate assumption about the viscous stress tensor so that the system of equations made by mass conservation and momentum balance becomes closed. One way to do this is that of assuming that the fluid is Newtonian, meaning that the viscous stresses are linearly proportional to the local strain rate through some properly defined viscous coefficients. The Newtonian model applies well to water, which is the working fluid we are considering. For Newtonian fluids the viscous stress tensor can be expressed as:

σ= 2µD − 2

3µ(∇ · u)I, (2.3.9)

where µ is the dynamic viscosity coefficient (which is constant in our case, as we said in the previous section) and D is the rate of strain tensor:

D = 1₂

h

∇u + (∇u)Ti

.

For incompressible flows the second term in equation (2.3.9) is null due to the in-comprimibility constraint so that:

σ= 2µD.

The divergence of the viscous stress tensor can then be computed as:

∇ · σ= ∇ ·hµ∇u + (∇u)T i= µh∇2u + ∇(∇ · u)i= µ∇2u,

where the incomprimibility constraint is exploited.

Substituting this in equations (2.3.7) and (2.3.8), and remembering that for in-compressible flow with uniform initial density the density is constant (ρ = ρ) we obtain, respectively: ∂u ∂t + ∇ · (uu) = − ∇p ρ + ν∇ 2u, ∂u ∂t + (u · ∇)u = − ∇p ρ + ν∇ 2u,

where the incomprimibility constraint has been used once again, ν = µ/ρ is the kine-matic viscosity coefficient (which is constant in our case), and the body forces have been neglected since the surface forces play a dominant role in the considered prob-lem. The last two equations represent the conservative and non conservative version, respectively, of the differential form of momentum balance for an incompressible flow.

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2.4 The incompressible Navier–Stokes equations

The system of equations that govern the incompressible flow of a viscous (newtonian) fluid, having uniform initial density, hence constant density, when body forces are neglected, in a domain Ω ⊂ R2, is then given by:

     ∂u ∂t + (u · ∇)u = −∇p + ν∇ 2u, x ∈ Ω, t > 0, ∇ ·u = 0, x ∈ Ω, t > 0, (2.4.1) where p is now the pressure divided by the density (but we still denote it as p). These equations form the so-called incompressible Navier–Stokes equations. Both are partial differential equations, but the first one is vectorial, while the second one is scalar. The unknowns are the velocity field u(x, t) and the pressure field p(x, t), while ρ and ν are given positive constant quantities. The nonlinear term (u · ∇)u

describes the process of convective transport, while ν∇2u the process of molecular

diffusion. It is worth observing that the pressure field p(x, t) plays the role of a Lagrangian multiplier associated to the constraint ∇ · u = 0, that the velocity field

u(x, t) must fulfill.

In order to obtain a well-posed problem it is necessary to specify proper initial and boundary conditions. Regarding initial conditions, we need to set only one (vectorial) initial condition for the velocity field and none for the pressure field since the Navier– Stokes equations comprehend a first-order in time equation for the velocity and no evolution equations for the pressure. Hence the initial velocity is specified as:

u(x, 0) = u0(x), ∀x ∈ Ω,

where u0(x) is a known divergence-free vector field, so that the incomprimibility

constraint is satisfied by it. The boundary conditions specification is a more delicate question, instead. A partition of the domain boundary ∂Ω is made so that we can define a Dirichlet boundary subset ΓD and a Neumann one ΓN such that ΓD∪ΓN =

∂Ω, ΓD∩ΓN = ∅. Suitable boundary conditions for the Navier–Stokes equations are

then written as:

   u(x, t) = gD(x, t), ∀x ∈ ΓD, t >0, h ν∇u + (∇u)Tn − pni(x, t) = g N(x, t), ∀x ∈ ΓN, t >0, (2.4.2) where gD and gN are given vector functions. The first is a Dirichlet type boundary

condition which prescribes the value of the velocity vector on ΓD: it is used either

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2.4. THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS onto a solid boundary which is moving. In the latter case it is said that a no-slip boundary condition is applied. Clearly, if gD = 0 we are referring to a non moving

solid boundary. It is worth noticing that if ΓD = ∂Ω, the pressure appears only

through its gradient in the equations; then if (u(x, t), p(x, t)) is a solution of the system (2.4.1), (u(x, t), p(x, t) + c(t)) is also a solution of the same system, where

c(t) is an arbitrary function of time, since ∇(p + c) = ∇p. In order to solve this

issue and determine uniquely the pressure, it is usually required that the pressure has a null average (R

ΩpdΩ = 0). Moreover, the prescribed Dirchlet data gD must

satisfy the following compatibility condition:

I

∂ΩgD

·ndS = 0,

since, if we apply the Gauss’ theorem to this equation we get:

I ∂ΩgD ·ndS = Z Ω ∇ ·udΩ = 0,

which is null due to the incomprimibility constraint.

If ΓN is not empty, the problem about the indetermination of the pressure

dis-appears, as on ΓN the pressure p appears without derivatives. The Neumann type

boundary condition, which is the second one appearing in equation (2.4.2), prescribes the normal component of the total stress (that take into account pressure and vis-cous stresses contributions) on ΓN. In particular when gN = 0 the subset ΓN is

called a free outflow. Moreover, when velocity gradients are null on ΓN, the

pre-scribed Neumann data gN correspond to the pressure on the same boundary. Here

the most common type of boundary conditions have been illustrated. For a more comprehensive discussion refer to [28, 29].

The Navier–Stokes problem is then formulated as follows: given a partition of the

domain boundary ∂Ω in two non empty sets ΓD and ΓN such that ΓD ∪ΓN = ∂Ω,

ΓD ∩ ΓN = ∅ and the vectorial functions u0 : Ω → R2, gD : ΓD × R+ → R2,

gN : ΓN × R+ → R2, find (u, p) such that:

                       ∂u ∂t + (u · ∇)u = −∇p + ν∇ 2u, (x, t) ∈ Ω × (0, T ], ∇ ·u = 0, (x, t) ∈ Ω × (0, T ], u(x, 0) = u0(x), x ∈ Ω, u(x, t) = gD(x, t), (x, t) ∈ ΓD ×(0, T ], h ν∇u + (∇u)Tn − pni(x, t) = g N(x, t), (x, t) ∈ ΓN ×(0, T ],

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2.4.1 Dimensionless form

The Navier–Stokes equations, equation (2.4.1), can be made dimensionless by intro-ducing a characteristic length L and a characteristic velocity U for the problem under exam. In this way 4 parameters appear in the incompressible Navier–Stokes prob-lem: L, U, ρ, µ. Nevertheless, it can be shown that, once that a particular shape for the domain has been chosen (the shape of the valve or micropump), all the possible incompressible flows that can develop in that particular domain are determined by a family of solutions where only one parameter is varying.

The following dimensionless quantities are defined:

e x ≡ x L, ∇f = L∇, g∇ 2 = L2_∇2_, ˜t≡ t T = U t L, ∂ ∂et = T ∂ ∂t, T = L U, e u ≡ u U, p ≡e p ρU2.

Using these definitions and doing the proper substitutions, the dimensionless form

of the incompressible Navier–Stokes equations is given by (for the sake of simplicity

the tilde symbol over the dimensionless quantities is not shown):

     ∂u ∂t + (u · ∇)u − 1 Re∇2u + ∇p = 0, x ∈ Ω, t > 0, ∇ ·u = 0, x ∈ Ω, t > 0,

where Re is a dimensionless number called Reynolds number, that determines the one-parameter family solutions satisfying the dimensionless incompressible Navier– Stokes equations, and is defined as:

Re = U L

ν = ρU L

µ .

This dimensionless number gives an estimate of the ratio of inertial forces to viscous forces and, consequently, quantifies the relative importance of these two types of forces, given ρ, U, L, µ.

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Chapter 3 Numerical discretization and

solution algorithms

In this chapter the process of numerical discretization applied on the previous equa-tions and the algorithms employed in order to obtain the solution of the problem will be described. The discretization process consists in transforming partial differential equations into a corresponding system of algebraic equations. This system is then solved in pre-determined locations in space and time, hence providing an approxi-mate solution to the starting equations. Moreover, the discretization procedure can be divided in two steps: solution domain discretization and equation discretization [30, 31].

The solution domain discretization can be further split in space and time domains discretization. The space domain discretization produces a numerical description of the space domain, dividing it into a finite number of control volumes (CV s), or cells, over which the equations will be numerically solved. The time domain (present in transient simulations) is also discretized into a finite number of time steps. The equation discretization, instead, transforms the terms inside partial differential equations into appropriate algebraic expressions.

The moving mesh problem is also considered for the case of shape changing fluid domains, as it happens in valveless micropumps. The differences about the results of the discretization with respect to the static mesh case are highlighted, and the defini-tion and automatic resoludefini-tion of the mesh deformadefini-tion problem through a particular mesh motion solver is described.

The CFD open-source software OpenFOAM is adopted to simulate the flow inside the valves and in the valveless micropump, and is here shortly described.

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3.1 OpenFOAM: a CFD open-source software

The simulation tool that will be used to simulate the flow inside the valves and in the valveless micropump is OpenFOAM (Open Field Operation and Manipulation). It is an open-source CFD software package that, nowadays, is used in different areas of engineering and science, and especially in the academic environment due to its usage flexibility. In fact OpenFOAM possesses different features thanks to which it is capable of solving different problems, such as those involving fluids, solid mechanics, electromagnetics and chemical reactions.

The structure of OpenFOAM is based on a flexible and efficient set of C++

modules, which also have the advantage of being extendable and modifiable, that are

used to build different application executables. The modules are needed to create tools accessible to the applications, such as libraries that implement constitutive laws and physical models (turbulence modelling and so on). The applications fall mainly into two categories:

• solvers, through which it is possible to simulate different problems in various engineering and science fields;

• utilities, that are designed to perform pre- and post-processing tasks involving, for example, data manipulations and visualization and mesh processing; OpenFOAM comes already with many pre-built libraries, solvers and utilities so that it can be used without bothering to the coding aspect. Nevertheless, the user can extend the collection of libraries, solvers and utilities, if he needs it (pre-requisite knowledge of physics and programming techniques, such as object-oriented program-ming, are clearly involved).

It is important to know that OpenFOAM uses the Finite Volume Method (FVM) to discretize systems of partial differential equations on 3D meshes that could be either structured or unstructured with polyhedral cells. Pre-built libraries that are capable to handle properly the dynamic mesh motion (including topological changes) are also present. Another fundamental aspect regards the possibility of running simulations in parallel: domain decomposition based parallelism is integrated at a very low level so that it is unnecessary to write any particular code for this purpose when dealing with the development of a solver.

One of the main reasons that lead to the choice of using OpenFOAM as the CFD simulation tool for the present thesis is that it is free and open-source, so that the source code can be modified in order to meet specific objectives. This is in contrast with commercial CFD softwares, that have very high license costs and whose source

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3.2. THE FINITE VOLUME METHOD codes cannot be accessed or modified. For these reasons OpenFOAM has a large user base.

3.2 The Finite Volume Method

The so-called Finite Volume Method (FVM) is employed for the discretization process and it is here described. This choice stems from the fact that OpenFOAM adopts this numerical method. The FVM adopted in this work is characterized by the following properties:

• it is based on the discretization of the integral form of the governing equations over every single CV . In this manner, quantities such as mass and momentum are conserved even at the discrete level. Hence the FVM is a conservative

method;

• equations are solved in a fixed Cartesian coordinate system on the mesh. The method can be employed both in steady-state and transient problems;

• meshes can be arbitrarily unstructured, meaning that CV s can be polyhedrals with a variable number of faces and neighbours. All dependent variables share the same CV s, that is to say that a colocated (or non-staggered) variable ar-rangement is adopted [32, 33];

• Systems of partial differential equations are treated in the segregated way [34, 35]. Hence they are solved one at a time, with the inter-equation coupling treated in an explicit manner. Nonlinear differential equations are linearized before the discretization and the nonlinear terms are lagged.

3.3 Solution domain discretization

As previously said, the discretization of the solution domain provides a computational domain on which the governing equations are solved. The process of discretization is split into two parts: space and time discretization. In this way the locations in space and time where the numerical solution is sought are determined.

Concerning time discretization, it suffices to prescribe the time interval and time step of the simulation. Regarding space discretization, the Finite Volume Method requires a subdivision of the physical domain into a finite number of CV s. These

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The numerical solution is sought at the centroid of the CV s, which is defined as the point P that satisfies:

Z

ΩP

(x − xP)dΩ = 0,

where ΩP is the CV whose centroid is P , x is a vector defining the coordinates of

a point belonging to ΩP and xP is the vector that contains the coordinates of the

point P .

Each CV is bounded by a set of flat faces, and each face is shared with only one neighbouring CV . Since the mesh is arbitrarily unstructured, it is assumed that the

CV has a generic polyhedral shape with a total number of nf faces. The CV faces

can be classified into two types: internal faces, that lie between two control volumes, and boundary faces, that lie on the boundaries of the physical domain.

For each CV face, a face area vector Sf can be defined. This vector is normal

to the face, has magnitude equal to the area of the face and points outwards from the cell with the lower label for internal faces, while it always points outwards for boundary faces. The cell with the lower label is called the owner while the other cell sharing the same face (with an higher label) is called the neighbour. Hence, the owner cell centroid corresponds to the point P and we call N the point that corresponds to the neighbour cell centroid. The face of a generic CV is marked with f, which also defines the centre of the face. A typical control volume (having a polyhedral shape and drawn in three dimensions) is shown in Fig. 3.1.

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3.4. DISCRETIZATION OF THE DIFFUSION-TRANSPORT EQUATION The Finite Volume Method behaves well with arbitrarily unstructured meshes, and this turns out to be useful when dealing with complex geometries, since struc-tured or block-strucstruc-tured grids are difficult or almost impossible to generate in that case, while grid-generating algorithms exist for unstructured meshes. Another ad-vantage that arbitrarily unstructured meshes have is that local grid refinement, which is needed in parts of the domain where high accuracy is requested (large gradients, for example), is easier to implement with respect to other types of meshes.

3.4 Discretization of the diffusion-transport

equa-tion

The discretization procedure is now illustrated for a generic diffusion-transport tion. The same procedure can be used in order to discretize the Navier–Stokes equa-tions, but, in that case, we have to pay particular attention to the nonlinearity of the momentum equation and to the pressure-velocity coupling. These issues will be addressed in the following sections.

Let’s consider now the generic diffusion-transport equation for the unknown scalar quantity φ(x, t): ∂(ρφ) ∂t | {z } unsteady term + ∇ · (ρUφ) | {z } convection term − ∇ ·(Γ∇φ) | {z } diffusion term = qφ(φ), | {z } source term (3.4.1) where ρ is the fluid density, U is the fluid velocity, Γ is the fluid diffusivity and

qφ(φ) is the source term. All the quantities are assumed to be known except for φ.

Equation (3.4.1) is a second-order equation, as the diffusion term presents the second derivative of φ in space. In order to obtain a good level of accuracy, it is necessary for the order of the discretization to be equal to, or higher than, the order of the equation that has to be discretized.

The accuracy of the discretization procedure depends on the assumed variation of the scalar function φ(x, t) in space and time around the point P . It can be shown (through a Taylor series expansion around P of φ) that the variation of φ(x, t) must be linear both in space and time in order to obatin a second-order accurate method. This requirement is translated as follows:

φ(x) = φP + (x − xP) · (∇φ)P,

φ(t + ∆t) = φt+ ∆t ∂φ ∂t

!t

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where φP = φ(xP), φt= φ(t), (∇φ)P is the gradient of φ computed in P ,

∂φ ∂t is

the time derivative of φ computed at t, t is the current time and ∆t is the chosen time step.

The Finite Volume Method requires that the integral form of equation (3.4.1) is satisfied over the control volume ΩP around the point P defined previously:

Z t+∆t t " ∂ ∂t Z ΩP ρφdΩ + Z ΩP ∇ ·(ρUφ)dΩ − Z ΩP ∇ ·(Γ∇φ)dΩ # dt = Z t+∆t t Z ΩP qφ(φ)dΩ ! dt. (3.4.3)

3.4.1 Spatial discretization

In this subsection, we examine the spatial discretization of the terms of equation (3.4.3), starting from the first term on the left hand side of it, and taking into account the assumed linear variation of φ in both space and time inside the control volume.

Unsteady term

The volume integral is replaced by the product of the mean value of the integrand and the CV volume. The former is then approximated with the value of the integrand at point P : ∂ ∂t Z ΩP ρφdΩ = Z ΩP ∂(ρφ) ∂t dΩ = ∂(ρφ) ∂t ΩP ≈ " ∂(ρφ) ∂t # P ΩP, (3.4.4) where " ∂(ρφ) ∂t # P

stands for the value of ∂(ρφ)

∂t at point P . By using equation (3.4.2)

it is possible to show that equation (3.4.4) is a second-order accurate approximation.

Convection term

Using Gauss’ divergence theorem in order to transform the volume integral into a surface one, and remembering that the control volume is bounded by nf flat faces,

the resulting surface integral can be written as a summation of integrals over all the faces: Z ΩP ∇ ·(ρUφ)dΩ = I ∂ΩP (ρU · nφ)dS = Xnf f =1 " Z Sf (ρU · nφ)dS # , (3.4.5)