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The Pennsylvania State University the Graduate School College of Engineering MICROACOUSTOFLUIDICS: an ARBITRARY LAGRANGIAN-EULER

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The Pennsylvania State University the Graduate School College of Engineering MICROACOUSTOFLUIDICS: an ARBITRARY LAGRANGIAN-EULER The Pennsylvania State University The Graduate School College of Engineering MICROACOUSTOFLUIDICS: AN ARBITRARY LAGRANGIAN-EULERIAN FRAMEWORK A Dissertation in Engineering Science and Mechanics by Nitesh Nama © 2017 Nitesh Nama Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2017 The dissertation of Nitesh Nama was reviewed and approved∗ by the following: Francesco Costanzo Professor of Engineering Science and Mechanics, Mathematics, and Mechanical & Nuclear Engineering, Center for Neural Engineering Intercollege Graduate Degree Program in Bioengineering Co-Chair of Dissertation Committee Dissertation Co-Advisor Tony Jun Huang Professor of Engineering Science and Mechanics Co-Chair of Dissertation Committee Dissertation Co-Advisor Jonathan Pitt Assistant Professor of Engineering Science and Mechanics Peter Butler Associate Dean for Education and Professor of Biomedical Engineering Judith A.Todd P.B. Breneman Department Head Chair Head of the Department of Engineering Science and Mechanics ∗Signatures are on file in the Graduate School. ii Abstract The lab-on-a-chip concept refers to the quest of integrating numerous functional- ities onto a single microchip for applications within medicine and biotechnology. To achieve precise fluid and particle handling capabilities required for such appli- cations, microacoustofluidics (i.e. the merger of acoustics and microfluidics) has shown great potential. The primary motivation of this dissertation is to revisit the formulation of the governing equations for microacoustofluidics in an fluid-structure interaction (FSI) context to develop a numerical formulation that is transparent with regards to the reference frames as well as the time-scale separation. In this context, we present a generalized Lagrangian formulation by posing our governing equations over a convenient mean configuration that does not coincide with the current configura- tions. The formulation stems from an explicit separation of time-scales resulting in two subproblems: a first-order problem, formulated in terms of the fluid dis- placement at the fast scale, and a second-order problem formulated in terms of the Lagrangian flow velocity at the slow time scale. Following a rigorous time-averaging procedure, the second-order problem is shown to be intrinsically steady, and with exact boundary conditions at the oscillating walls. Also, as the second-order prob- lem is solved directly for the Lagrangian velocity, the formulation does not need to employ the notion of Stokes drift, or any associated post-processing, thus fa- cilitating a direct comparison with experiments. Because the first-order problem is formulated in terms of the displacement field, our formulation is directly appli- cable to more complex fluid-structure interaction problems in microacoustofluidics. We also present a comparison of the generalized Lagrangian formulation with the typically employed Eulerian formulation and highlight the superior numerical performance of our formulation to aid easier comparison with the experimental observations. iii Next, we describe the fluid and particle motion in acoustically-actuated, con- fined, leaky systems. The investigated model system is a microchip typically used for biomedical analyses; a liquid-filled polydimethylsiloxane microchannel driven acoustically by inter-digital transducers. Through a combination of quantitative experimental measurements and numerical results, we reveal the full three di- mensional motion of the fluids and suspended particles. By varying the size of suspended particles, we capture the two limits for which the particle motion is dominated by the acoustic streaming drag to which it is dominated by the acous- tic radiation force. The observed fluid and particle motion is captured numerically without any fitting parameter by a reduced-fluid model based on a thermoviscous, Lagrangian velocity-based formulation. Through a combination of experimental observations and precise numerical boundary conditions, we remove the existing ambiguity in the literature concerning the acoustic streaming direction as well as the critical particle size for which the particle motion switches between the afore- mentioned two limits. Further, combining experimental and numerical results, we indirectly determine the first-order acoustic field of the vertically-propagating pseudo-standing wave as well as give an estimate of the actual displacement on the piezoelectric substrate. Through a combination of a simplified analytical model and our numerical results, we demonstrate that these “pseudo-standing” waves owe their origin to the small yet significant reflections from the fluid and channel wall interface and are the primary reason for the vertical focusing of particles. We present numerical results for the vertical focusing of both positive as well as neg- ative contrast neutrally-buoyant particles and provide the design criteria for the microchannels to achieve desired vertical focusing locations. We also demonstrate the ability to tune the focusing locations of particles in both horizontal as well as vertical direction by tuning the phase difference between the incoming surface acoustic waves (SAWs) and the applied acoustic power. Further, we outline the extension of the presented generalized Lagrangian for- mulation for FSI problems in microacoustofluidics. To this end, we present “proof- of-concept” results concerning the tracking of a fluid sub-domain inside a microa- coustofluidic device. iv Table of Contents List of Figures viii List of Tables xiv List of Symbols xv List of Acronyms xxi Acknowledgments xxii Chapter 1 Introduction to microacoustofluidics 1 1.1 Basic concepts of microacoustofluidics . 2 1.1.1 Acoustic streaming . 2 1.1.2 Acoustic radiation force . 3 1.2 Need for numerical studies . 4 1.3 Modeling approaches for microacoustofluidics . 5 1.3.1 FSI in microacoustofluidics . 6 1.4 Dissertation outline . 7 Chapter 2 Literature review 9 2.1 Basic notation . 9 2.2 Governing equations . 10 2.3 Perturbation expansion approach . 11 2.4 Concluding remarks . 17 Chapter 3 Generalized Lagrangian formulation 19 3.1 Basic notation and kinematics . 19 3.2 Governing equations . 23 v 3.3 Reformulation of the governing equations . 23 3.4 Time average, time scale separation, and orders of magnitude . 24 3.5 Mean fields and Lagrangian-mean fields . 28 3.6 Hierarchical boundary value problems . 31 3.6.1 Remarks on the balance of mass . 32 3.6.2 Zeroth-order subproblem . 33 3.6.3 Acoustic subproblem . 33 3.6.4 Mean dynamics subproblem . 34 3.7 Numerical scheme implementation . 38 3.8 Verification by the method of manufactured solution . 40 3.8.1 2D verification . 43 3.8.2 3D verification . 46 3.9 Concluding Remarks . 48 Chapter 4 Application to sharp-edge system 49 4.1 Introduction . 49 4.2 Microacoustofluidic channel with sharp edges . 50 4.3 Numerical formulation . 52 4.3.1 Model system and boundary conditions . 52 4.3.2 Mean trajectories . 54 4.4 Results . 56 4.4.1 Constitutive parameters . 56 4.4.2 Comparison of different formulations . 56 4.4.3 Remarks on the numerical implementation . 58 4.5 Concluding remarks . 61 Chapter 5 Application to standing surface acoustic wave (SSAW) 62 5.1 Introduction . 63 5.2 Physical effects . 65 5.3 Experiments . 68 5.3.1 Experimental setup . 68 5.4 Analytical model . 71 5.4.1 Analytical estimation of radiation force . 72 5.5 Numerics . 75 5.5.1 Governing equations . 75 5.5.2 Model system and computational domain . 78 5.5.3 Boundary conditions . 79 5.5.4 Single-particle acoustophoretic trajectories . 82 vi 5.6 Results . 83 5.6.1 Comparison of different formulations . 83 5.6.2 Experimental validation . 86 5.6.3 First-order fields . 89 5.6.3.1 Wall impedance sweep . 90 5.6.3.2 Phase sweep . 90 5.6.4 Particle trajectories . 92 5.6.5 Particle manipulation in vertical direction . 97 5.7 Concluding remarks . 99 Chapter 6 Extension to the IFEM 100 6.1 Brief review of FSI approaches . 100 6.2 Immersed finite element method . 102 6.3 Immersed finite element method for microacoustofluidics . 104 6.4 Fluid body tracking . 105 6.5 Concluding remarks . 107 Chapter 7 Future work and perspective 108 7.1 Summary of dissertation . 108 7.2 Future directions . 110 Appendix A Representation formulas 112 Appendix B Non-Technical Abstract 115 Bibliography 117 vii List of Figures 3.1 Schematic of the domains we consider. B0 denotes the reference configuration of our system. B(t) denotes the current configuration of the system. We view B(t) as the outcome of a deformation χξ from an intermediate configuration B(T ), which we view as the configuration of the body whose motion is observed at the slow time scale. The dashed line represents the trajectory of a material particle P (shown in the reference configuration). Adapted from ref. [1] . 20 3.2 2D Verification: Plot of error vs. mesh size for the first-order ver- ification analysis for (a) Q2 elements and (b) Q3 elements for the displacement, ξ.............................. 45 3.3 2D Verification: Plot of error vs. mesh size for the second-order ver- ification analysis for (a) Q2−Q1 elements and (b) Q3−Q2 elements for second-order Lagrangian velocity, vL and pressure, hq(x, y)i. 45 3.4 3D Verification: Plot of error vs mesh size for the first-order ver- ification analysis for (a) Q2 elements and (b) Q3 elements for the displacement, ξ.............................. 47 3.5 3D Verification: Plot of error vs mesh size for the second-order veri- fication analysis for (a) Q2−Q1 elements and (b) Q3−Q2 elements for second-order Lagrangian velocity, vL and pressure, hq(x, y)i. 47 4.1 (a) Schematic of the device showing a microfluidic channel with sharp edge structures on its side walls. The channel walls are sub- jected to a time-harmonic excitation produced by a piezoelectric transducer placed on one side of the channel. (b) Typical micro streaming patterns produced in the fluid occupying the channel as a response to piezoelectric excitation. (c) Typical geometric dimen- sions of the corrugated channel.
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