This dissertation investigates the fluid–structure dynamics involved in squeeze-film levitation, in pursuit of a theoretical explanation for the anomalously large, attractive load-bearing forces observed in recent experiments.

Consider a rigid disk vibrating rapidly along its axis of symmetry near a parallel wall, inducing oscillatory airflow in the thin film of air separating the disk and the wall as well as its periphery. Due to the nonlinear effects of fluid inertia and compressibility, the air pressure within the film varies in time asymmetrically about its ambient value, providing a nonzero time-averaged pressure force on the disk. Prior studies of this “squeeze-film” effect report that the time-averaged force typically repels the disk from the wall, making it a suitable mechanism for bearing lubrication. The strong attractive forces reported recently thus constitute a radical departure from this historical norm, expanding the range of possible applications to include wall-climbing robots and versatile contactless grippers. This dissertation aims to understand the fluid–structure physics underlying the emergence of strong attraction, through the derivation of reduced mathematical formulations that may ultimately aid the development of such novel systems.

Gas-lubricated systems represent an exceptional family of slender fluid flows for which the effects of compressibility and viscous shear enter simultaneously while maintaining small values of the Knudsen number, thereby guaranteeing applicability of the continuum hypothesis in describing the flow. Thus, the classical Navier–Stokes equations are used here to investigate the gas dynamics in the thin film as well as the effectively incompressible discharge and entrainment of air across a small region surrounding its edge. An approximate solution obtained using the method of matched asymptotic expansions indicates that the augmented attractive forces observed in recent experiments can be attributed to the pronounced dynamic bending of the highly flexible disks utilized. The theoretical formulation is then generalized to describe two-way fluid–structure coupling between the undulating disk and the thin-film airflow, thereby yielding predictions of system behavior that exhibit greatly improved agreement with experimental data. The canonical problems outlined herein can be readily extended to describe more complex configurations of practical interest—in particular, those involving transportation of the levitated object.

Flow that develops over a semi-infinite horizontal plate has been studied over the past decades. In this particular case, a boundary-layer flow over a horizontal hot plate is chosen and its stability characteristics at a finite distance from the leading edge are analyzed. A critical value of the Grashof number Gr based on the local boundary-layer thickness is defined and used to analyze the resulting instability.

Due to the nature of the hot plate, the Boussinesq approximation used in previous linear stability analyses becomes less desirable since wall-to-ambient temperature differences are not close to unity. And as a result, a non-Boussinesq analysis is needed and presented here for two instability modes: vortex, i.e. Gortler-like streamwise vortices, and wave, i.e. spanwise traveling waves.

Numerical results are presented such as the neutral stability curve, and critical Grashof number for Prandtl numbers of 0.7 over a wide range of wall-to-ambient temperature ratios. It is found that as this ratio increases, the susceptibility of the flow to the vortex mode of instability decreases while the wave mode instability becomes more prominent. The present study provides an approach suitable for both the vortex and wave instability modes with different wall-to-ambient temperature ratio. The results for the two modes are compared with each other as well as to other available instability data.

Cerebrospinal fluid (CSF) is a water-like fluid that surrounds the brain and the spinal cord, together known as the central nervous system (CNS). CSF provides a physical cushion for the CNS and also plays an important physiological role by maintaining the electrolytic environment, transporting hormones, circulating nutrients and chemicals filtered from the blood, and removing waste products from cell metabolism of the CNS. It is generally accepted that the absence of CSF circulation may compromise the normal physiological functions of the CNS. CSF circulation also provides a mechanism for the delivery of potent analgesics and chemotherapy to the CNS, a drug delivery procedure often referred to as intrathecal drug delivery (ITDD). Despite significant efforts, most of these processes remain poorly understood.

This dissertation analyzes CSF flow and solute transport along the spinal canal by using asymptotic methods based on the disparity of length and time scales associated with this problem. In addition to the oscillatory flow induced by the cardiac and respiratory cycles, with zero time- averaged velocity at any location, it is found that small corrections associated with the convective acceleration and canal deformation lead to a nonzero steady-streaming velocity. This small steady velocity, together with Stokes drift caused by the non-uniform oscillatory flow, determines the slow time-averaged Lagrangian motion of the CSF, which is found to be responsible for the transport of solutes along the canal. A key outcome of the analysis is a time-averaged transport equation that describes solute dispersion in the long-time scale. The use of this simplified equation circumvents the need to compute concentration fluctuations resulting from the fast oscillatory motion in the short-time scale, drastically reducing the associated computational times. The accuracy and limitations of the time-averaged description are tested by comparison to the results of direct numerical simulations spanning hundreds of oscillation cycles, as needed to generate significant dispersion of the solute. The analysis is extended to study effects of buoyancy-induced motion, arising when the injected solute has a density that differs from that of the CSF. For the small density differences that characterize ITDD drugs, buoyancy is found to have a significant effect on the convective transport, leading to large changes in the resulting solute-dispersion patterns. Finally, the classical problem of oscillatory flow past a circular cylinder is extended to the case of a streamwise periodic array of cylinders, providing insight regarding the effects of spinal-canal micro-anatomical features, such as trabeculae, ligaments, and nerve roots, on the flow and transport in the spinal canal.

Theoretical and numerical techniques are used to investigate three different problems involving non-isothermal fluid flow.

The first part of the dissertation investigates flame dynamics in a strained mixing layer established between two-dimensional counterflowing streams of fuel and oxidizer. When a one-step Arrhenius chemistry model is employed for the chemistry description, the numerical computations for small values of the stoichiometric mixture fraction yield a C-shaped premixed front with a trailing diffusion flame attached to one of its ends, a structure that is markedly different to the tri-brachial structure found in previous studies. Analytic predictions from the G equation are found to agree well with the numerical results. An explanation for these unexpected shapes is obtained by analyzing the variation with dilution of the burning velocity of planar premixed flames. Detailed-chemistry results are presented next for H$_2$-O$_2$-N$_2$ systems with high degrees of N$_2$ dilution, such that the resulting flame temperature lies close to the crossover value. Under these near-critical conditions, the flame dynamics is found to be strongly dependent on the initial conditions. A bifurcation diagram is presented in the stoichiometric mixture-fraction vs. strain-rate plane that identifies six different combustion regimes involving four different flame types, namely, one-dimensional diffusion flames, propagating/retreating edge flames, broken flame tubes, and isolated flame tubes.

A spherically symmetric heterogeneous fuel-oxidizer system is investigated next as an ignition model for hypergolic gelled propellants. Depending on the conditions, the fuel-oxidizer system may approach an explosive mode or it may settle into a steady-state mode. The critical condition defining the transition between these two states is determined analytically and the ignition time for the explosive mode is approximated by solving an integral equation for the interface temperature, employing activation-energy asymptotics.

A non-Boussinesq stability analysis of natural-convection boundary-layer flows over hot inclined surfaces is presented in the last part of the dissertation. Depending on the inclination angle, the disturbances are seen to evolve into either streamwise vortices or spanwise traveling waves. The critical inclination angle at which transition between the two instability modes occurs is calculated as a function of wall-to-ambient temperature ratio. For sufficiently large values of this ratio, wave modes are found to be always predominant, regardless of the inclination angle.

Asymptotic methods and experimental techniques have been used in combination with direct numerical simulations and stability analyses to investigate effects of buoyancy-driven motion in two different reactive problems, namely, flickering of nonpremixed diffusion flames in open atmospheres and slowly reacting combustion of confined mixtures. A brief description of these two separate topics is given below.

Part I of this dissertation deals with the flickering of jet diffusion flames and the puffing of liquid-fuel pool fires. Both phenomena are a manifestation of an axisymmmetric global hydrodynamic instability driven by the interactions of the buoyancy force with the

density differences induced by the chemical heat release, which occurs in the flame sheet separating the fuel and oxidizer domains. For these flows, the predictive capability of local quasi-parallel stability analyses is limited by the non-slender character of the resulting

eigenmodes, so that a biglobal analysis is needed to accurately determine marginal instability conditions and resulting frequencies. The results for jet diffusion flames, including the Froude number/Reynolds number instability boundaries for different fuel-feed dilutions,

are compared with direct numerical simulations, giving good agreement for the range of conditions explored in our study. For liquid-fuel pool fires, the stability analysis provides the critical value of the Rayleigh number at the onset of the puffing instability. The predictions for different liquid fuels are compared with the results obtained in small-scale laboratory experiments.

Part II is concerned with the “slowly reacting” mode of combustion, and its thermal-explosion limits, of an initially cold gaseous mixture enclosed in a spherical vessel with a constant wall temperature, a relevant problem in connection with the safe storage

and transportation of reactant gas mixtures. Following Frank-Kamenetskii’s seminal analysis of this problem, the strong temperature dependence of the effective overall reaction rate is taken into account by using a single-reaction model with an Arrhenius rate having

a large activation energy, resulting in a critical value Da_c of the controlling Damköhler number above which the slowly reacting mode of combustion no longer exists. A Rayleigh number Ra based on the relevant density difference is seen to measure the relative effect of

natural convection. Our numerical computations indicate that the value of Da_c increases with Ra as a result of the enhanced heat-transfer rate. Specific consideration is given to the flow structure in the asymptotic limits Ra << 1 and Ra >> 1, which yield accurate

predictions of critical explosion conditions. For completeness, the application of Frank-Kamenetskii’s ideas to the problem of flow of a reactive mixture in a pipe is presented in an appendix for the case of buoyancy-free conditions.

Asymptotic techniques are used to investigate two different phenomena, namely, acoustically driven counterflows and the flow field surrounding fire whirls.

In Part I of the dissertation, the interaction of non-premixed flamelets with acoustic waves of large characteristic wavelength, central to the development of acoustic instabilities in liquid-propellant rocket engines, is investigated using as model the counterflow diffusion flame subject to harmonic pressure and strain variations, with the presentation given in this work proceeding with increasing levels of complexity, outlined below.

In order to relate to typical experimental realizations of counterflow diffusion flames, the presentation begins with the investigation of the high-Reynolds and low-Mach number collision of two chemically frozen gaseous streams of different density. The self-similarity of the stagnation-point region is analyzed, with the strain-rate and stagnation point location, amongst other properties relevant for counterflow-flame studies, given as functions of the macroscopic properties of the experimental setup, including the nozzle-separation to semi-width ratio, for irrotational and rotational flows, with explicit formulas given for the former.

A general formulation is then provided for reacting mixing layers in counterflows subject to both time-varying strain and pressure using an inverse-thermal-conductivity-weighted coordinate which is shown to have benefits when compared to the classic Howarth-Dorodnitzyn variable. The formulation is applied to the interaction of acoustic waves with non-premixed flamelets by consideration of small amplitude harmonic oscillations of the pressure or strain-rate, with the amplitude serving as small parameter in the perturbative analysis for model one-step Arrhenius chemistry. First, the limit of infinitely fast reaction for non-unity Lewis numbers is considered. It is shown that differential-diffusion effects promote fluctuations of the flame location and reactant consumption rates. In connection with acoustic instabilities characterized by the Rayleigh index, the analysis predicts acoustic amplification for all frequencies in the pressure response, whereas a critical crossover frequency is identified in the strain response demarcating a transition between amplification and attenuation. Next, finite-rate effects are considered for systems with large activation energies. The results indicate the response for typical propellant combinations leads to acoustic amplification, not attenuation, the amplification being larger at higher strain rates.

These results, drawn on the basis of one-step model chemistry, are supplemented with numerical computations of hydrogen-air systems employing realistic chemical-kinetic mechanisms. The Rayleigh index is employed as a vehicle for quantifying inaccuracies of predictions caused by the introduction of reduced chemistry to decrease computation times. The computations indicate that inaccuracies of a systematically reduced 2-step mechanism, derived from a detailed 12-step mechanism for hydrogen-air systems, are small at low strain rates but become appreciable as extinction is approached.

Part II of the dissertation describes the steady axisymmetric structure of the cold boundary-layer flow surrounding fire whirls developing over localized fuel sources lying on a horizontal surface. The structure is shown to consist of three separate regions, including an outer inviscid swirling region, a near-wall boundary layer and a near-axis non-slender collision region, each described sequentially. Particular attention is given to the terminal shape of the boundary-layer velocity near the axis, displaying a three-layered structure described by matched asymptotic expansions. The resulting composite expansion, dependent on the level of ambient swirl, is useful in mathematical formulations of localized fire-whirl flows, providing consistent boundary conditions for further numerical investigations.

Neutrophil extravasation, a critical component in the innate immune response, comprises two sequential processes: transendothelial migration (TEM) and interstitial migration. TEM requires intimate physical contact between leukocytes and vascular endothelial cells, triggering a cascade of biochemical and biomechanical interactions whose effect on leukocyte migration in 3-dimensional (3D) matrices is currently unclear. To address this question, we employ a novel transwell (TW)-based leukocyte treatment assay in which human umbilical vascular endothelial cells (HUVEC) are cultured on a fibronectin-treated TW insert, mimicking the endothelium and basement membrane in vitro. Neutrophil-like differentiated HL60 (dHL-60) cells are loaded in the upper chamber of the insert and cells that cross the membrane-supported monolayer are collected from the lower chamber for further experimental use. TW treated cells are subsequently seeded in a well-characterized, custom-built chemotaxis device containing a collagen gel matrix to observe 3D migration in the presence or absence of an externally imposed chemotactic gradient. Analysis of nearly 50,000 dHL-60 and primary human neutrophil (PMN) trajectories, leveraging sequential and variational Bayesian inference algorithms, demonstrate that increases in neutrophil activity and concomitant loss of migratory persistence after TEM are modulated respectively and quasi-independently by HUVEC interaction and junctional trafficking. Moreover, neutrophils exhibit two distinct migratory subpopulations: one characterized by a fast, directed, chemotactically efficient high motility phenotype, and one characterized by a slower, tortuous, almost chemotactically agnostic low motility phenotype. These migratory subpopulations are conserved across treatment conditions and recapitulate the behavior of PMNs. The relative prevalence of these subpopulations controls the population level changes in neutrophil migration observed following extravasation and are modulated by TEM-mediated changes in neutrophil expression of GRK2, a kinase closely associated with chemokine receptor trafficking, migrational persistence, effector functionality, and collective neutrophil response to invading pathogens. In conjunction with observed population level increases in neutrophil effector functionality after TEM, these results suggest an important role for the low motility migration phenotype in promoting efficient pathogen clearance and motivate future investigation into the connection between neutrophil motility phenotype and effector functionality.