(\u00a3', r) = ( \u00a3 + C T , T ) . We may then eliminate any correction to the background equilibrium profile using C, and remove the term quj^ by suitably selecting the frame speed c. The result is our final amplitude equation, (1 - 2di)( are periodicity, the choice of origin (equation (2.57) is trans-lationally invariant) and the integral constraint, f Jo <_>(s)ds = 0 (2.58) Chapter 2. Dynamics of roll waves (il Figure 2.19: Top panel: Steadily propagating roll-wave solutions of the amplitude equa-tion for L = 4 and u = 0.04 (dotted) and fi = 0 (solid). The lower panel shows the real (solid) and imaginary (dashed) parts of an unstable eigenfunction with twice the spatial period as the basic roll wave. We use the integral of (p to display the eigenfunc-tion because

> (2-62) ( 2 ^ - l ) V - ^ - w \u00ab \u00ab = 0, (2.63) with ip an auxiliary variable and a the sought growth-rate. The solution proceeds by introducing another Bloch wavenumber, K, to gauge stability with respect to pertur-bations with longer spatial scale than the steady wave train. Numerical computations then provide the growth rate, Re(cr), as a function of K; an example eigenfunction of the weakly viscous solution shown in figure 2.19 is also displayed in a second panel of that picture. In the inviscid problem, the stability theory is complicated by the shock, which, in general, shifts in space under any perturbation. The shifted shock contributes a delta function to the linear solution. We take this singular component into account using suitable jump conditions: Integrating (2.62) and (2.63) with p \u2014 0 across the discontinuity, and allowing for an arbitrary delta-function of amplitude A in tp, gives: J a - ^ i T ( $ + - # + + fl = ip (2.64) ip+ -rp~ = IA. (2.65) A boundary-layer analysis based on the weakly viscous stability problem provides ex-actly these relations, except as matching conditions across the boundary layer. The regularity condition, ip \u2014 a0, must also be imposed at the singular point, $ = c. De-spite the lower-order of the linear stability problem, an analytical solution is not possible and we again solve the system numerically. Figure 2.19 once more compares inviscid and weakly viscous solutions. Typical results for the dependence of the growth rate of the most unstable mode on wave spacing, L, are shown in figure 2.20. Four values of the Bloch wave number are shown, corresponding to steady wave-trains with n = 1, 2, 3 and 4 waves, each a distance L apart, in a periodic domain of length, nL. As we increase the wave spacing, there is a critical value beyond which periodic trains with multiple waves become stable. This stabilization of multi-wave trains applies to general values of K and p, as illustrated by the neutral stability curves shown in figure 2.21. Thus, wave-trains with sufficiently Chapter 2. Dynamics of roll waves 63 Figure 2.20: Linear stability results of roll-waves using the amplitude equation for p = 1 (top) and u = 0 (bottom). Growth rate is plotted against wave spacing (L) for perturbations having a Bloch wavenumber of K = 2n\/nL (except for n = 1, where K = 0.) Chapter 2. Dynamics of roll waves 61 0.9 0.8 unstable 0.7 (1=2 \/ n-4 0.6 a. 0.5 0.4 0.3 stable 0_> 0 7.5 8 8.5 9 L 9.5 0.9 0.8 0.7 0.6 >-0.5 0.4 0.3 0.2 0.1 0 Figure 2.21: Stability boundaries for nonlinear roll waves on the (L, pt)-plane. The first panel shows the stability curves for n = 2. 3 and 4 (corresponding to roll-wave trains with n peaks in a periodic domain of size nL). The second panel shows the stability boundary for a single roll wave in much longer periodic domains. wide spacing become stable to subharmonic perturbations, removing any necessity for coarsening. Figures 2.20 and 2.21 also illustrate that at yet larger wave spacing, a different instability appears which destabilizes a single roll wave in a periodic box (n = 1). For these wavelengths, the nonlinear wave develops a long, flat tail resembling the unstable uniform flow. Hence, we interpret the large-L instability to result from perturbations growing on that plateau. We verify this character of the instability by solving the amplitude equation numerically, beginning from an initial condition close to the unstable nonlinear wave. Figure 2.22 illustrates how small disturbances grow and disrupt the original wave; eventually further peaks appear and four roll waves are present by the end of the computation, of which two are about to merge. Later still, the system converges to a steady train of three waves. In other words, trains with spacings that are too wide suffer wave-spawning instabilities that generate wavetrains with narrower separations. The combination of the destabilization of trains of multiple waves at lower spacing and the wave-spawning instability at higher spacing provides a wavelength selection mechanism for nonlinear roll waves. We illustrate this selection mechanism further in figure 2.23, which shows the results of many initial value problems covering a range Chapter 2. Dynamics of roll waves 65 Figure 2.22: A solution of the amplitude equation, beginning with an initial condition near an unstable roll wave. (L = 62 and p = 1). The dotted line in the final picture shows the initial condition. Chapter 2. Dynamics of roll waves 66 of domain lengths, d. Each computation begins with a low-amplitude initial condition with relatively rapid and irregular spatial variation. The figure catalogues the final wave spacing and displays the range over which trains of a given wave separation are linearly stable. Also shown is the wavelength of the most unstable linear eigenmode of the uniform equilibrium, which typically outruns the other unstable modes to create a first nonlinear structure in the domain. At lower viscosities (p), the most unstable mode is too short to be stable, and the inception of the associated nonlinear wave is followed by coarsening until the wave separation falls into the stable range. As we raise p, however, the most unstable mode falls into the stable range, and the nonlinear wave-trains that appear first remain stable and show no coarsening. Thus, viscosity can arrest coarsening altogether. 2 . 7 C o m p a r i s o n w i t h e x p e r i m e n t s We verified the predictions of the long wave equation (2.57) using laboratory experi-ments. Water was poured down an channel, inclined to the horizontal at an angle of about 7\u00b0. The channel was 18 m long, 10 cm wide and 3 cm deep. At the end of the channel, water was collected and re-circulated using a small centrifugal pump and the flow rate was controlled using a valve. A flow rate of 20 liters\/min was used which corresponds to a steady depth of 7 mm, speed of 65 cm\/s and a Froude number of 2.5. A video camera mounted above the channel recorded the propagation of these waves. Water was dyed red so that through the camera, deeper regions appeared darker. Thus color is a proxy for depth. By extracting columns of pixels from different frames of the recording, images similar to shown in figure 2.24 can be assembled. The figure shows the growth of small random perturbations to the water surface. Dark lines in the figure are crests, which are moving with a speed of 1 m\/s. The speed of the waves from the weakly nonlinear theory is about the same. The wavelength that appears first is roughly 0.57 cm corresponding to a nondimensional wavenumber of about 0.63. If one is to believe that the random perturbations at the inlet do not have any preferred frequency, then one would expect the fastest growing mode to be observed downstream. Under this assumption the observed wavelength corresponds to a value of p = 1 from (2.57), whereas it suggests p = 0.25 from the St. Venant equations themselves. This Chapter 2. Dynamics of roll waves 67 ( a ) n=1 d Figure 2.23: Final roll-wave spacings (crosses) in a suite of initial-value problems with varying domain size d and two values of \/z. The shaded region shows where nonlinear wave trains are linearly stable. Also shown are the stability boundaries of the uniform flow (dashed line) and the fastest growing linear mode from that equilibrium (dotted line). Chapter 2. Dynamics of roll waves 6S Figure 2.24: Roll waves appearing spontaneously on the flow on an incline. Color intensity shows perturbation from the mean, darker values representing deeper regions. variance in the value of p is expected as F = 2.5 may not be in the weakly nonlinear regime of the St. Venant equations, where (2.57) is valid. Since in the analysis we have used periodic waveforms, more controlled experiments were performed to better correspond with the theory. Periodic waves were forced at the inlet of the channel at different frequencies. A small paddle attached to a pendulum carried out this forcing. The swinging of the pendulum caused the paddle to carve out periodic waveforms on the flow. The forcing frequency could be changed by changing the length of the pendulum. This allowed us the generate almost-periodic wavetrains. To compare the steadily propagating roll wave profiles calculated from equation (2.57) with observed ones, we needed the instantaneous perturbation of the water surface at different locations in the channel. Measurement of the instantaneous height profile is extremely difficult (e.g. see [145]) and we resort instead to measuring a time series of the depth at a given location. The spatial dependence can then be inferred by assuming Chapter 2. Dynamics of roll waves 69 0 1 2 3 4 5 6 7 8 9 x Figure 2.25: Experimentally measured roll wave profiles (diamonds) compared with steadily propagating solutions of (2.57) for fi = 0.1 (solid line) and \\i = 0 (dashed line). The flow rate was set to 25 liters\/min and the average water depth was measured to be about 6 mm for an average speed of approximately 60 cm\/s. This corresponds to a Froude number of 2.5. This profile is obtained from a time series measured at a distance of 6 m downstream of the wave-generating paddle. that the waves are steadily propagating. The perturbation to the water surface was measured using a conductivity device similar to the ones used by Brock [126], which consists of a set of electrodes mounted above the water surface. As the water level rises, the electrode makes contact with the water surface and completes a circuit. By using 20 such electrodes at the same downstream location, but mounted at slightly different elevations compared to the mean, the instantaneous water level could be bracketed. The time series in figure 2.1 were obtained in this way. The measured profile is compared with periodic solution of (2.57) in figure 2.25. The magnitudes and the shapes of the measured profiles seem to be predicted very well by (2.57). Another estimate for the parameter \/x can be made using the \"width\" of the hydraulic jump as measured through the profile to be approximately 0.5. However, the measurement procedure used here is obtrusive to the flow, especially near sharp gradients, and hence the measurement of the width of hydraulic jump may not be accurate. Nevertheless, the measured profiles can be well approximated by using u. = 0 \u2014 0.1. A l l these estimates of fi suggest that the typical value is around 0 to 1. Chapter 2. Dynamics of roll waves 70 0 10 20 30 40 50 60 70 80 90 100 Time (sec) Figure 2.26: Intensity data from an experiments with periodic inlet perturbations. The forcing frequency is 1.57 s - 1 , which corresponds to a wavelength of 0.68 m. Chapter 2. Dynamics of roll waves 71 We turn our attention towards experimental observation of the coarsening instability. The fate of the waves formed by periodically forcing them with different frequencies at the inlet recorded in the form of pictures similar to figure 2.24. Results for some sample forcing frequencies is shown in figure 2.26-2.28. If the forcing frequency is sufficiently fast (e.g. figure 2.26), the distance between waves is short and coarsening instability disrupts the periodicity of the waves. As the wavelength is increased (e.g. figure 2.27), the periodic waves generated seem to be fairly robust. Small random perturbations do not disturb the flow very much, indicating stability for this wavelength. As the distance between the waves is increased further, as seen in figure 2.28, The periodic waves get disrupted again, but in this case via new waves spawning in between. The results from all the forcing frequencies are summarized in figure 2.29. The critical ripple distance at which waves become stable to coarsening is much larger than the value predicted by the weakly nonlinear theory. The reason for the disagreement could be that the experimental flow is not in the weakly nonlinear regime any more. On the other hand, the transition to nucleation of new waves seems to be better predicted. 2.8 Discussion In this article, we have investigated turbulent roll waves in flows down planes with topography. We combined numerical computations of both the linear and nonlinear problems with an asymptotic analysis in the vicinity of the onset of instability. The results paint a coherent picture of the roll-wave dynamics. The addition of low-amplitude bottom topography tends to destabilize turbulent flows towards long-wave perturbations, depressing the stability boundary to smaller Froude number. At moderate topographic amplitudes, an eye of instability also appears at much smaller Froude number, a feature connected to the development of topograph-ically induced hydraulic jumps in the background equilibrium flow (at least for the St. Venant model). At larger amplitudes, the topography appears to be stabilizing, and the onset of roll waves occurs at higher Froude numbers than expected for a flat bot-tom. This is consistent with observations of hydraulic engineers, who traditionally have exploited structure in the bed to eliminate roll waves in artificial water conduits, albeit Chapter 2. Dynamics of roll waves 72 Chapter 2. Dynamics of roll waves 73 0 10 20 30 40 50 60 70 80 90 100 Time (sec) Figure 2.28: Intensity data for a forcing frequency of 0.2 s\" 1 and a wavelength of 5.9 m. 20 40 60 80 100 L Fi gure 2.29: Experimental observations are compared with those predicted from weakly nonlinear theory (shaded region). The circles correspond to cases where coarsening was observed, the plus signs denote cases where periodicity was not disrupted, whereas the crosses denote observations of nucleation. Chapter 2. Dynamics of roll waves 74 usually in the direction transverse to the flow [146, 151]. We have also found that the reduced model furnished by asymptotic theory re-produces the nonlinear dynamics of roll waves. The model indicates that roll waves proceed through an inverse cascade due to coarsening by wave mergers, as found previ-ously [130, 144, 160]. This phenomenon was also observed in the experiments conducted by Brock [126] However, the cascade does not continue to the longest spatial scale, but instead becomes interrupted over intermediate wavelengths. Moreover, wave-trains with longer scale are unstable to wave-nucleation events. Thus, roll-wave trains emerge with a range of selected spatial scales. Although our results for low-amplitude topograpy are quite general, the discussion of instabilities caused by the hydraulic jump has surrounded a sinusoidal topographic profile and one may wonder how the results differ when the bed is more complicated. To answer this question we have made further explorations of the linear stability problem with a less regular form for In particular, we have tested the linear stability of equi-libria flowing over \"random topography\". Here, \u00a3 is constructed using a Fourier series representation; the coefficients of the series are chosen randomly from a normal distri-bution whose mean and standard deviations depend on the order of the Fourier mode. In this way, the topography conforms to a specific spectral distribution, as sometimes used in descriptions of the ocean's floor [123]. An example is shown in figure 2.30, which displays the realization of \u00a3, an inviscid equilibrium solution on the (n, H)\u2014 phase plane, and inviscid and weakly viscous stability boundaries on the (F, a)-plane. The overall conclusions are much the same as for the sinusoidal case: the inclusion of topography lowers the stability boundary below F = 2, and there is a close association with the formation of hydraulic jumps in the equilibrium. We close by remarking on the application of our results. We have considered shallow-water equations with drag and viscosity, focussing mostly on the St. Venant parame-terization for turbulent flows and briefly on the Shkadov model for laminar flows. We found that introduction of small, periodic, but otherwise arbitrary, topography desta-bilizes turbulent roll waves but stabilizes the laminar ones. For both kind of flows, the formation of a hydraulic jump in the equilibrium can further destabilize the flow (at least near the F2 \u2014 curve, if not near F = F i ) , a feature that may play a role in other physical settings. For example, carefully fabricated periodic ribbing in the elastic wall Chapter 2. Dynamics of roll waves 75 Figure 2.30: A computation with \u2022\"random\" topography: Panel (a) shows the realization of the topography and its derivative, constructed as follows: C, is built from a Fourier series in which real and imaginary parts of the amplitude, \u00a3\u201e, are drawn randomly from normal probability distributions with zero mean and standard deviation, (n 2 + 1 6 ) - 5 ' 4 , for n = 1,2, ...,32, and then a reality condition is imposed. In panel (b), we show the inviscid equilibrium for a = 3, kb = 10 and F = 1.9, together with the organizing curves, H3 = F2 and (1 \u2014 C,X)H3 = 1; the solution is about to form a hydraulic jump (and is marked by a star in panel (c)). Panel (c) shows the (shaded) instability region on the (a, F)\u2014 plane for v = 0; to the right of this region, the periodic equilibria cease to exist, and weakly viscous solutions develop hydraulic jumps. Also indicated are the viscous stability boundaries for v = 0.25 and 0.5; viscous equilibria are unstable above this curve. The dashed lines show the corresponding stability boundaries predicted by asymptotics (with theory A for v = 0.25 and 0.5 and theory B for v = 0). Chapter 2. Dynamics of roll waves 76 of a conduit may promote instability in the' related physiological and engineering prob-lems. In contrast, our results on the nonlinear dynamics of roll waves are more general; whatever the underlying physical setting, this model should equally apply. 2.9 References [122] S. V . Alekseenko, V. E. Nakoryakov, and B. G. Pokusaev. Wave formation on a vertical falling liquid film. AIChE J., 31:1446-1460, 1985. [123] N . J. Balmforth, G. R. Ierley, and W. R. Young. Tidal conversion by subcritical topography. J. Phys. Oceanogr., 32:2900-2914, 2002. [124] T. B. Benjamin. Wave formation in laminar flow down an inclined plane. J. Fluid Mech., 2:554-574, 1957. [125] D. J. Benney. 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Math., 105:143, 2000. 80 C h a p t e r 3 Flow induced elastic oscillations 3.1 Introduction Steadily forced flows interacting with elastic structures can spontaneously induce time-periodic oscillations. A commonly observed instance of such oscillations is the fluttering of a flag [162, 190]. In the pulp and paper industry, such oscillations are important to thin-film coating and paper production processes [167, 197, 198]. The disastrous Tacoma Narrows bridge collapse in 1940 and many others are also thought to be due to aeroelastic oscillations excited by a strong wind [183]. The flutter of an airplane wing or any of its other parts is yet another example where fluid-structure interaction can have severe consequences [192]. A brief review of some of these and other examples from engineering can be found in the articles by Shubov [193, 194]. In this chapter we study the oscillations excited by a fluid flow through a narrow channel interacting with an elastic structure. An example demonstrating this phe-nomenon is shown in figure 3.1. A through-cut is made in a freshly set block of gelatin and air is passed through it. The gelatin block starts to vibrate. Similar vibrations are seen in physiological systems, where these oscillations manifest themselves as audible acoustic signals [177]. Perhaps the most commonly experienced example of such oscil-lations is speech. Air flowing through the vocal cords causes them to vibrate producing sound. The dynamics of this process is of interest to the physiological community as well as computer scientists interested in speech synthesis. Lumped parameter models, pioneered by Ishizaka & Flanagan [180], have become popular to describe speech gen-eration, but more sophisticated one and two-dimensional models [179, 199] have also Chapter 3. Flow induced elastic oscillations 81 been solved numerically to understand the phenomenon. Other notable examples are the sounds made by blood flowing through partially open arteries. These sounds are called Korotkoff sounds and are routinely used by physicians in the measurement of blood pressure [165, 170, 173, 184]. One of the contending theories is that these sounds generated are due to an instability of the steady flow. Another motivation for studying flow induced elastic oscillations comes from what geologists term as \"volcanic tremor\". It is a sustained ~1 Hz seismic signal measured near volcanic sites, sometimes lasting for as long as months. The signal itself is some-times very harmonic and its spectrum has sharp peaks, although at other times it is broadband and noisy. A clear explanation of this tremor remains elusive, although sev-eral theories have been proposed [181]. One of these theories postulates that tremor is caused by magma or magmatic fluids flowing through cracks in rocks. Lumped param-eter models, similar in principle to those used for phonation, were employed by Julian [181], even the validity of this mechanism as a candidate is questionable [163]. A more careful analysis is needed to verify the feasibility of such models to explain volcanic tremor. There is also a considerable amount of interest in understanding the excitation mech-anism of wind-driven musical instruments. Fluid-structure interaction is an important factor for instruments involving reeds, e.g. clarinet, saxophone, etc. Understanding their mechanism is crucial to computationally synthesizing realistic music. The current state of research is a set of lumped parameter models [175, 176], in which a detailed modeling of the fluid dynamics is missing. An analogous problem is the excitation of acoustic modes in flutes and organ pipes by an air jet, referred to as air-reed instruments, similar to the sound made by blowing over beverage bottles. The role of elasticity in this problem is played by the compressibility of the resonating air column. In this case as well, lumped parameter models to explain the excitation exist however an accurate modelling of the jet from first principles is required [168, 174, 176]. These oscillations can be rationalized as a case of oscillatory instability of a steady equilibrium flow. We investigate one such mechanism for a linear instability, the one that excites the natural modes of elastic oscillations. In the absence of an externally driven flow and any significant damping, an initial disturbance causes the elastic structure to exhibit time-dependent oscillations. For example, when a tuning fork is struck, the Chapter 3. Flow induced elastic oscillations 82 prongs of the fork start to vibrate. These oscillations eventually decay because energy is lost due to radiation of sound to the surroundings, viscosity of surrounding fluid and any damping present in the elastic medium. However, if the fluid is now forced to flow by an external agency, it can exert additional hydrodynamic forces on the elastic structure and provide an energy source to the elastic oscillations. This can cause the elastic oscillations to grow, constituting an instability mechanism. This mechanism is central to the lumped-parameter analysis used for modelling phonation and musical instruments, and though questionable, a promising candidate for explaining volcanic tremor. However, certain assumptions about the flow or the elastic structure had to be made ad hoc in the lumped-parameter models. Moreover, a lot of detail was used in their construction to achieve quantitative accuracy [174, 176, 180, 181]. This obfuscated the underlying physical mechanism for exciting the oscillations. The equations had to be numerically analyzed to reveal the oscillatory instability of the steady state and that left the underlying mechanism unclear to intuition. Motivated by these shortcomings, we present an account of the fluid and solid me-chanics from first principles with the objective of isolating and demonstrating the un-derlying instability mechanism. The mechanism involving lumped parameter models alluded to in figure 1.2 was only uncovered to us as a result of the present analysis. This mechanism provides a unified approach to explaining the elastic oscillations seen in the various examples. As a specific example for demonstrating the instability and the accompanying analysis, we consider a fluid flowing through a channel of finite length, with the channel walls made up of a block of rectangular elastic material (the details are provided in \u00a73.2). This conceptual setup is motivated by and similar to the experiments with vibrations of the gelatin block depicted in figure 3.1. The elastic deformation is modelled by a Hookean elastic law, while the Navier-Stokes equations govern the fluid flow. Thus, this model is qualitatively and quantitatively faithful, albeit more compli-cated to analyse than lumped parameter models. Of course, our aspiration of uncovering the instability mechanism analytically is not possible for the problem in its full generality. We have to appeal to certain features of the setup that simplify the mathematics and allow us to make progress. The most important assumption we make is that the channel is long and narrow. This allows us to exploit certain models which are rigorously derived as approximations of the Navier-Chapter 3. Flow induced elastic oscillations to recording 1 | Microphone air < 1 1.5 Flow rate (lit\/min) 2.5 Figure 3.1: Details of the experiment on elastic oscillations in a gelatin block. A schematic setup of the experiment involving tremor of a gelatin block is shown in the upper panel. The base of the block is 9\" x 9\" and it was 3\" high. Compressed air is forced from the bottom to top through a knife-cut in the block (dimensions 2\" perpen-dicular to the plane of the paper in the top panel). As a critical flow rate is exceeded, the block starts to vibrate at a frequency of about 70 Hz. The microphone located over the block records the sound generated by these vibrations. The amplitude of the signal recorded is plotted in the lower panel as a function of the air flow rate. Chapter 3. Flow induced elastic oscillations 84 Stokes equations for the flows of thin films. This derivation is briefly outlined in \u00a73.3. As a result of these simplifications, the channel can be treated as one-dimensional with the only unknown flow quantities being the local channel width and the flow rate. A second considerable simplification comes from the assumption that the elastic structure is stiff as compared to the stresses in the fluid. This assumption renders the hydrodynamic forces weak in comparison to the elastic stresses. As a result, the dominant motion of the elastic structure is decoupled from the flow and can be explained in terms of its natural modes of oscillations. This description, to leading order is true irrespective of the precise details of the elastic structure. Be it an elastic beam, a stretched membrane or an extended elastic body, it possesses a set of elastic modes which determines its dynamics. As shown in \u00a73.4, an appropriately constructed asymptotic expansion then allows us to study the action of hydrodynamic forces on the normal modes of a the elastic structure. These two assumptions allow a unified treatment for all the previously mentioned examples and many more. When the elastic structure is not very stiff, its motion is coupled with the flow. Such a situation can not be studied analytically in general. Insight can still be gained through a computational solution with a simpler elastic structure. In \u00a73.5, we explore such a solution for the special case of the channel walls being formed by a stretched membrane. Finally, we exploit the analogy between the elastic and acoustic oscillations to de-velop a theory for the latter. The excitation of acoustic oscillations has long been attributed to the sinuous instability of an inviscid jet. This mechanism is reviewed in detail in \u00a73.6 and a simple experiment is devised to show that further investigation into the mechanism is required. The theory we develop is crude owing to the lack of a rigorous but simple model for the jet, unlike the model for thin films. The other feature that obscures the analogy is the absence of a clearly defined interface between the air jet and the air in the resonant cavity. Nevertheless, an ad hoc model is proposed based on the similarity with the elastic model, that acts as a proof of concept of the analogy and serves as a stepping stone to further experimental and theoretical analysis. With this picture in mind, we start with the mathematical formulation and non-dimensionalization of the governing equations. Chapter 3. Flow induced elastic oscillations 85 X (u,w,p) t 2H ' L Figure 3.2: Schematic setup for the mathematical model. 3.2 Mathematical formulation and non-dimensionalization A fluid of density p and kinematic viscosity v is flowing through a channel of average width 2H and length L (see figure 3.2). The flow is represented by the fluid velocity, u = (u, w) and a dynamic pressure p. The channel wall is located by the function 2 = h(x. t) and it separates the fluid from an linear elastic material. The displacement in this elastic medium, denoted by \u00a3 = (\u00a3, rj), is represented in a Lagrangian frame using the (X. Z)-coordinate system. The gradient operator acting on these displacements is denoted with a subscript 'X'. The displacement field is governed by the momentum balance law, Pstu = V* - TE + Vx - *-\u201e (3.1) where ps is the density of the solid, TE is an elastic stress and TV is a viscous stress. The elastic stress is given by the Hookean law r e = A e \/ ( V x - \u00a3 ) + \/ J e(V J C\u00a3 + V x \u00a3 T ) , (3.2) Here A e and pe are the Lame constants for the elastic material. They are related to the Young's modulus and the Poisson ratio as Poisson ratio = .. ^ e 7. Young's modulus = ffiS^f \u2014 (3.3) 2(Ae + pe) 6 Ae + Pe Chapter 3. Flow induced elastic oscillations 86 The viscous stress is given by the Newtonian constitutive relation T \u201e = AVI(VX \u2022 \u00a3t) + pvd(Vx$t + Vx&), (3.4) where Av and pVd are the coefficients of bulk and shear viscosities respectively. The total stress will be denoted by r = TV + re. The fluid in the channel is governed by the mass and momentum conservation equa-tions ut + uux + wuz + \u2014 = i \/V 2 u , (3-5) P wt + uwx + wwz + \u2014 = vV2w, (3.6) P ux + wz= 0. (3.7) These equations are accompanied by a set of boundary conditions and interface matching conditions. On the boundary of the solid we either have a no-displacement condition (\u00a3 = V \u2014 0) or the stress-free condition (r \u2022 n = 0, where n is the normal to the boundary). At the channel inlet, the fluid velocity specified. The fluid exit boundary condition is specified later in this chapter. The fluid and solid satisfy continuity of stress and velocity at the interface, which behaves as a material boundary; i.e., tt = \u00ab . (3-8) TZZ = . 1 , 0 (~P + 1pvwz - puhx(uz + wx)) (3.9) TXZ = , ((p - 2pvux)hx + pv{uz + wx)) (3.10) V1 + hx where the elastic displacements are evaluated at (X,0) and the fluid velocities and pressures at the Eulerian counterpart (a; = X + \u00a3, z = h(x)). For the fluid equations, we are heading towards a thin film approximation with H ~~ ( 3 - 2 9 ) cht + qx=0 (3.30) where RCQ = eR\/r is a rescaled Reynolds number, q = JQ udz is the volume half-flux and a = 12\/5, (3 = 6\/5, and r = 1 are constants. The above equation, sometimes called the Shkadov equation [191], models thin film flows over inclined planes [166, 178] qualitatively well in spite of the ad iioc nature of the assumption. This assumption is accurate when eR ~~

~~ a n d ip. The flux is related to the channel-width through (3.40), which gives q0(x) = -iu)0c(p{x), (3.51) where we have used the fixed flux condition at the inlet. As a matter of notation, subscript \"in\" refers to the inlet (x = \u20141\/2) and \"out\" to refer to the exit (x = 1\/2). Chapter 3. Flow induced elastic oscillations 95 Finally, substituting ho and qo in (3.41) yields Po(x) = poout +-{iw_e(3 + iuJocRc^ip + (iu)0caReq + 9)(<\u00a3 - 0 o u t) (3.52) + \/3i_eq(\/l0 - \/lOout) + (3 + fleqi-Jo)^0^} , where poout is the exit pressure of the fluid, used as an arbitrary constant of integration. Using the Bernoulli-like condition, it can be evaluated to po o ut = ^ o q 7 ( ^ o o u t + ^ o c ^ o u t ) -The frequency correction can be found at the next order of <5 and its imaginary part will indicate the stability of the steady state. At 0(<5), we have <^o\u00a3i + v x ' Tei + V x \u2022 T \u201e I - -2GJ0WI\u00a3 0, (3.53) with Telxz = -Tvixz and Teizz = ~Po ~ Tylzz at Z = 0 (3.54) and the homogeneous conditions at the other boundaries. The correction in frequency, CJI, is the object of our interest here, which can be found by taking a dot product of (3.53) with \u00a3 0 and integrating over the whole elastic domain (denoted by V). The left hand side of this product can be simplified by multiple applications of divergence theorem to give -2ccwi ( |\u00a3o| 2) = (\u00a3o ' (<^o\u00a3i + V x \u2022 r e i + V x \u2022 TV1)) (3.55) = p0h0 - ioJo (A\u201ei (V x \u2022 \u00a3 0 ) 2 + f i v l D 0 : D0) (3.56) where J{x) = J\\ fdx a n d (3-57) (f(X,Z))= f fdV, (3.58) Jv A> = V x \u00a3 0 + V x t f . (3.59) Only the imaginary part of wi contributes to the instability, the real part merely perturbs the frequency of free oscillations. The real part of the linear growth rate can then be written as <5<^i(V x \u2022 \u00a3 0 ) 2 + fiviDo \u2022 D0) ^^m^-V\" \u00ab.n ( 3'6 0 ) Chapter 3. Flow induced elastic oscillations 96 which, if positive, means instability. Thus the whole analysis boils down to the sign of the integral $$(po)ho, which depends on the fluid inlet and exit boundary conditions. This integral can be interpreted as the work done by the fluid pressure on the elastic boundary, alluded in the mechanism mentioned earlier. Substituting (3.52) in (3.60) and performing some integrations by parts, we get ( ) _ w>r\u00b0 q v 2) 0 J 2 ooo + '\"' (3.61) The first term on the right hand side of (3.61) represents the contribution of hydrody-namic forces, while the second term arises because of the viscous damping in the elastic material. In the absence of any fluid, the second term is responsible for decaying of the elastic normal modes. Even if the fluid in the channel is flowing, the steady solution is stable at Req = R = 0. The growth rate has a linear dependence on R. If the exit pressure is assumed to be fixed (7 = 0), then the coefficient of this linear dependence is negative and the growth rate becomes more negative with an increase in R. However, if 7 > a\/2, this coefficient can be positive for a sufficiently large value of R. The critical value of R in this case is given by (* c qW = 6 = + 2 ^ \u00ab V \u00ab jo)a> + *i~~