Predicting Breakup Characteristics of Liquid Jets Disturbed by Practical Piezoelectric Devices
|
|
- Merilyn Campbell
- 6 years ago
- Views:
Transcription
1 ILASS Americas 2th Annual Conference on Liquid Atomization and Spray Systems, Chicago, IL, May 27 Predicting Breakup Characteristics of Liquid Jets Disturbed by Practical Piezoelectric Devices M. Rohani, D. Dunn-Rankin, and F. Jabbari Department of Mechanical and Aerospace Engineering University of California Irvine, Irvine, CA Abstract In this paper, we study the breakup characteristics of a jet of liquid. To break a capillary jet into droplets, a piezoelectric is often used to generate disturbances growing along the jet. Rayleigh s linear theory predicts that uniform droplets are produced when the jet is perturbed by a single wavenumber disturbance. However, studies limited to linear behavior are unable to predict how the jet behaves if it is subjected to a multiplefrequency input. Measurements of an actual piezoelectric s dynamics show that driven with single frequency harmonic signal, it disturbs the jet with three output sine waves: one steady state response and two very lightly damped modes corresponding to the structural resonances of the device. In this paper, we study the interaction of these output waves to estimate the range of frequencies where irregularity effects are likely to occur on the surface of the jet that might lead to nonuniform droplet formation. Control strategies designed to eliminate these unwanted dynamics can then retain uniform breakup over a wide range of input frequencies.
2 Introduction Spray and atomization has been an important topic of research for a long time because of its broad applications in combustion, inkjet printing, surface coating, and many others. In particular, hydrodynamic instability of capillary jets has been studied to control the resulting droplets sizes and spacings. Theories on the instability of the jets are mainly inspired by the pioneering work of Lord Rayleigh in 878 []. Rayleigh studied the capillary instability that grows temporally in an infinitely long jet. Rayleigh s work was modified in 93 by Weber, [8], who took ambient fluid and liquid viscosity effects into account. In 973, Keller et al. studied the spatial instability of a capillary jet. They introduced an instability governed by a spatially growing disturbance that is initiated at the base of the jet. This spatially oriented configuration is more applicable to our experiment compared to the temporal instability configuration which would require a jet of infinite length simultaneously excited along its entire length. They also showed that temporal and spatial theories coincide at high Weber numbers [2]. Later in 986, Leib and Goldstein argued that the Keller theory is not valid for Weber numbers below a critical value and that the jet is absolutely unstable in the low Weber number range [3, 4]. Since studies limited to linear theories are unable to explain some phenomena such as satellite droplet formation, much emphasis has been placed on studying nonlinearities. In 974, H. C. Lee used nonlinear theories to study satellite formation [5], and Pimbley et al. used spatial instability analysis to explain nonuniform droplet formation, both experimentally and theoretically [6]. Explaining satellite formation to a high degree of accuracy requires nonlinear studies. However, in this paper, the goal is to avoid nonlinear studies by determining the interval of wavenumbers in which irregularities on the surface of the jet tend to occur. Subsequently, we introduce certain control strategies to eliminate satellites in that interval. In the first section, we discuss theories that explain the disturbance growth on the surface of the jet stream. Then, we consider the breakup characteristics of a jet perturbed by a composition of disturbances. Finally, we suggest a control method to eliminate the undesirable droplets that would be formed due to nonlinear effects. Theory Disintegration of a jet of fluid is a result of the Kelvin-Helmholtz instability and capillary instability [7]. The Kelvin-Helmholtz instability results from differences in velocities of the jet and its surrounding fluid. It can be shown that the effect of the Kelvin-Helmholtz instability is absent in a jet ejected into the vacuum due to the lack of hydrodynamic interaction between the jet of fluid and its ambient [7]. In this case, only the capillary instability acts on the jet. Capillary force, which is the surface tension over the radius of the jet, increases by narrowing the stream and results in a capillary collapse. Therefore, in the presence of both an ambient and surface tension, a combination of Kelvin- Helmholtz and capillary instability leads the jet to breakup. Rayleigh considered capillary instability in an infinitely long incompressible inviscid fluid cylinder, neglecting the ambient. He showed that any disturbance with a wavelength larger than the circumference of the cylinder jet grows exponentially until it pinches into droplets. For small disturbances, the problem can be solved linearly and the solution for the radius of the jet in time and space is: r(x,t) r s = Re[ε e i(kx ωt) ], () where x is the axis along the jet, r s is the unperturbed jet stream radius, and ε is the initial amplitude of an axisymmetric disturbance applied to the stream. Rayleigh studied the disturbances that grow with time while having a sinusoidal oscillation in space. The assumption that Rayleigh made represents a temporal instability. This requires that k, the wave number, be real and ω be a complex number consisting of the disturbance frequency and the temporal growth rate as real and imaginary parts, respectively, ω = ω R + iω I. In the case of an unstable perturbation, ω I, the growth rate, is greater than zero. For practical purposes, spatial instability is taken into account by assuming a complex k (k = k R + ik I ) and real ω in Equation () [2]. The imaginary part of k brings about the disturbance growth in space. Since ω is real, the disturbance oscillates without any amplification in time. Equation (2) can be rewritten for the spatial instability as: r(x,t) r s = Re[ε e i(ωt) e x(β ik) ], (2) where k R is the wave-number and k I is assumed to be the temporal growth rate when it is large relative to the Weber number [2, 7]. To study breakup characteristics of the jet, it is of interest to understand how the growth rate varies with the disturbance wavenumber. Considering the
3 stabilizing effect of viscosity and the surrounding fluid, Weber derived an expression for the growth rate in terms of the wavenumber and the fluid characteristics [8]: β 2 + 3µ(k ) 2 ρ s rs 2 β σ(k ) 2 2ρ s rs 3 ( (k ) 2 ) V s 2 ρ a (k ) 2 K (k ) 2ρ s rsk 2 (k, ) (3) where µ is the stream dynamic viscosity, ρ s is the density of the liquid in the stream, k = 2πrs λ is the non-dimensional wave number of the disturbance, r s is the unperturbed jet radius, σ is the surface tension of the stream fluid, V s is the stream speed, ρ s is the ambient fluid density, and K n is the n th order Bessel function of the second kind. Fig. shows the plot of the growth rate expression derived by Weber as a function of input frequency. Growth rate (s ) 2.5 x Input Frequency (Hz) x 4 Figure. Variation of growth rate as a function of input frequency. For the disturbance to be unstable, the nondimensional wavenumber has to be less than one. The peak represents the fastest growth rate that causes the shortest pinch-off time when the amplitude of the disturbance is uniform for all wavenumbers. This is the desirable range of operation for practical systems, since it is the region most likely to produce uniform droplets. Breakup Analysis As discussed earlier, linear theory can be used to investigate breakup characteristics of a jet, subjected to a single frequency disturbance. In this section, we explain how the jet behaves in the presence of a combination of distortions applied to the surface of the jet, based on actual piezoelectric forcing behavior. In general, practical oscillating devices are such that, induced with a single frequency signal, their output consists of other modes with different frequencies representing structural responses of the device. As shown in previous work [9], it is observed experimentally that for the device used in our study, there always exist two very lightly damped modes corresponding to the structural resonances of the device. A 4 th order function that represents the observed response is determined based on experimental frequency response data. Therefore, the output of the transfer function driven with a harmonic signal oscillating at driving frequency, f d, consists of three components as shown in Fig. 2. The first component, called the input mode, corresponds to the permanent response of the transfer function and oscillates with the driving frequency. Two other output components, known as the structural modes, correspond to very lightly damped resonant modes that oscillate at natural frequencies of the actuator, (6kHz and 6kHz, for the device we used, [9]). Fig. 3 shows the amplitudes of A(f d ), B(f d ), and C(f d ) of Fig. 2 as the frequency of input is swept from 4kHz to 8.5kHz. We have scaled the amplitudes of the output components by dividing them by a constant value equal to the maximum amplitude occurring at 6kHz. According to the linear theory, all of the perturbation components should behave independently of the effects of other components. Hence, we can use superposition of a total of N growing disturbances to express the radius of the jet. This operation is formulated as: r(x,t) r s = N Re[ε j e i(ωjt).e x(βj ikj) ] (4) j= Unfortunately, although linear theory agrees well with experimental results for a single frequency perturbation, it does not explain the jet behavior in the case of multiple components over the entire frequency range. The interaction of disturbance components gives rise to nonlinearities, especially close to the pinch-off, and the result is no longer predictable with linear theory. Superposition can provide a good estimation result only when there is one dominant wave which dominates the effect of other components. Having a dominant disturbance causes the jet to act as if it is disturbed by a single frequency disturbance. Without any interference from the other modes, linear theory predicts a uniform breakup for the jet. Here, we investigate the interaction of waves and describe how they compete with each other for growth. We expect to observe nonuniform droplets sizes and spacings in most of the regions where most competitions between modes occur.
4 Figure 2. Three harmonic components are produced at the output of the actuator induced by a single frequency input. (a) Scaled amplitude (b) Scaled amplitude (c) Scaled amplitude x x Input mode oscillating with driving frequency First structural mode oscillating at 6Hz Second structural mode oscillating at 6Hz x 4 Figure 3. Amplitude of the disturbance components as a function of input frequency. (a), (b), and (c) respectively show A(f d ), B(f d ), and C(f d ), which are the amplitudes of the output components shown in Fig. 2. Interaction among disturbances As mentioned earlier, the actual disturbance consists of three harmonics with initial amplitudes varying with input frequency. According to Equation (3), the growth rate is a function of the wavenumber. Therefore, for the input mode, the growth rate varies with driving frequency variation, whereas structural modes have constant growth rates as depicted in Fig. 4. Considering that growth rate has a more significant effect on breakup compared to the initial disturbance amplitude, one can conservatively say that the mode with growth rate near the peak, i.e., the second structural mode, tends to grow much faster than the disturbance at the first structural mode. Therefore, we only need to study the interaction of the second structural mode and the driving mode, except at frequencies very close to the first natural frequency, where the first structural mode has an effect on the pinch-off due to its large amplitude. It is only near this resonant frequency that the amplitude has a dominant effect on breakup. In this case, the resonant mode and the driving mode have equal frequencies, amplitudes, and growth rates. This is thus similar to disturbing the jet with a single frequency perturbation that has a very large amplitude. Hence, the pinch-off times due to both modes coincide and a uniform breakup is likely to occur at the first natural frequency. Fig. 4 shows a superimposition of the growth rate (Fig. ) over the amplitude (Fig. 3(a)), for the input mode. The purpose of this curve is to display both the amplitude and the growth rate of the input mode at each frequency. The combination of the disturbance amplitude and the growth rate produces a pinch-off or breakup time. As mentioned earlier, the strongest resonance of the device occurs around 6Hz, but there is a second structural resonance around 6Hz. The latter is in the region of very high growth rate while the former is in a region of more modest growth rate. We now increase the driving frequency up to 8kHz. The amplitude of the second structural mode grows much faster compared to the other two modes, even though its absolute value is initially much smaller than the other two. Fast growth results in
5 Growth rate (s ) Input mode growth rate Input mode amplitude Input Frequency [Hz] Figure 4. Superimposition of scaled amplitude over scaled growth rate for the input mode, as a function of driving frequency. a fast pinch-off. It is possible for the short pinch-off time of the structural mode to dominate the effects of the other components. With this competition in mind, Fig. 5 displays pinch-off times versus frequency. In this figure, the input mode and both structural modes are considered independently to determine the pinch-off time of the same jet. Combining the growth rates and the amplitudes discussed in the previous section, pinch-off time at each frequency can be derived using Equation 2. Short pinch-off time of the second structural mode is clear from the figure at 7-8kHz. x 4 than that of the second structural mode. However, since the growth rate of input mode is somewhat smaller in this range (see Fig. 3), the combination of the amplitude and growth rate leads to close pinchoff times as shown in Fig. 5. As a result, competition can occur between the input mode and the second structural mode, which can lead to pronounced nonlinear effects and eventual satellite formation. At high frequencies around the peak of the growth rate dome, approximately from khz to 5kHz, we can see that the input mode has the shortest pinch-off time. In this region we would predict that the input mode is the dominant disturbance component. This can be explained by the very high growth rate of the input mode as well as its amplitude. Therefore, the combination of the amplitude and the growth rate results in a short pinch-off time for the input mode as shown in Fig. 5. The input mode dominates the effect of the structural mode and the jet can be assumed to be disturbed by a single frequency disturbance that can lead to generation of uniform droplets in the downstream region. The existence of a dominant disturbance wave allows us to use linear theory for simulating the breakup in this region of the frequency spectrum. Fig. 6 shows the simulation results of the jet breakup at the frequency corresponding to the peak of the growth rate. The jet is disturbed at 4.5kHz, which is the frequency where the growth rate has its maximum value. Substituting the growth rate, the wave number, and the frequency in Equation 2, the radius of the jet is determined along the jet, at time instance. 2.6 x Pinch off times Input mode First structural mode Second structural mode Breakup simulation 2.2 First natural frequency: 6kHz Second natural frequency: 6kHz 2 Pinch off time x Figure 5. Pinch-off time as a function of driving frequency. Each mode is considered independently. At slightly higher frequencies, but still prior to the peak of the growth rate dome (i.e., 8-kHz), the amplitude of the input mode is considerably larger Figure 6. Breakup simulation results derived from applying linear theory. Uniform circles in the picture demonstrate uniform pinch-off at the frequency where the growth rate has its maximum value. An experimental image of the jet breakup at the same frequency is shown in Fig. 7. Uniform breakup shown in this picture confirms that the linear assumptions made in the simulation are valid for this frequency range.
6 Figure 7. Experimental image of uniform droplets generation at 4.5kHz, the frequency of the peak of the growth rate. What happens at inputs near the second natural frequency, is similar to what was described at the first natural frequency. In this case the second structural mode has the same frequency, amplitude, and growth rate as the input mode. This results in equal pinch-off times, as shown in Fig. 5. That is, the input mode and the second structural mode coincide strongly in both pinch-off time and in the expected frequency of the resulting droplets. Going even higher in frequency and passing the second natural frequency, the growth rate of the driving mode decreases quickly and the second structural mode again becomes the dominant one. This is represented in Fig. 5 where the pinch-off time for the second resonant mode is shorter than for the input mode. Considering the initial disturbance of the piezoelectric as a function of time confirms to some extent what we concluded from the analysis of the interaction of disturbance components. The initial disturbance is the superposition of the output components of the piezoelectric device (Fig. 2) before they start growing. Studying the initial amplitude profile provides us with an approximation of how the disturbance is distributed on the surface of the jet. In Fig. 8, variation of the initial disturbances with time is demonstrated at the first natural frequency (6kHz) and at 9kHz. The beat frequency in the first subplot in Fig. 8 is 43Hz. the piezoelectric (even though the amplitude varies somewhat) suggests a uniform dispersion of pinchoffs. The frequency is uniform so that the instability growth will still pinch off the same size fluid parcel each time. This situation should be compared with Fig., which shows the initial disturbance variation at 9Hz where competition between modes was predicted based on the breakup time analysis shown in Fig. 5. The effect of a high frequency disturbance is obvious from Fig. and it is plausible that the driving mode competes with the highfrequency structural resonance mode. Initial amplitude 6 Hz x 3 Figure 9. Initial disturbance amplitude variation at 6kHz. Initial amplitude 9 Hz Initial amplitude x 3 Amplitude Amplitude Figure. Initial disturbance amplitude variation at 9kHz. An experimental image from the jet stream at 9Hz, (Fig. ), clearly shows a satellite formation that can result from competing breakup modes. Figure 8. Initial disturbance amplitudes at 6Hz and 9Hz. Fig. 9 shows a close up view of the initial disturbance variation at 6Hz through the window shown in left side of the figures. Uniform motion of Figure. Experimental image of satellite formation at 9Hz.
7 Control Approach We are interested in avoiding nonuniform droplet formation by suppressing the extraneous disturbances. For this purpose, control methods are proposed that manipulate the piezoelectric input signal to provide us with a single wavenumber perturbation over the entire range of driving frequencies. This system can be assumed to be a droplet generator followed by a capillary stream function. Figure 3. The output of the actuator is tracked and the controller is designed to reduce the error in the closed loop system. Figure 2. Open loop block diagram of the system. Fig. 2, is an open loop model of the system. The input of the droplet generator is the voltage oscillating at the driving frequency and the output is the summation of structural and driving harmonic functions of the input frequency. One simple approach to control the droplets to breakup uniformly is to find the input needed to get a single frequency at the output of the actuator. The input can easily be calculated from an inverse Laplace transform of, I(s) = R(s) H(s), where R(s) is different at each frequency since it is the Laplace transform of the desired single frequency disturbance. H(s), the transfer function of the actuator, is experimentally determined as discussed earlier, using the procedure described in [9]. Open loop control, however, is not likely to be a good solution, since it is sensitive to parameter variation. In other words, if characteristics of the system change due to temperature or pressure variations, the results might not be valid. To have a less sensitive system to variation of parameters, a closed loop feedback control can be used. It can provide us with uniform breakup even if the model does not match exactly the real system. Droplet size, average droplet frequency, or output of the piezoelectric can be used as output feedback of the system. The closed loop system is shown in Fig. 3, schematically. This is the subject for our future activity. Summary and Conclusion According to Rayleigh s linear theory, any disturbance imposed on a jet of fluid, grows in an exponential manner if its wavelength exceeds the circumference of the jet stream. One common practical device that exerts this distortion is a piezoelectric actuator that responds to an input voltage by applying a distortion onto the surface of the jet. In this paper we show that in a practical device, typical piezoelectric actuators produce multiple wavenumber disturbances which compete with each other in some frequencies. Disturbance components competition results in nonuniform droplet formation. Analyzing the interaction of waves allows us to recognize the range of frequencies prone to satellite formation. Also, an output feedback control model is proposed whose job would be to eliminate satellites by tracking the output of the actuator and preventing disturbances away from the desired input frequency. References [] Lord Rayleigh, On the instability of jets, Proc. London Math. Soc., :4-3, 878. [2] J. S. Keller, S. I. Rubinow, and Y. O. Tu, Spatial instability of a jet, Phys. Fluids, 6: , 973. [3] S. J. Leib and M. E. Goldstein, The generation of capillary instabilities on a liquid jets, J. Fluid Mech., 68:479-5, 985. [4] S. J. Leib and M. E. Goldstein, Convective and absolute instability of a viscous liquid jet, Phys. Fluids, 29: , 986. [5] H. C. Lee Drop formation in a liquid jet, IBM J. Res. Develop., 8: , 974. [6] W. T. Pimbley and H. C. Lee, Satellite droplet formation in a liquid jet, IBM J. Res. Develop., 2:2-3, 977. [7] T. Funada, D. D. Joseph, and S. Yamashita, Stability of a liquid jet into incompressible gases and liquids, Int. J. Multiphase Flow, 3:279-3, 24. [8] C. Weber, Zum Zerfall eines Flüssigkeitsstrahles, Ztschr. f. angew. Math und Mech, :36-54, 93. [9] D. K. Iobbi, Controlling piezoelectric generated droplets, Master thesis, University of california, Irvine,, 24.
Liquid Jet Breakup at Low Weber Number: A Survey
International Journal of Engineering Research and Technology. ISSN 0974-3154 Volume 6, Number 6 (2013), pp. 727-732 International Research Publication House http://www.irphouse.com Liquid Jet Breakup at
More informationPaper ID ICLASS EXPERIMENTS ON BREAKUP OF WATER-IN-DIESEL COMPOUND JETS
ICLASS-2006 Aug.27-Sept.1, 2006, Kyoto, Japan Paper ID ICLASS06-047 EXPERIMENTS ON BREAKUP OF WATER-IN-DIESEL COMPOUND JETS Sheng-Lin Chiu 1, Rong-Horng Chen 2, Jen-Yung Pu 1 and Ta-Hui Lin 1,* 1 Department
More informationHigher Orders Instability of a Hollow Jet Endowed with Surface Tension
Mechanics and Mechanical Engineering Vol. 2, No. (2008) 69 78 c Technical University of Lodz Higher Orders Instability of a Hollow Jet Endowed with Surface Tension Ahmed E. Radwan Mathematics Department,
More informationNumerical Studies of Droplet Deformation and Break-up
ILASS Americas 14th Annual Conference on Liquid Atomization and Spray Systems, Dearborn, MI, May 2001 Numerical Studies of Droplet Deformation and Break-up B. T. Helenbrook Department of Mechanical and
More informationFigure 11.1: A fluid jet extruded where we define the dimensionless groups
11. Fluid Jets 11.1 The shape of a falling fluid jet Consider a circular orifice of a radius a ejecting a flux Q of fluid density ρ and kinematic viscosity ν (see Fig. 11.1). The resulting jet accelerates
More informationCapillary Instability of a Jet Induced by Surface Tension Modulations
Capillary Instability of a Jet Induced by Surface Tension Modulations B. Barbet,2, P. Atten 2 and A. Soucemarianadin,3 TOXOT Science & Applications (IMAJE Group), BP, 265 Bourg les Valence (France) 2 L.E.M.D
More informationLinear analysis of three-dimensional instability of non-newtonian liquid jets
J. Fluid Mech. (2006), vol. 559, pp. 451 459. c 2006 Cambridge University Press doi:10.1017/s0022112006000413 Printed in the United Kingdom 451 Linear analysis of three-dimensional instability of non-newtonian
More informationDynamics of Transient Liquid Injection:
Dynamics of Transient Liquid Injection: K-H instability, vorticity dynamics, R-T instability, capillary action, and cavitation William A. Sirignano University of California, Irvine -- Round liquid columns
More informationLiquid Jet Instability Under Gravity Effects
st AIAA Aerospace Sciences Meeting including the New Horions Forum and Aerospace Exposition 7 - January, Grapevine (Dallas/Ft. Worth Region), Texas AIAA -9 Liquid Jet Instability Under Gravity Effects
More information9 Fluid Instabilities
9. Stability of a shear flow In many situations, gaseous flows can be subject to fluid instabilities in which small perturbations can rapidly flow, thereby tapping a source of free energy. An important
More informationPressure corrections for viscoelastic potential flow analysis of capillary instability
ve-july29-4.tex 1 Pressure corrections for viscoelastic potential flow analysis of capillary instability J. Wang, D. D. Joseph and T. Funada Department of Aerospace Engineering and Mechanics, University
More informationStability of a Liquid Jet into Incompressible Gases and Liquids
LiquidJet-06-25bw.tex 1 Stability of a Liquid Jet into Incompressible Gases and Liquids T. Funada, 1 D.D. Joseph 23 and S. Yamashita 1 1 Department of Digital Engineering, Numazu College of Technology,
More informationViscous contributions to the pressure for potential flow analysis of capillary instability of two viscous fluids
PHYSICS OF FLUIDS 17, 052105 2005 Viscous contributions to the pressure for potential flow analysis of capillary instability of two viscous fluids. Wang and D. D. oseph a Department of Aerospace Engineering
More informationPlateau-Rayleigh Instability of a Cylinder of Viscous Liquid (Rayleigh vs. Chandrasekhar) L. Pekker FujiFilm Dimatix Inc., Lebanon NH USA
Plateau-Rayleigh Instability of a Cylinder of Viscous Liquid (Rayleigh vs. Chandrasekhar) L. Pekker FujiFilm Dimatix Inc., Lebanon NH 03766 USA Abstract In 1892, in his classical work, L. Rayleigh considered
More informationHydromagnetic Self-gravitating Stability of Streaming Fluid Cylinder. With Longitudinal Magnetic Field
Applied Mathematical Sciences, Vol. 8, 2014, no. 40, 1969-1978 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.312721 Hydromagnetic Self-gravitating Stability of Streaming Fluid Cylinder
More informationKelvin Helmholtz Instability
Kelvin Helmholtz Instability A. Salih Department of Aerospace Engineering Indian Institute of Space Science and Technology, Thiruvananthapuram November 00 One of the most well known instabilities in fluid
More informationCorrection of Lamb s dissipation calculation for the effects of viscosity on capillary-gravity waves
PHYSICS OF FLUIDS 19, 082105 2007 Correction of Lamb s dissipation calculation for the effects of viscosity on capillary-gravity waves J. C. Padrino and D. D. Joseph Aerospace Engineering and Mechanics
More informationEXPERIMENTS OF CLOSED-LOOP FLOW CONTROL FOR LAMINAR BOUNDARY LAYERS
Fourth International Symposium on Physics of Fluids (ISPF4) International Journal of Modern Physics: Conference Series Vol. 19 (212) 242 249 World Scientific Publishing Company DOI: 1.1142/S211945128811
More informationSpatial Instability of Electrically Driven Jets with Finite Conductivity and Under Constant or Variable Applied Field
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 4, No. 2 (December 2009), pp. 249 262 Applications and Applied Mathematics: An International Journal (AAM) Spatial Instability of
More informationBlack holes and the leaking faucet in your bathroom
Black holes and the leaking faucet in your bathroom Nicolas Vasset Journal club May 5th, 2011 Nicolas Vasset (Basel) Black holes and leaking faucets 05/11 1 / 17 Based on the following articles [Lehner
More informationINTERFACIAL WAVE BEHAVIOR IN OIL-WATER CHANNEL FLOWS: PROSPECTS FOR A GENERAL UNDERSTANDING
1 INTERFACIAL WAVE BEHAVIOR IN OIL-WATER CHANNEL FLOWS: PROSPECTS FOR A GENERAL UNDERSTANDING M. J. McCready, D. D. Uphold, K. A. Gifford Department of Chemical Engineering University of Notre Dame Notre
More informationPHYS 432 Physics of Fluids: Instabilities
PHYS 432 Physics of Fluids: Instabilities 1. Internal gravity waves Background state being perturbed: A stratified fluid in hydrostatic balance. It can be constant density like the ocean or compressible
More informationEffect of an electric field on the response of a liquid jet to a pulse type perturbation
Effect of an electric field on the response of a liquid jet to a pulse type perturbation Guillermo Artana and Bruno Seguin CONICET, Dept. Ingenieria Mecanica, Fac. Ingenieria, Universidad de Buenos Aires,
More informationGeneral introduction to Hydrodynamic Instabilities
KTH ROYAL INSTITUTE OF TECHNOLOGY General introduction to Hydrodynamic Instabilities L. Brandt & J.-Ch. Loiseau KTH Mechanics, November 2015 Luca Brandt Professor at KTH Mechanics Email: luca@mech.kth.se
More informationPurely irrotational theories of the effects of viscosity and viscoelasticity on capillary instability of a liquid cylinder
ve-march17.tex 1 Purely irrotational theories of the effects of viscosity and viscoelasticity on capillary instability of a liquid cylinder J. Wang, D. D. Joseph and T. Funada Department of Aerospace Engineering
More informationChapter 1. Introduction to Nonlinear Space Plasma Physics
Chapter 1. Introduction to Nonlinear Space Plasma Physics The goal of this course, Nonlinear Space Plasma Physics, is to explore the formation, evolution, propagation, and characteristics of the large
More informationFORMATION OF UNIFORMLY-SIZED DROPLETS FROM CAPILLARY JET BY ELECTROMAGNETIC FORCE
Seventh International Conference on CFD in the Minerals and Process Industries CSIRO, Melbourne, Australia 9-11 December 2009 FORMATION OF UNIFORMLY-SIZED DROPLETS FROM CAPILLARY JET BY ELECTROMAGNETIC
More informationMagnetohydrodynamics Stability of a Compressible Fluid Layer Below a Vacuum Medium
Mechanics and Mechanical Engineering Vol. 12, No. 3 (2008) 267 274 c Technical University of Lodz Magnetohydrodynamics Stability of a Compressible Fluid Layer Below a Vacuum Medium Emad E. Elmahdy Mathematics
More informationInvestigation of an implicit solver for the simulation of bubble oscillations using Basilisk
Investigation of an implicit solver for the simulation of bubble oscillations using Basilisk D. Fuster, and S. Popinet Sorbonne Universités, UPMC Univ Paris 6, CNRS, UMR 79 Institut Jean Le Rond d Alembert,
More informationSimulation of Liquid Jet Breakup Process by Three-Dimensional Incompressible SPH Method
Seventh International Conference on Computational Fluid Dynamics (ICCFD7), Big Island, Hawaii, July 9-13, 212 ICCFD7-291 Simulation of Liquid Jet Breakup Process by Three-Dimensional Incompressible SPH
More informationChE 385M Surface Phenomena University of Texas at Austin. Marangoni-Driven Finger Formation at a Two Fluid Interface. James Stiehl
ChE 385M Surface Phenomena University of Texas at Austin Marangoni-Driven Finger Formation at a Two Fluid Interface James Stiehl Introduction Marangoni phenomena are driven by gradients in surface tension
More informationWilliam A. Sirignano Mechanical and Aerospace Engineering University of California, Irvine
Combustion Instability: Liquid-Propellant Rockets and Liquid-Fueled Ramjets William A. Sirignano Mechanical and Aerospace Engineering University of California, Irvine Linear Theory Nonlinear Theory Nozzle
More informationThe Harmonic Oscillator
The Harmonic Oscillator Math 4: Ordinary Differential Equations Chris Meyer May 3, 008 Introduction The harmonic oscillator is a common model used in physics because of the wide range of problems it can
More informationCHAPTER 5. RUDIMENTS OF HYDRODYNAMIC INSTABILITY
1 Lecture Notes on Fluid Dynamics (1.63J/.1J) by Chiang C. Mei, 00 CHAPTER 5. RUDIMENTS OF HYDRODYNAMIC INSTABILITY References: Drazin: Introduction to Hydrodynamic Stability Chandrasekar: Hydrodynamic
More informationLecture notes Breakup of cylindrical jets Singularities and self-similar solutions
Lecture notes Breakup of cylindrical jets Singularities and self-similar solutions by Stephane Popinet and Arnaud Antkowiak June 8, 2011 Table of contents 1 Equations of motion for axisymmetric jets...........................
More informationPrototype Instabilities
Prototype Instabilities David Randall Introduction Broadly speaking, a growing atmospheric disturbance can draw its kinetic energy from two possible sources: the kinetic and available potential energies
More informationFlow control. Flow Instability (and control) Vortex Instabilities
Flow control Flow Instability (and control) Tim Colonius CDS 101 Friday, Oct 15, 2004 Many control problems contain fluid systems as components. Dashpot in mass-spring-damper systems HVAC system that thermostat
More informationUniversity of Bristol - Explore Bristol Research. Link to publication record in Explore Bristol Research PDF-document.
Dobra, T., Lawrie, A., & Dalziel, S. B. (2016). Nonlinear Interactions of Two Incident Internal Waves. 1-8. Paper presented at VIIIth International Symposium on Stratified Flows, San Diego, United States.
More informationNumerical simulations of the edge tone
Numerical simulations of the edge tone I. Vaik, G. Paál Department of Hydrodynamic Systems, Budapest University of Technology and Economics, P.O. Box 91., 1521 Budapest, Hungary, {vaik, paal}@vizgep.bme.hu
More informationDamped Harmonic Oscillator
Damped Harmonic Oscillator Wednesday, 23 October 213 A simple harmonic oscillator subject to linear damping may oscillate with exponential decay, or it may decay biexponentially without oscillating, or
More informationPrimary, secondary instabilities and control of the rotating-disk boundary layer
Primary, secondary instabilities and control of the rotating-disk boundary layer Benoît PIER Laboratoire de mécanique des fluides et d acoustique CNRS Université de Lyon École centrale de Lyon, France
More information2 D.D. Joseph To make things simple, consider flow in two dimensions over a body obeying the equations ρ ρ v = 0;
Accepted for publication in J. Fluid Mech. 1 Viscous Potential Flow By D.D. Joseph Department of Aerospace Engineering and Mechanics, University of Minnesota, MN 55455 USA Email: joseph@aem.umn.edu (Received
More informationSpray formation with complex fluids
Journal of Physics: Conference Series Spray formation with complex fluids To cite this article: S Lustig and M Rosen 2011 J. Phys.: Conf. Ser. 296 012019 View the article online for updates and enhancements.
More informationPinching threads, singularities and the number
Pinching threads, singularities and the number 0.0304... Michael P. Brenner Department of Mathematics, MIT, Cambridge, Massachusetts 02139 John R. Lister DAMTP, Cambridge CB3 9EW, England Howard A. Stone
More information1. Comparison of stability analysis to previous work
. Comparison of stability analysis to previous work The stability problem (6.4) can be understood in the context of previous work. Benjamin (957) and Yih (963) have studied the stability of fluid flowing
More informationDispersion relations, stability and linearization
Dispersion relations, stability and linearization 1 Dispersion relations Suppose that u(x, t) is a function with domain { < x 0}, and it satisfies a linear, constant coefficient partial differential
More informationDifferential criterion of a bubble collapse in viscous liquids
PHYSICAL REVIEW E VOLUME 60, NUMBER 1 JULY 1999 Differential criterion of a bubble collapse in viscous liquids Vladislav A. Bogoyavlenskiy* Low Temperature Physics Department, Moscow State University,
More informationOn the breakup of fluid films of finite and infinite extent
PHYSICS OF FLUIDS 19, 072107 2007 On the breakup of fluid films of finite and infinite extent Javier A. Diez a Instituto de Física Arroyo Seco, Universidad Nacional del Centro de la Provincia de Buenos
More informationNon-Linear Plasma Wave Decay to Longer Wavelength
Non-Linear Plasma Wave Decay to Longer Wavelength F. Anderegg 1, a), M. Affolter 1, A. Ashourvan 1, D.H.E. Dubin 1, F. Valentini 2 and C.F. Driscoll 1 1 University of California San Diego Physics Department
More informationMagnetically Induced Transparency and Its Application as an Accelerator
Magnetically Induced Transparency and Its Application as an Accelerator M.S. Hur, J.S. Wurtele and G. Shvets University of California Berkeley University of California Berkeley and Lawrence Berkeley National
More informationInterfacial waves in steady and oscillatory, two-layer Couette flows
Interfacial waves in steady and oscillatory, two-layer Couette flows M. J. McCready Department of Chemical Engineering University of Notre Dame Notre Dame, IN 46556 Page 1 Acknowledgments Students: M.
More informationSimulation of a Pressure Driven Droplet Generator
Simulation of a Pressure Driven Droplet Generator V. Mamet* 1, P. Namy 2, N. Berri 1, L. Tatoulian 1, P. Ehouarn 1, V. Briday 1, P. Clémenceau 1 and B. Dupont 1 1 DBV Technologies, 2 SIMTEC *84 rue des
More informationNonlinear shape evolution of immiscible two-phase interface
Nonlinear shape evolution of immiscible two-phase interface Francesco Capuano 1,2,*, Gennaro Coppola 1, Luigi de Luca 1 1 Dipartimento di Ingegneria Industriale (DII), Università di Napoli Federico II,
More informationTransverse wave - the disturbance is perpendicular to the propagation direction (e.g., wave on a string)
1 Part 5: Waves 5.1: Harmonic Waves Wave a disturbance in a medium that propagates Transverse wave - the disturbance is perpendicular to the propagation direction (e.g., wave on a string) Longitudinal
More informationPurely irrotational theories of the effects of viscosity and viscoelasticity on capillary instability of a liquid cylinder
. Non-Newtonian Fluid Mech. 19 (005 106 116 Purely irrotational theories of the effects of viscosity and viscoelasticity on capillary instability of a liquid cylinder. Wang a, D.D. oseph a,, T. Funada
More informationAnumerical and analytical study of the free convection thermal boundary layer or wall jet at
REVERSED FLOW CALCULATIONS OF HIGH PRANDTL NUMBER THERMAL BOUNDARY LAYER SEPARATION 1 J. T. Ratnanather and P. G. Daniels Department of Mathematics City University London, England, EC1V HB ABSTRACT Anumerical
More informationJet, Wave, and Droplet Velocities for a Continuous Fluid Jet
Jet, Wave, and Droplet Velocities for a Continuous Fluid Jet Randy Fagerquist Scitex Digital Printing, Inc., Dayton, Ohio Abstract Presented at IS&T s Eleventh International Congress on Advances in Non-Impact
More informationChapter 4. Gravity Waves in Shear. 4.1 Non-rotating shear flow
Chapter 4 Gravity Waves in Shear 4.1 Non-rotating shear flow We now study the special case of gravity waves in a non-rotating, sheared environment. Rotation introduces additional complexities in the already
More information2 The incompressible Kelvin-Helmholtz instability
Hydrodynamic Instabilities References Chandrasekhar: Hydrodynamic and Hydromagnetic Instabilities Landau & Lifshitz: Fluid Mechanics Shu: Gas Dynamics 1 Introduction Instabilities are an important aspect
More informationProcess Dynamics, Operations, and Control Lecture Notes 2
Chapter. Dynamic system.45 Process Dynamics, Operations, and Control. Context In this chapter, we define the term 'system' and how it relates to 'process' and 'control'. We will also show how a simple
More information12.1 Viscous potential flow (VPF)
1 Energy equation for irrotational theories of gas-liquid flow:: viscous potential flow (VPF), viscous potential flow with pressure correction (VCVPF), dissipation method (DM) 1.1 Viscous potential flow
More informationWaves & Oscillations
Physics 42200 Waves & Oscillations Lecture 22 Review Spring 2013 Semester Matthew Jones Midterm Exam: Date: Wednesday, March 6 th Time: 8:00 10:00 pm Room: PHYS 203 Material: French, chapters 1-8 Review
More informationCapillary-gravity waves: The effect of viscosity on the wave resistance
arxiv:cond-mat/9909148v1 [cond-mat.soft] 10 Sep 1999 Capillary-gravity waves: The effect of viscosity on the wave resistance D. Richard, E. Raphaël Collège de France Physique de la Matière Condensée URA
More informationTheory of Ship Waves (Wave-Body Interaction Theory) Quiz No. 2, April 25, 2018
Quiz No. 2, April 25, 2018 (1) viscous effects (2) shear stress (3) normal pressure (4) pursue (5) bear in mind (6) be denoted by (7) variation (8) adjacent surfaces (9) be composed of (10) integrand (11)
More informationPoint Vortex Dynamics in Two Dimensions
Spring School on Fluid Mechanics and Geophysics of Environmental Hazards 9 April to May, 9 Point Vortex Dynamics in Two Dimensions Ruth Musgrave, Mostafa Moghaddami, Victor Avsarkisov, Ruoqian Wang, Wei
More informationPAPER 345 ENVIRONMENTAL FLUID DYNAMICS
MATHEMATICAL TRIPOS Part III Monday, 11 June, 2018 9:00 am to 12:00 pm PAPER 345 ENVIRONMENTAL FLUID DYNAMICS Attempt no more than THREE questions. There are FOUR questions in total. The questions carry
More informationLecture 9 Laminar Diffusion Flame Configurations
Lecture 9 Laminar Diffusion Flame Configurations 9.-1 Different Flame Geometries and Single Droplet Burning Solutions for the velocities and the mixture fraction fields for some typical laminar flame configurations.
More informationAnalysis of Hermite s equation governing the motion of damped pendulum with small displacement
Vol. 10(12), pp. 364-370, 30 June, 2015 DOI: 10.5897/IJPS2015.4364 Article Number: 9A709F454056 ISSN 1992-1950 Copyright 2015 Author(s) retain the copyright of this article http://www.academicjournals.org/ijps
More informationInstabilities due a vortex at a density interface: gravitational and centrifugal effects
Instabilities due a vortex at a density interface: gravitational and centrifugal effects Harish N Dixit and Rama Govindarajan Abstract A vortex placed at an initially straight density interface winds it
More informationTemperature ( o C)
Viscosity (Pa sec) Supplementary Information 10 8 10 6 10 4 10 2 150 200 250 300 Temperature ( o C) Supplementary Figure 1 Viscosity of fibre components (PC cladding blue; As 2 Se 5 red; CPE black) as
More informationMODELING ON THE BREAKUP OF VISCO-ELASTIC LIQUID FOR EFFERVESCENT ATOMIZATION
1446 THERMAL SCIENCE, Year 2012, Vol. 16, No. 5, pp. 1446-1450 MODELING ON THE BREAKUP OF VISCO-ELASTIC LIQUID FOR EFFERVESCENT ATOMIZATION by Li-Juan QIAN * China Jiliang University, Hangzhou, China Short
More informationLiquid-Rocket Transverse Triggered Combustion Instability: Deterministic and Stochastic Analyses
Liquid-Rocket Transverse Triggered Combustion Instability: Deterministic and Stochastic Analyses by W. A. Sirignano Mechanical and Aerospace Engineering University of California, Irvine Collaborators:
More information7.2.1 Seismic waves. Waves in a mass- spring system
7..1 Seismic waves Waves in a mass- spring system Acoustic waves in a liquid or gas Seismic waves in a solid Surface waves Wavefronts, rays and geometrical attenuation Amplitude and energy Waves in a mass-
More informationFlow Focusing Droplet Generation Using Linear Vibration
Flow Focusing Droplet Generation Using Linear Vibration A. Salari, C. Dalton Department of Electrical & Computer Engineering, University of Calgary, Calgary, AB, Canada Abstract: Flow focusing microchannels
More informationFormation and Long Term Evolution of an Externally Driven Magnetic Island in Rotating Plasmas )
Formation and Long Term Evolution of an Externally Driven Magnetic Island in Rotating Plasmas ) Yasutomo ISHII and Andrei SMOLYAKOV 1) Japan Atomic Energy Agency, Ibaraki 311-0102, Japan 1) University
More informationSolution Set 2 Phys 4510 Optics Fall 2014
Solution Set Phys 4510 Optics Fall 014 Due date: Tu, September 16, in class Scoring rubric 4 points/sub-problem, total: 40 points 3: Small mistake in calculation or formula : Correct formula but calculation
More informationDefense Technical Information Center Compilation Part Notice
UNCLASSIFIED Defense Technical Information Center Compilation Part Notice ADPO 11342 TITLE: The Meniscus Oscillation of Ink Flow Dynamics in Thermal Inkjet Print DISTRIBUTION: Approved for public release,
More informationCombined Effect of Magnetic field and Internal Heat Generation on the Onset of Marangoni Convection
International Journal of Fluid Mechanics & Thermal Sciences 17; 3(4): 41-45 http://www.sciencepublishinggroup.com/j/ijfmts doi: 1.11648/j.ijfmts.1734.1 ISSN: 469-815 (Print); ISSN: 469-8113 (Online) ombined
More informationThe Magnetorotational Instability
The Magnetorotational Instability Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics March 10, 2014 These slides are based off of Balbus & Hawley (1991), Hawley
More informationLecture 12. AO Control Theory
Lecture 12 AO Control Theory Claire Max with many thanks to Don Gavel and Don Wiberg UC Santa Cruz February 18, 2016 Page 1 What are control systems? Control is the process of making a system variable
More informationEffect of Liquid Viscosity on Sloshing in A Rectangular Tank
International Journal of Research in Engineering and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Print): 2320-9356 Volume 5 Issue 8 ǁ August. 2017 ǁ PP. 32-39 Effect of Liquid Viscosity on Sloshing
More informationStability of Liquid Metal Interface Affected by a High-Frequency Magnetic Field
International Scientific Colloquium Modelling for Electromagnetic Processing Hannover, March 4-6, 3 Stability of Liquid Metal Interface Affected by a High-Frequency Magnetic Field J-U Mohring, Ch Karcher,
More informationAn optimum design of a double pendulum in autoparametric resonance for energy harvesting applications
An optimum design of a double pendulum in autoparametric resonance for energy harvesting applications Taizoon Chunawala 1, Maryam Ghandchi-Tehrani 2, Jize Yan 2 1 Birla Institute of Technology and Science-Pilani,
More informationFundamentals of Fluid Dynamics: Waves in Fluids
Fundamentals of Fluid Dynamics: Waves in Fluids Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/ tzielins/ Institute
More informationAcoustic streaming around a spherical microparticle/cell under ultrasonic wave excitation
Acoustic streaming around a spherical microparticle/cell under ultrasonic wave excitation Zhongheng Liu a) Yong-Joe Kim b) Acoustics and Signal Processing Laboratory, Department of Mechanical Engineering,
More informationELECTROSTATIC ION-CYCLOTRON WAVES DRIVEN BY PARALLEL VELOCITY SHEAR
1 ELECTROSTATIC ION-CYCLOTRON WAVES DRIVEN BY PARALLEL VELOCITY SHEAR R. L. Merlino Department of Physics and Astronomy University of Iowa Iowa City, IA 52242 December 21, 2001 ABSTRACT Using a fluid treatment,
More information10.52 Mechanics of Fluids Spring 2006 Problem Set 3
10.52 Mechanics of Fluids Spring 2006 Problem Set 3 Problem 1 Mass transfer studies involving the transport of a solute from a gas to a liquid often involve the use of a laminar jet of liquid. The situation
More informationAC & DC Magnetic Levitation and Semi-Levitation Modelling
International Scientific Colloquium Modelling for Electromagnetic Processing Hannover, March 24-26, 2003 AC & DC Magnetic Levitation and Semi-Levitation Modelling V. Bojarevics, K. Pericleous Abstract
More informationApplication of the immersed boundary method to simulate flows inside and outside the nozzles
Application of the immersed boundary method to simulate flows inside and outside the nozzles E. Noël, A. Berlemont, J. Cousin 1, T. Ménard UMR 6614 - CORIA, Université et INSA de Rouen, France emeline.noel@coria.fr,
More informationDisplacement at very low frequencies produces very low accelerations since:
SEISMOLOGY The ability to do earthquake location and calculate magnitude immediately brings us into two basic requirement of instrumentation: Keeping accurate time and determining the frequency dependent
More informationChapter 9. Electromagnetic Waves
Chapter 9. Electromagnetic Waves 9.1 Waves in One Dimension 9.1.1 The Wave Equation What is a "wave?" Let's start with the simple case: fixed shape, constant speed: How would you represent such a string
More informationOne-dimensional Spray Combustion Optimization with a Sequential Linear Quadratic Algorithm
One-dimensional Spray Combustion Optimization with a Sequential Linear Quadratic Algorithm Justin A. Sirignano, Luis Rodriguez, Athanasios Sideris, and William A. Sirignano Department of Mechanical and
More informationDispersion relations, linearization and linearized dynamics in PDE models
Dispersion relations, linearization and linearized dynamics in PDE models 1 Dispersion relations Suppose that u(x, t) is a function with domain { < x 0}, and it satisfies a linear, constant coefficient
More informationSuppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber
Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber J.C. Ji, N. Zhang Faculty of Engineering, University of Technology, Sydney PO Box, Broadway,
More informationHeriot-Watt University. A pressure-based estimate of synthetic jet velocity Persoons, Tim; O'Donovan, Tadhg. Heriot-Watt University.
Heriot-Watt University Heriot-Watt University Research Gateway A pressure-based estimate of synthetic jet velocity Persoons, Tim; O'Donovan, Tadhg Published in: Physics of Fluids DOI: 10.1063/1.2823560
More informationChapter 10. Solids and Fluids
Chapter 10 Solids and Fluids Surface Tension Net force on molecule A is zero Pulled equally in all directions Net force on B is not zero No molecules above to act on it Pulled toward the center of the
More informationNUMERICAL MODELING FOR ATOMIZATION OF COAXIAL LIQUID/GAS JETS
90 Journal of Marine Science and Technology, Vol. 1, No. 4, pp. 90-99 (004) NUMERICAL MODELING FOR ATOMIZATION OF COAXIAL LIQUID/GAS JETS Stephen Gen-Ken Chuech*, Andrzej J. Przekwas**, and Chih-Yuan Wang***
More informationNIMROD simulations of dynamo experiments in cylindrical and spherical geometries. Dalton Schnack, Ivan Khalzov, Fatima Ebrahimi, Cary Forest,
NIMROD simulations of dynamo experiments in cylindrical and spherical geometries Dalton Schnack, Ivan Khalzov, Fatima Ebrahimi, Cary Forest, 1 Introduction Two experiments, Madison Plasma Couette Experiment
More informationSurface Tension Effect on a Two Dimensional. Channel Flow against an Inclined Wall
Applied Mathematical Sciences, Vol. 1, 007, no. 7, 313-36 Surface Tension Effect on a Two Dimensional Channel Flow against an Inclined Wall A. Merzougui *, H. Mekias ** and F. Guechi ** * Département de
More informationTORSION PENDULUM: THE MECHANICAL NONLINEAR OSCILLATOR
TORSION PENDULUM: THE MECHANICAL NONLINEAR OSCILLATOR Samo Lasič, Gorazd Planinšič,, Faculty of Mathematics and Physics University of Ljubljana, Slovenija Giacomo Torzo, Department of Physics, University
More information