Oscillatory measurements of silicone oils —Loss and storage modulus master curves

The work describes a way to obtain loss modulus and storage modulus master curves from oscillatory measurements of silicone oils.The loss modulus master curve represents the dependence of the viscous flow behavior on · ₀ ^* and the storage modulus master curve — the dependence of the elastic flow behavior on · ₀ ^* .The relation between the values of the loss modulus and storage modulus master curves (at a certain frequency) is a measurement of the viscoelastic behavior of a system. The G/G-ratio depends on · ₀ ^* which leads to a viscoelastic master curve. The viscoelastic master curve represents the relation between the elastic and viscous oscillatory flow behavior.     

Nonlinear rheological behavior of a concentrated spherical silica suspension

Linear and nonlinear viscoelastic properties were examined for a 50 wt% suspension of spherical silica particles (with radius of 40 nm) in a viscous medium, 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The effective volume fraction of the particles evaluated from zero-shear viscosities of the suspension and medium was 0.53. At a quiescent state the particles had a liquid-like, isotropic spatial distribution in the medium. Dynamic moduli G^* obtained for small oscillatory strain (in the linear viscoelastic regime) exhibited a relaxation process that reflected the equilibrium Brownian motion of those particles. In the stress relaxation experiments, the linear relaxation modulus G(t) was obtained for small step strain (0.2) while the nonlinear relaxation modulus G(t, ) characterizing strong stress damping behavior was obtained for large (>0.2). G(t, ) obeyed the time-strain separability at long time scales, and the damping function h() (–G(t, )/G(t)) was determined. Steady flow measurements revealed shear-thinning of the steady state viscosity () for small shear rates (< ^–1; = linear viscoelastic relaxation time) and shear-thickening for larger (>^–1). Corresponding changes were observed also for the viscosity growth and decay functions on start up and cessation of flow, + (t, ) and _– (t, ). In the shear-thinning regime, the and dependence of ₊(t,) and _–(t,) as well as the dependence of () were well described by a BKZ-type constitutive equation using the G(t) and h() data. On the other hand, this equation completely failed in describing the behavior in the shear-thickening regime. These applicabilities of the BKZ equation were utilized to discuss the shearthinning and shear-thickening mechanisms in relation to shear effects on the structure (spatial distribution) and motion of the suspended particles.Dedicated to the memory of Prof. Dale S. Parson     

Quantitative analysis of flow visualizations in an unsteady channel flow

Quantitative results concerning the modulation of the ejection and bursting frequency in an unsteady channel flow obtained by flow visualizations are presented and compared with probe measurements. The frequency of the imposed velocity oscillations f covers a large range going from the quasi steady limit to the time mean bursting frequency in the corresponding steady flow. The imposed amplitudes of the velocity oscillations are 13% and 20% of the centerline velocity. The bursting process is identified by the intermittent lift up of the dye injected at the wall. Qualitative analysis of the flow visualizations show that the ejection activity at a given phase of the oscillation cycle is repetitive from one cycle to the other. The modulation amplitude of the ejection frequency f _e is sensitive to the imposed frequency. At low imposed frequency f _e is modulated as the wall shear stress, but the inner scaling does not hold when f ⁺ is high. Here, (+) corresponds to the quantities normalized with the inner variables, i.e. the friction velocity u and the viscosity . The grouping of the ejections into bursts show the coexistence of two categories of events which react differently to the forcing. The groups of ejections (Multiple Ejection Bursts) are governed by the modulation of the wall shear stress in the whole imposed frequency range. The solitary ejections (or the Single Ejection Bursts) have modulation amplitudes and phases which differ significantly from those of in the intermediate and high imposed frequency range. There is a good agreement between the flow visualization data and the probe measurements.     

Rheology/morphology/flow conditions relationships for polymethylmethacrylate/rubber blend

Viscoelastic behavior, phase morphology and flow conditions relationships in polymer/rubber blends have been investigated. The importance of such correlations is illustrated on polymethylmethacrylate (PMMA)/rubber blends subjected to different flow conditions both under small and large deformations. In small-amplitude oscillatory shear (the morphology does not change during the flow) the elastic modulus G of the concentrated blends shows a secondary plateau, G _p , in the low frequency region. This solid-like behavior appears for rubber particle contents beyond the percolation threshold concentration (15%). Morphological observations revealed that for concentrations higher than 15%, the particles are dispersed in a three-dimensional network-type structure.In capillary flow it was found that the network-type structure was destroyed and replaced by an alignment of particles in the flow direction. This morphological modification resulted in a decrease in both viscosity and post-extrusion swell of the blends. Morphological observations revealed that the ordered structure in the flow direction was concentrated only in the skin region of the extrudate, where the shear stress is higher than the secondary plateau, G _p . A simple kinetic mechanism is proposed to explain the observed morphology.Similarly, steady shear measurements performed in the cone-and-plate geometry revealed alignment of particles in the flow direction for shear stress values higher than G_p.Presented in part at the Symposium Recent Developments in Structured Continua Montréal (Canada) 26–28 May 1993 and at the 45th Canadian Chemical Engineering Conference, Quebec, October 15–18 (1995)     

Evaluation of the dynamic-mechanical properties of solids from a cooperative model for stress relaxation

For many solid materials the stress relaxation process obeys the universal relationF = – (d/d lnt)_max = (0.1 ± 0.01) ( ₀ — _i ), regardless of the structure of the material. Here denotes the stress,t the time, ₀ the initial stress of the experiment and _i the internal stress. A cooperative model accounting for the similarity in relaxation behaviour between different materials was developed earlier. Since this model has a spectral character, the concepts of linear viscoelasticity are used here to evaluate the corresponding prediction of the dynamic mechanical properties, i.e. the frequency dependence of the storageE () and lossE () moduli. Useful numerical approximations ofE () andE () are also evaluated. It is noted that the universal relation in stress relaxation had a counterpart in the frequency dependence ofE (). The theoretical prediction of the loss factor for high-density polyethylene is compared with experimental results. The agreement is good.     

Finite difference analysis of plane Poiseuille and Couette flow developments

Summary The problem of flow development from an initially flat velocity profile in the plane Poiseuille and Couette flow geometry is investigated for a viscous fluid. The basic governing momentum and continuity equations are expressed in finite difference form and solved numerically on a high speed digital computer for a mesh network superimposed on the flow field. Results are obtained for the variations of velocity, pressure and resistance coefficient throughout the development region. A characteristic development length is defined and evaluated for both types of flow.Nomenclature h width of channel - L ratio of development length to channel width - p fluid pressure - p ₀ pressure at channel mouth - P dimensionless pressure, p/ ² - P ₀ dimensionless pressure at channel mouth - P pressure defect, P ₀–P - (P)₀ pressure defect neglecting inertia - Re Reynolds number, uh/ - u fluid velocity in x-direction - mean u velocity across channel - u ₀ wall velocity - U dimensionles u velocity u/ - U _c dimensionless centreline velocity - U ₀ dimensionless wall velocity - v fluid velocity in y-direction - V dimensionless v velocity, hv/ - x coordinate along channel - X dimensionless x-coordinate, x/h ² - y coordinate across channel - Y dimensionless y-coordinate, y/h - resistance coefficient, - ₀ resistance coefficient neglecting inertia - fluid density - fluid viscosity     

The secondary flow induced around a sphere in an oscillating stream of elastico-viscous liquid

The present paper is devoted to the theoretical study of the secondary flow induced around a sphere in an oscillating stream of an elastico-viscous liquid. The boundary layer equations are derived following Wang's method and solved by the method of successive approximations. The effect of elasticity of the liquid is to produce a reverse flow in the region close to the surface of the sphere and to shift the entire flow pattern towards the main flow. The resistance on the surface of the sphere and the steady secondary inflow increase with the elasticity of the liquid.Nomenclature a radius of the sphere - b ^ik contravariant components of a tensor - e contravariant components of the rate of strain tensor - F() see (47) - G total nondimensional resistance on the surface of the sphere - g _ik covariant components of the metric tensor - f, g, h secondary flow components introduced in (34) - k ₀ measure of relaxation time minus retardation time (elastico-viscous parameter) - K =k ₀ ²/V ₀ ² , nondimensional parameter characterizing the elasticity of the liquid - n measure of the ratio of the boundary layer thickness and the oscillation amplitude - N, T defined in (44) - p arbitrary isotropic pressure - p _ik covariant components of the stress tensor - p ^ik contravariant components of the stress tensor associated with the change of shape of the material - R =V ₀ a/v, the Reynolds number - S =a/V ₀, the Strouhall number - r, , spherical polar coordinates - u, v, w r, , component of velocity - t time - V(, t) potential velocity distribution around the sphere - V ₀ characteristic velocity - u, v, t, y, P nondimensional quantities defined in (15) - reciprocal of s - density - defined in (32) - defined in (42) - ₀ limiting viscosity for very small changes in deformation velocity - complex conjugate of - oscillation frequency - = ₀/, the kinematic coefficient of viscosity - , defined in (52) - (, y) stream function defined in (45) - =(NT/2n)^1/2 y - /t convective time derivative (1) _ik      

Die swell of filled polymer melts

The Barus effect in polypropylene and polystyrene blended with a variety of fillers at various concentrations was investigated using a capillary extrusion rheometer. If the die swell is defined as the square of the ratio of the extrudate diameterd to the die diameterD, it is found to depend on the apparent shear stress _W . Below a certain value of _w the relation =B _B ^A applies. The die swell, _M , of a filled polymer depends on the type, size and volume fraction of the filler. In particular,A increases as the volume fraction increases and is largest for powders, smaller for flakes and smallest for fibres, whereasB shows the opposite trend but to a lesser extent.     

On some nearly viscometric flows of viscoelastic liquids

Superposition of oscillatory shear imposed from the boundary and through pressure gradient oscillations and simple shear is investigated. The integral fluid with fading memory shows flow enhancement effects due to the nonlinear structure. Closed-form expressions for the change in the mass transport rate are given at the lowest significant order in the perturbation algorithm. The elasticity of the liquid plays as important a role in determining the enhancement as does the shear dependent viscosity. Coupling of shear thinning and elasticity may produce sharp increases in the flow rate. The interaction of oscillatory shear components may generate a steady flow, either longitudinal or orthogonal, resulting in increases in flow rates akin to resonance, and due to frequency cancellation, even in the absence of a mean gradient. An algorithm to determine the constitutive functions of the integral fluid of order three is outlined.Nomenclature A _n Rivlin-Ericksen tensor of order . - A _k Non-oscillatory component of the first order linear viscoelastic oscillatory velocity field induced by the kth wave in the pressure gradient - d Half the gap between the plates - e _x, e _z Unit vectors in the longitudinal and orthogonal directions, respectively - G(s) Relaxation modulus - G History of the deformation - Stress response functional - I() Enhancement defined as the ratio of the frequency dependent part of the discharge to the frequencyindependent part of it at the third order - I ^*() Enhancement defined as the ratio of the increase in discharge due to oscillations to the total discharge without the oscillations - k Power index in the relaxation modulus G(s) - k _i ^–1 Relaxation times in the Maxwell representation of the quadratic shear relaxation modulus (s ₁, s ₂) - m _i ^–1, n _i ^–1 Relaxation times in the Maxwell representations of the constitutive functions ₁(s ₁,s ₂,s ₃) and ₄ (s ₁, s ₂,s ₃), respectively - P Constant longitudinal pressure gradient - p Pressure field - _mx ,⁽³⁾ _nz ,⁽³⁾ Mean volume transport rates at the third order in the longitudinal and orthogonal directions, respectively - ₀,⁽³⁾, ₁,⁽³⁾ Frequency independent and dependent volume transport rates, respectively, at the third order - s = t- Difference between present and past times t and      

On slip effect in free coating of non-Newtonian fluids

The failure of the current theories to predict the coating thickness of non-Newtonian fluids in free coating operations is shown to be a result of the effective slip at the moving rigid surface being coated. This slip phenomenon is a consequence of stress induced diffusion occurring in flow of structured liquids in non-homogeneous flow fields. Literature data have been analysed to substantiate the slip hypothesis proposed in this work. The experimentally observed coating thickness is shown to lie between an upper bound, which is estimated by a no-slip condition for homogeneous solution and a lower bound, which is estimated by using solvent properties. Some design considerations have been provided, which will serve as useful guidelines for estimating coating thickness in industrial practice.fa exponent in eq. (15) - b n/(4 –n)(n + 1) - Ca Capillary number - D diffusivity - De Deborah number - g acceleration due to gravity - G Goucher number - h thickness profile - h ₀ final coating thickness - K consistency index - L length available for diffusion - L _t tube length - n power-law index - P pressure drop - Q flow rate - R cylinder radius - R _t tube radius - t time available for diffusion - T ₀ dimensionless thickness without slip - T _s dimensionless thickness with slip - U _c theoretically calculated withdrawal velocity to match the film thickness - u _s slip velocity - U withdrawal velocity - U _w theoretically calculated withdrawal velocity based on solvent properties - U ^* effective withdrawal velocity - x distance in the direction of flow - y distance transverse to the flow direction - curvature coefficient - slip coefficient - curvature coefficient - rate of deformation tensor - u _s /U - relaxation time - density - surface tension - shear stress in tube flow - _w wall shear stress in tube flow - stress tensor - _w wall shear stress - T _s /T ₀ NCL-Communication No. 2818     

Pulsatile blood flow through arterioles

An analytical study was made to examine the effect of vascular deformability on the pulsatile blood flow in arterioles through the use of a suitable mathematical model. The blood in arterioles is assumed to consist of two layers — both Newtonian but with differing coefficients of viscosity. The flow characteristics of blood as well as the resistance to flow have been determined using the numerical computations of the resulting expressions. The applicability of the model is illustrated using numerical results based on the existing experimental data. r, z coordinate system - u, axial/longitudinal velocity component of blood - p pressure exerted by blood - _b density of blood - µ viscosity of blood - t time - , displacement components of the vessel wall - T _t0,T ₀ known initial stresses - density of the wall material - h thickness of the vessel wall - T _t,T stress components of the vessel - K _l,K _r components of the spring coefficient - C _l,C _r components of the friction coefficient - M _a additional mass of the mechanical model - r ₁ outer radius of the vessel - thickness of the plasma layer - r ₁ inner radius of the vessel - circular frequency of the forced oscillation - k wave number - E ₀,E _t, , _t material parameters for the arterial segment - µ _p viscosity of the plasma layer - Q total flux - Q _p flux across the plasma zone - Q _h flux across the core region - Q mean flow rate - resistance to flow - P pressure difference - l length of the segment of the vessel     

Heat transfer and equilibrium temperature of a sphere in a supersonic rarefied gas flow

Some results are presented of experimental studies of the equilibrium temperature and heat transfer of a sphere in a supersonic rarefied air flow.The notations D sphere diameter - u, , T,,l, freestream parameters (u is velocity, density, T the thermodynamic temperature,l the molecular mean free path, the viscosity coefficient, the thermal conductivity) - T₀ temperature of the adiabatically stagnated stream - T_e mean equilibrium temperature of the sphere - T_w surface temperature of the cold sphere (T_we) - mean heat transfer coefficient - _e air thermal conductivity at the temperature T_e - P Prandtl number - M Mach number     

On the effects of a periodic, spanwise variation of suction upon asymptotic suction flow

Summary The effects of superposing streamwise vorticity, periodic in the lateral direction, upon two-dimensional asymptotic suction flow are analyzed. Such vorticity, generated by prescribing a spanwise variation in the suction velocity, is known to play an important role in unstable and turbulent boundary layers. The flow induced by the variation has been obtained for a freestream velocity which (i) is steady, (ii) oscillates periodically in time, (iii) changes impulsively from rest. For the oscillatory case it is shown that a frequency can exist which maximizes the induced, unsteady wall shear stress for a given spanwise period. For steady flow the heat transfer to, or from a wall at constant temperature has also been computed.Nomenclature (x, y, z) spatial coordinates - (u, v, w) corresponding components of velocity - (, , ) corresponding components of vorticity - t time - stream function for v and w - v _w mean wall suction velocity - nondimensional amplitude of variation in wall suction velocity - characteristic wavenumber for variation in direction of z - T temperature - P pressure - density - coefficient of kinematic viscosity - coefficient of thermal diffusivity - (/v _w)² - frequency of oscillation of freestream velocity - nondimensional amplitude of freestream oscillation - /v _w ² - z z - y –v _w y/ - v _w ² t/4 - /v _w - U ₀ characteristic freestream velocity - u/U ₀ - coefficient of viscosity - _w wall shear stress - Prandtl number (/) - q heat transfer to wall - T _w wall temperature - T (T _w–T)/(T _w–)     

The value of the critical exponent for reaction-diffusion equations in cones

Let D R ^N be a cone with vertex at the origin i.e., D = (0, )x where S ^N–1 and x D if and only if x = (r, ) with r=¦x¦, . We consider the initial boundary value problem: u _t = u+u ^p in D×(0, T), u=0 on Dx(0, T) with u(x, 0)=u ₀(x) 0. Let ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on and let ₊ denote the positive root of (+N–2) = ₁. Let p ^* = 1 + 2/(N + ₊). If 1 < p < p ^*, no positive global solution exists. If p>p ^*, positive global solutions do exist. Extensions are given to the same problem for u _t=+¦x¦ u ^p .This research was supported in part by the Air Force Office of Scientific Research under Grant # AFOSR 88-0031 and in part by NSF Grant DMS-8 822 788. The United States Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright notation therein.     

Measurements of thermally stratified pipe flow using image-processing techniques

The cross-correlation technique and Laser Induced Fluorescence (LIF) have been adopted to measure the time-dependent and two-dimensional velocity and temperature fields of a stably thermal-stratified pipe flow. One thousand instantaneous and simultaneous velocity and temperature maps were obtained at overall Richardson numberRi = 0 and 2.5, from which two-dimensional vorticity, Reynolds stress and turbulent heat flux vector were evaluated. The quasi-periodic inclined vortices (which connected to the crest) were revealed from successive instantaneous maps and temporal variation of vorticity and temperature. It has been recognized that these vortices are associated with the crest and valley in the roll-up motion.List of symbols A Fraction of the available light collected - C Concentration of fluorescence - D Pipe diameter - I Fluorescence intensity - L Sampling length along the incident beam - I ₀ Intensity of an excitation beam - I _c (T) Calibration curve between temperature and fluorescence intensity - I _ref Reference intensity of fluorescence radiation - Re _b Reynolds number based on bulk velocity,U _b D/v - Ri Overall Richardson number based on velocity difference,gDT/U ² - t Time - t Time interval between the reference and corresponding matrix - T Temperature - T ₁,T ₂ Temperature of lower and upper layer - T ^* Normalized temperature, (T–T ₁)/T - T _c (I) Inverse function of temperature as a function ofI _c - T _ref Reference temperature - T Temperature difference between upper and lower flow,T ₂ –T ₁ - U ₁ Velocity of lower stream - U ₂ Velocity of upper stream - U _b Bulk velocity - U _c Streamwise mean velocity atY/D=0 - U Streamwise velocity difference between upper and lower flow,U ₁–U ₂ - u, v, T Fluctuating component ofU, V, T - U, V Velocity component of X, Y direction - X Streamwise distance from the splitter plate - Y Transverse distance from the centerline of the pipe - Z Spanwise distance from the centerline of the pipe - Quantum yield - Absorptivity - vorticity calculated from a circulation - Kinematic viscosity - circulation     

A visualization study of the interaction of a free vortex with the wake behind an airfoil

The two-dimensional interaction of a single vortex with a thin symmetrical airfoil and its vortex wake has been investigated in a low turbulence wind tunnel having velocity of about 2 m/s in the measuring section. The flow Reynolds number based on the airfoil chord length was 4.5 × 10³. The investigation was carried out using a smoke-wire visualization technique with some support of standard hot-wire measurements. The experiment has proved that under certain conditions the vortex-airfoil-wake interaction leads to the formation of new vortices from the part of the wake positioned closely to the vortex. After the formation, the vortices rotate in the direction opposite to that of the incident vortex.List of symbols c test airfoil chord - C vortex generator airfoil chord - TA test airfoil - TE test airfoil trailing edge - TE _G vortex generator airfoil trailing edge - t dimensionless time-interval measured from the vortex passage by the test airfoil trailing edge: gDt=(T-T- _TE)·U/c - T time-interval measured from the start of VGA rotation - U free stream velocity - U vortex induced velocity fluctuation - VGA vortex generator airfoil - y distance in which the vortex passes the test airfoil - Z vortex circulation coefficient: Z=/(U · c/2) - vortex generator airfoil inclination angle - vortex circulation - vortex strength: =/2     

Chaos and integrability in a nonlinear wave equation

We consider the parametrized family of equations _tt ,u- _xx u-au+u ₂ ² u=O,x(0,L), with Dirichlet boundary conditions. This equation has finite-dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invariant sets with chaotic dynamics, provided that =2, 3, 4,.... For =1 we prove the existence of infinitely many first integrals pairwise in involution.     

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

Correlations between porous medium flow data and turbulent tube flow results for aqueous hydroxypropylguar solutions (Part 2)

Turbulent tube flow and the flow through a porous medium of aqueous hydroxypropylguar (HPG) solutions in concentrations from 100 wppm to 5000 wppm is investigated. Taking the rheological flow curves into account reveals that the effectiveness in turbulent tube flow and the efficiency for the flow through a porous medium both start at the same onset wall shear stress of 1.3 Pa. The similarity of the curves = ( _w ) and = ( _w ), respectively, leads to a simple linear relation / =k, where the constantk or proportionality depends uponc. This offers the possibility to deduce (for turbulent tube flow) from (for flow through a porous medium). In conjunction with rheological data, will reveal whether, and if yes to what extent, drag reduction will take place (even at high concentrations).The relation of our treatment to the model-based Deborah number concept is shown and a scale-up formula for the onset in turbulent tube flow is deduced as well.     

The theory of a vibrating-rod viscometer

The paper presents a complete theory for a viscometer based upon the principle of a circular-section rod, immersed in a fluid, performing transverse oscillations perpendicular to its axis. The theory is established as a result of a detailed analysis of the fluid flow around the rod and is subject to a number of criteria which subsequently constrain the design of an instrument. Using water as an example it is shown that a practical instrument can be designed so as to enable viscosity measurement with an accuracy of ±0.1%, although it is noted that many earlier instruments failed to satisfy one or more of the newly-established constraints.Nomenclature A, D constants in equation (46) - A _m , B _m , C _m , D _m constants in equations (50) and (51) - A _j , B _j constants in equation (14) - a _j ⁺ , a _j ^– wavenumbers given by equation (15) - C _f drag coefficient defined in equation (53) - c speed of sound - D _b drag force of fluid b - D ₀ coefficient of internal damping - E extensional modulus - f(z) initial deformation of rod - f(), F _m () functions of defined in equation (41) - F force in the rod - force per unit length near t=0 - F dimensionless force per unit length near t=0 - g _m amplitude of transient force - G modulus of rigidity - h, h* functions defined by equations (71) and (72) - H functions defined by equation (69) and (70) - I second moment of area - I _0,1, J _0,1, K _0,1 modified Bessel functions - k, k functions defined in equations (2) - L half-length of oscillator - Ma Mach number - m _b added mass per unit length of fluid b - m _s mass per unit length of solid - n _j eigenvalue defined in equations (15) and (16) - R radius of rod - R _c radius of container - r radial coordinate - T tension - T _visc temperature rise due to heat generation by viscous dissipation - t time - v _r , v radial and angular velocity components - y lateral displacement - y ₀ initial lateral displacement - y ₁, y ₂ successive maximum lateral displacement - z axial coordinate - dimensionless tension - dimensionless mass of fluid - dimensionless drag of fluid - amplification factor - logarithmic decrement in a fluid - _a , _b logarithmic decrement in fluids a and b - ₀ logarithmic decrement in vacuo - _j logarithmic decrement in mode j in a fluid - spatial resolution of amplitude - _v voltage resolution - r, , , _s, , increments in R, , , _s , , - dimensionless amplitude of oscillation - dimensionless axial coordinate - angular coordinate - _f thermal conductivity of fluid - viscosity of fluid - viscosity of fluid calculated on assumption that * - _a , _b viscosity of fluids a and b - _m constants in equation (10) - dimensionless displacement - _j j the component of - density of fluid - _a , _b density of fluids a and b - _s density of tube or rod material - dimensionless radial coordinate - * dimensionless radius of container - dimensionless times - spatial component of defined in equation (11) - _j , _tm jth, mth component of - dimensionless streamfunction - ⁰, ¹ components of in series expansion in powers of - streamfunction - dimensionless frequency (based on ) - angular frequency - ₀ angular frequency in absence of fluid and internal damping - _j angular frequency in mode j in a fluid - _a , _b frequencies in fluids a and b