SOME EXTENDED RESULTS OF “SUBHARMONIC RESONANCE BIFURCATION THEORY OF NONLINEAR MATHIEU EQUATION”

The authors of [1] discussed the subharmonic resonance bifurcation theory of nonlinear Mathieu equation and obtained six bifurcation diagrams in (α,β )-plane. In this paper, we extended the results of[1] and pointed out that there may exist as many as fourteen bifurcation diagrams which are not topologically equivalent to each other.     

Electrical charging of bodies as a consequence of charged-particle extraction by a hydrodynamic flow

We investigate the problem of electrical charging of bodies as a result of charged-particle extraction by a hydrodynamic flow. The analysis is performed in view of the application to the problem of motion electrification of aircraft caused by a stream of charged particles into the surrounding space. We formulate the appropriate system of nonstationary electrohydrodynamic equations. It is shown that in many applications the charging of electrically insulated bodies consists of two successive intermediate processes. The first process is the formation of charge Q on the body in time T₁ The second process consists of a change of the body potential (with a constant charge Q) as a consequence of the stream of charged particles into the outside space noted above. At the end of the second process (with duration T₂) the body potential is at . We also investigate the problem of charging a spherical source of neutral and charged particles. Using the analytical solution we find the quantities Q and and the characteristic times T₁ and T₂. It is shown that the time T₂ can exceed T₁ by several orders of magnitude. We formulate the problems of nonstationary electric fields during the extraction of several types of particles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 94–103, September–October, 1977.     

A note on the tan δ(T) peak as a glass transition indicator in biosolids

The plasticization of many biosolids can take place over a fairly broad temperature range. The resulting loss of stiffness is primarily expressed by a drastic drop of G(T) whose magnitude is usually higher than G(T) by one or two orders of magnitude. Both G(T) and G(T) have characteristic properties that can vary widely among biomaterials. Consequently, the tan (T) peak need not be a mark of the transition center and it can be observed at temperatures where different materials have undergone a very different degree of plasticization as judged by the magnitude of G(T). This is demonstrated by computer simulations using typical functions that describe G(T) and G(T) at the glass transition region and with published data on the dynamic mechanical behavior of a variety of biosolids.     

Motion of a small sphere in a nonhomogeneous flow of an incompressible liquid

We consider the motion of a small sphere in an arbitrary potential flow of an ideal liquid. For the general case we obtain an integral of the equations of motion and a particular solution. We find flows in which the force acting on the sphere is central. We also obtain exact solutions of the equations of motion of the sphere for the cases of stationary flows around a cylinder and around a body of revolution when the forces are noncentral. N. E. Zhukovskii [1] calculated the force acting on a fixed sphere in an arbitrary nonstationary flow. Kelvin [2] obtained the equations of motion of a sphere in a stationary flow of a liquid circulating through a hole in a solid. A formula for the force F, acting on a fixed small body of volume V in a stationary flow with speed v, was obtained by Taylor [3]: F = (T ₀ / v)Vv + 1/2_V v ² Here T₀ is the kinetic energy of an unbounded liquid in which a body moves with velocity v. This problem was solved in [3] through a direct integration of the pressure forces over the surface of the body in a flow defined by multipoles of the first and second orders at infinity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 57–61, September–October, 1973.     

Calculation of shear influences on the phase separation of polymer blends exhibiting upper critical solution temperatures

Calculations were performed on the basis of a generalized Gibbs energy of mixing G , which is the sum of the Gibbs energy of mixing of the stagnant system and E _s, the energy stored in the system during stationary flow. With increasing shear rate , the demixing temperatures shift to lower values (shear-induced mixing; diminution of the heterogeneous area), then to higher values (shear-induced demixing), and finally to lower values again before the effects fade out. The details of the rather complex phase diagrams resulting for a given shear rate are primarily determined by a band in the T/ plane ( = mole fraction) within which (² E _s/²) _T <0 (i.e., E _S acts towards phase separation). There are two ranges of within which closed miscibility gaps can exist: The more common outer islands are partly or totally situated outside the equilibrium gap (and within the above mentioned band). As is raised they break away from the mainland at the upper end of the first region of shear-induced mixing and shift to T>UCST where they submerge. Bound to a suitable choice of parameters, a second kind of closed miscibility gaps, the inner islands, which always remain within the equilibrium solubility gap (and outside the band of negative curvature of E _S) is additionally observed. This time the islands break away from the mainland at the lower end of the first region of shear-induced mixing where they also submerge. The present findings are compared with the results of previous calculations for LCSTs.     

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     

The Hopf bifurcation with symmetry for the Navier-Stokes equations in (Lp(Ω)) n,with application to plane poiseuille flow

The existence of periodic solutions of the Navier-Stokes equations in function spaces based upon (L _p())ⁿis proved. The paper has three parts, (a) A proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of (L _p())ⁿ. (b) The evolution equation is restricted to a space of time periodic functions and a Fredholm integral equation on this space is formed. The Lyapunov-Schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry. (c) The theory is applied to the bifurcation of periodic solutions from planar Poiseuille flow in the presence of symmetry (SO(2) x O(2) x S ¹) yielding new results for this classic problem. The O(2) invariance is in the spanwise direction. With the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction. There are however points of degeneracy on the neutral surface. A numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi-periodic solutions.     

Wave angle for oblique detonation waves

The flow field associated with a steady, planar, oblique detonation wave is discussed. A revision is provided for- diagrams, where is the wave angle and is the ramp angle. A new solution is proposed for weak underdriven detonation waves that does not violate the second law. A Taylor wave, encountered in unsteady detonation waves, is required. Uniqueness and hysteresis effects are also discussed.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Das erweiterte Korrespondenzgesetz für die dynamische Idealviskosität und die Idealwärmeleitfähigkeit reiner Stoffe

Zusammenfassung Zur Berechnung der dynamischen Idealviskosität Ideal (T) und der Idealwärmeleitfähigkeit ideal (T) benötigt man die kritische TemperaturT _kr, das kritische spezifische Volum _kr, die MolmasseM, den kritischen Parameter _kr und die molare isochore WärmekapazitätC _v(T). Sowohl das theoretisch, als auch das empirisch abgeleitete erweiterte Korrespondenzgesetz ergeben eine für praktische Zwecke ausreichende Genauigkeit für die Meßwertwiedergabe, die bei den assoziierenden Stoffen und den Quantenstoffen jedoch geringer ist als bei den Normalstoffen.

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The extended correspondence law for the ideal dynamic viscosity and the ideal thermal conductivity of pure substances

For the calculation of the ideal dynamic viscosity Ideal (T) and the ideal thermal conductivity ideal (T) the critical temperatureT _kr, the critical specific volumev _kr, the molecular massM, the critical parameter _kr, and the molar isochoric heat capacityC _v(T) is needed. Not only the theoretically determined but also the empirically determined extended correspondence law gives for practical use a good representation of the measured data, which for the associating substances and the quantum substances is not so good as for the normal substances.

     

Codimension two bifurcations of symmetric cycles in Hamiltonian systems with an antisymplectic involution

Consider a Hamiltonian system with parameters, such that there exists an involution which reverses this Hamiltonian system. Let us assume the linear part L at =0 has only nonzero purely imaginary eigen-values ±ib₁,..., ±ib_n. In this paper, we classify the typical bifurcations of families of symmetric periodic solutions of this system at resonance if b_i/b_j=±1, ±2, or ±1/2 and the number of parameters needed is one or two. First, one puts the Hamiltonians into a convenient normal form. Next, applying a Lyapunov-Schmidt reduction and making further manipulations, one can geta reduced bifurcation equation which can possess certain symmetry. Finally, by using elementary methods from singularity theory or isotopy methods, one obtains the desired bifurcation diagrams.     

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     

Mixed convection on vertical plate for fluids of any prandtl number

A mixed convection parameter=(Ra) ^1/4/(Re)^1/2, with=Pr/(1+Pr) and=Pr/(1 +Pr)^1/2, is proposed to replace the conventional Richardson number, Gr/Re², for combined forced and free convection flow on an isothermal vertical plate. This parameter can readily be reduced to the controlling parameters for the relative importance of the forced and the free convection,Ra ^1/4/(Re ^1/2 Pr ^1/3) forPr 1, and (RaPr)^1/2/(RePr ^1/2 forPr 1. Furthermore, new coordinates and dependent variables are properly defined in terms of, so that the transformed nonsimilar boundary-layer equations give numerical solutions that are uniformly valid over the entire range of mixed convection intensity from forced convection limit to free convection limit for fluids of any Prandtl number from 0.001 to 10,000. The effects of mixed convection intensity and the Prandtl number on the velocity profiles, the temperature profiles, the wall friction, and the heat transfer rate are illustrated for both cases of buoyancy assisting and opposing flow conditions.

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Mischkonvektion an einer vertikalen Platte für Fluide beliebiger Prandtl-Zahl

Zusammenfassung Für die kombinierte Zwangs- und freie Konvektion an einer isothermen senkrechten Platte wird ein Mischkonvektions-Parameter=( Ra) ^1/4 (Re)^1/2, mit=Pr/(1 +Pr) und=Pr/(1 +Pr)^1/2 vorgeschlagen, den die gebräuchliche Richardson-Zahl, Gr/Re², ersetzen soll. Dieser Parameter kann ohne weiteres auf die maßgebenden Kennzahlen für den relativen Einfluß der erzwungenen und der freien Konvektion reduziert werden,Ra ^1/4/(Re ^1/2 Pr ^1/3) fürPr 1 und (RaPr)^1/4/(RePr)^1/2 fürPr 1. Weiterhin werden neue Koordinaten und abhängige Variablen als Funktion von definiert, so daß für die transformierten Grenzschichtgleichungen numerische Lösungen erstellt werden können, die über den gesamten Bereich der Mischkonvektion, von der freien Konvektion bis zur Zwangskonvektion, für Fluide jeglicher Prandtl-Zahl von 0.001 bis 10.000 gleichmäßig gültig sind. Der Einfluß der Intensität der Mischkonvektion und der Prandtl-Zahl auf die Geschwindigkeitsprofile, die Temperaturprofile, die Wandreibung und den Wärmeübergangskoeffizienten werden für die beiden Fälle der Strömung in und entgegengesetzt zur Schwerkraftrichtung dargestellt.

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Nomenclature C _f local friction coefficient - C _p specific heat capacity - f reduced stream function - g gravitational acceleration - Gr local Grashoff number,g T _w –T )x³/v² - Nu local Nusselt number - Pr Prandtl number,v/ - Ra local Rayleigh number,g T _w –T x ³/( v) - Re local Reynolds number,u x/v - Ri Richardson number,Gr/Re ² - T fluid temperature - T _w wall temperature - T free stream temperature - u velocity component in thex direction - u free stream velocity - v velocity component in they direction - x vertical coordinate measuring from the leading edge - y horizontal coordinate Greek symbols thermal diffusivity - thermal expansion coefficient - mixed convection parameter (Ra)1/4/Re)^1/2 - pseudo-similarity variable,(y/x) - ₀ conventional similarity variable,(y/x)Re ^1/2 - dimensionless temperature, (T–T T _W –T - unified mixed-flow parameter, [(Re) ^1/2 + (Ra)^1/4] - dynamic viscosity - kinematic viscosity - stretched streamwise coordinate or mixed convection parameter, [1 + (Re)^1/2/(Ra) ^1/4]^–1=/(1 +) - density - Pr/(1 + Pr) _w wall shear stress - stream function - Pr/(l+Pr)^1/3 This research was supported by a grand from the National Science Council of ROC     

A Degenerate Bifurcation Structure in the Dynamics of Coupled Oscillators with Essential Stiffness Nonlinearities

We study the degenerate bifurcations of the nonlinear normal modes(NNMs) of an unforced system consisting of a linear oscillator weaklycoupled to a nonlinear one that possesses essential stiffnessnonlinearity. By defining the small coupling parameter , we study thedynamics of this system at the limit 0. The degeneracy in the dynamics ismanifested by a 'bifurcation from infinity' where a bifurcation point isgenerated at high energies, as perturbation of a state of infiniteenergy. Another (nondegenerate) bifurcation point is generated close tothe point of exact 1:1 internal resonance between the linear andnonlinear oscillators. The degenerate bifurcation structure can bedirectly attributed to the high degeneracy of the uncoupled system inthe limit 0, whose linearized structure possesses a double zero, and aconjugate pair of purely imaginary eigenvalues. First we construct localanalytical approximations to the NNMs in the neighborhoods of thebifurcation points and at other energy ranges of the system. Then, we`connect' the local approximations by global approximants, and identifyglobal branches of NNMs where unstable and stable mode and inverse modelocalization between the linear and nonlinear oscillators take place fordecreasing energy.     

Interaction between thermocapillary and buoyancy driven convection

Convective flows driven by the variation of surface tension due to a radial temperature gradient along a liquid-gas interface were studied. Three liquids of different viscosities were applied, so that a wide range of Marangoni numbers was encountered. Light sheet technique and differential interferometry were taken to analyse the thermal flows. The mechanism of stationary thermocapillary convection, the influence of the radial temperature gradient and the kinematic viscosity on the Marangoni boundary layer thickness are discussed. Transitions from the steady to the oscillatory Marangoni convection are discovered and the oscillations are visualized with differential interferometry.List of symbols a thermal diffusivity - D cell diameter - f tangential stress - H cell height - Mg Marangoni number, Mg = U · R/a - Pr Prandtl number, Pr = v/a - r radial coordinate tangential to the interface - R cell radius - Re Reynolds number, Re = UR/v - T temperature - T _b, T_m temperature at the boundary and in the centre of the cell, respectively - T temperature difference, T — T _b — T_m - U reference velocity, U = ¦d/dT¦(T/R) R/ - v _r radial stream velocity - v _x velocity at the interface - z axial coordinate normal to the interface - dynamic viscosity - kinematic viscosity - surface tension - d/dT thermal coefficient of surface tension A version of this paper was presented at the 7th Physico-Chemical Hydrodynamics, PCH Conference, June 25–29, 1989, Cambridge, MA, USA     

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–)     

Random perturbations of bifurcation diagrams

Random perturbations of one dimensional bifurcation diagrams can exhibit qualitative behavior that is quite different from that of the unperturbed, deterministic situation. For Markov solutions of one dimensional random differential equations with bounded ergodic diffusion processes as perturbations, effects like disappearance of stationary Murkov solutions (break through), slowing down, bistability, and random symmetry breaking can occur. These effects are partially the results of local considerations, but as the perturbation range increases, global dynamics can alter the picture as well. The results are obtained via the analysis of stationary solutions of degenerate Markov diffusion processes, of stationary, non-Markovian solutions of stochastic flows, and of Lyapunov exponents of stochastic flows with respect to steady states.     

Solidification behavior of the theta system 2-propanol/poly(n-butyl methacrylate)

Thermoreversible gelation of the system 2-propanol/poly (n-butyl methacrylate) — as detected by D'SC or dielectric experiments — does not manifest itself in a straightforward manner in the dynamic-mechanical properties. Its occurrence can, however, be seen in many ways: i) For constant composition of the system and a reference temperature lower than T _gel, the storage modulus G is larger than the loss modulus G in the glass transition zone of the master curve and both vary in an almost parallel manner with the angular frequency over almost two decades (whereas this feature is normally found for other gelling systems within the rubber plateau or the flow region). ii) The entanglement molecular weight obtained from G_max is markedly less max than the entanglement molecular weight in the melt divided by ₂, the volume fraction of the polymer. iii) The temperature influences change from WLF like to Arrhenius-like behavior as T is lowered in the case of highly concentrated polymer solutions; analogous considerations hold true as ₂ is increased at constant T. iv) For sufficiently low temperatures, the activation energy of flow exhibits a maximum in the concentration range where the gelation is — according to DSC experiments — most pronounced. Like with ordinary non-gelling systems it is possible to construct master curves. On the basis of Graessley's theory identical dependencies are obtained for the variation of the entanglement parts of the stationary viscosity with shear rate and for the dependence of the entanglement part of the complex viscosity on the frequency of oscillation. Zero shear viscosity and limiting value of the complex viscosity for vanishing as a function of ₂ match smoothly and exhibit two points of inflection.     

Symmetry-Breaking Bifurcations of Charged Drops

It has been observed experimentally that an electrically charged spherical drop of a conducting fluid becomes nonspherical (in fact, a spheroid) when a dimensionless number X inversely proportional to the surface tension coefficient is larger than some critical value (i.e., when <_c). In this paper we prove that bifurcation branches of nonspherical shapes originate from each of a sequence of surface-tension coefficients ), where ₂=_c. We further prove that the spherical drop is stable for any >₂, that is, the solution to the system of fluid equations coupled with the equation for the electrostatic potential created by the charged drop converges to the spherical solution as t provided the initial drop is nearly spherical. We finally show that the part of the bifurcation branch at =₂ which gives rise to oblate spheroids is linearly stable, whereas the part of the branch corresponding to prolate spheroids is linearly unstable.     

Thermal stability of composite superconducting tape under the effect of a two-dimensional dual-phase-lag heat conduction model

Thermal stability of composite superconducting tape subjected to a thermal disturbance is numerically investigated under the effect of a two-dimensional dual-phase-lag heat conduction model. It is found that the dual-phase-lag model predicts a wider stable region as compared to the predictions of the parabolic and the hyperbolic heat conduction models. The effects of different design, geometrical and operating conditions on superconducting tape thermal stability were also studied.a conductor width, (m) - A conductor cross sectional area of, (m²) - As conductor aspect ratio, (a/b) - b conductor thickness, (m) - Bi Biot number - B dimensionless disturbance Intensity - C heat capacity, (J m^–3 K^–1) - D disturbance energy density, (W m^–3) - f volume fraction of the stabilizer in the conductor - g(T) steady capacity of the Ohmic heat source, (W m^–3) - g_max Ohmic heat generation with the whole current in the stabilizer, (W m^–3) - G_max dimensionless maximum Joule heating - h convective heat transfer coefficient, (W m^–2 K^–1) - J current density, (A m^–2) - k thermal conductivity of conductor, (W m^–1 K^–1) - q conduction heat flux vector, (W m^–2) - Q dimensionless Joule heating - R relaxation times ratio (_T/2_q) - t rime, (s) - T temperature, (K) - T_c critical temperature, (K) - T_c1 current sharing temperature, (K) - T_i initial temperature, (K) - T_o ambient temperature, (K) - x, y co-ordinate defined in Fig. 1, (m) - thermal diffusivity (m² s^–1) - dimensionless time - _i dimensionless duration time - dimensionless y-variable - _o superconductor dimensionless thickness - dimensionless temperature - _c1 dimensionless current sharing temperature - 1 dimensionless maximum temperature - dimensionless disturbance energy - numerical tolerance - x width of conductor subjected to heat disturbances, (m) - y thickness of conductor subjected to heat disturbances, (m) - dimensionless x-variable - _o superconductor dimensionless width - stabilizer electrical resistivity, () - _q relaxation time of heat flux, (s) - _T relaxation time of temperature gradient, (s) - i initial - sc current sharing - max maximum - o ambient     

Concentration dependence of the critical viscoelastic properties of gelatin at the gel point

Gelatin gel properties have been studied through the evolution of the storage [G()] and the loss [G()] moduli during gelation or melting near the gel point at several concentrations. The linear viscoelastic properties at the percolation threshold follow a power-law G()G() and correspond to the behavior described by a rheological constitutive equation known as the Gel Equation. The critical point is characterized by the relation: tan = G/G = cst = tan ( · /2) and it may be precisely located using the variations of tan versus the gelation or melting parameter (time or temperature) at several frequencies. The effect of concentration and of time-temperature gel history on its variations has been studied. On gelation, critical temperatures at each concentration were extrapolated to infinite gel times. On melting, critical temperatures were determined by heating step by step after a controlled period of aging. Phase diagrams [T = f(C)] were obtained for gelation and melting and the corresponding enthalpies were calculated using the Ferry-Eldridge relation. A detailed study of the variations of A with concentration and with gel history was carried out. The values of which were generally in the 0.60–0.72 range but could be as low as 0.20–0.30 in some experimental conditions, were compared with published and theoretical values.