Relaxation time, Maxwellian

Keywords self-diffusion coefficient, thermal hydrodynamic fluctuations, dynamic viscosity, kinematic viscosity, Maxwellian relaxation time... 

Figure 4a. Real reduced modulus plotted with the Maxwellian relaxation time-frequency product, f VsJoo for PPO (A), 59° scattering angle (O), 20°...

The time derivative can be replaced with the Maxwellian relaxation time t, which is the time during which the stresses in the metal are damped after the applied forces are removed (11). 

It has been a common practice to describe visco-elastic fluid behaviour in steady shear in terms of a shear stress Ty and the first normal stress difference (N ) both of which are functions of shear rate. Generally, a fluid relaxation or characteristic time, Xf, (or a spectrum) is defined to quantify the viscoelastic behaviour. There are several ways of defiiung a characteristic time by combining shear stress and the first normal stress difference, e.g. the so-called Maxwellian relaxation time is given by ... 

Capillary forces in mixed fluid phase conditions are inversely proportional to the curvature of the interface. Therefore, menisci introduce elasticity to the mixed fluid, and mixtures of two Newtonian fluids exhibit global Maxwellian response. For more details see Alvarellos [1], his behavior is experimentally demonstrated with a capillary tube partially filled with a water droplet. The tube is tilted at an angle (3 smaller than the critical angle that causes unstable displacement. Then, a harmonic excitation is applied to the tube in the axial direction. For each frequency, the amplitude of the vibration is increased until the water droplet becomes unstable and flows in the capillary. Data in Figure 3 show a minimum required tube velocity between 40 and 50 Hz. This behavior indicates resonance of the visco-elastic system. The ratio of the relaxation time and characteristic time for pure viscous effect is larger than 11.64. 

Let us consider an ensemble containing a large number of isolated doublets exposed to a heat bath of temperature T at time t = 0. If the time scale for the doublet to reach thermal equilibrium with the heat bath is much smaller than the time scale of dissociation of the doublet, the initial energy distribution can be assumed to be Maxwellian. Since the depth of the potential well is of the order of or greater than kT, even for doublets of sufficiently small particles, the time scale to reach thermal equilibrium (which is of the order of the relaxation time for Brownian motion) is much smaller than the time scale of dissociation. Consequently, it is reasonable to approximate the initial energy distribution as Maxwellian, i.e.,... 

In modeling the interaction of a liquid with plate modes, the high frequency of operation necessitates the consideration of viscoelastic response by the liquid. For the simple liquids examined, good agreement was obtained by modeling the liquid as a Maxwellian fluid with a single relaxation time r. When the Maxwellian fluid is driven in oscillatory flow with cot < 1, it responds as a Newtonian fluid characterized by the shear viscosity, rj. For wt > 1, the oscillation rate approaches the rate of molecular motion in the liquid and energy ceases to be dissipated in... 

The rheological consequences of the Maxwell model are apparent in stress relaxation phenomena. In an ideal solid, the stress required to maintain a constant deformation is constant and does not alter as a function of time. However, in a Maxwellian body, the stress required to maintain a constant deformation decreases (relaxes) as a function of time. The relaxation process is due to the mobility of the dashpot, which in turn releases the stress on the spring. Using dynamic oscillatory methods, the rheological behavior of many pharmaceutical and biological systems may be conveniently described by the Maxwell model (for example, Reference 7, Reference 17, References 20 to 22). In practice, the rheological behavior of materials of pharmaceutical and biomedical significance is more appropriately described by not one, but a finite or infinite number of Maxwell elements. Therefore, associated with these are either discrete or continuous spectra of relaxation times, respectively (15,18). 

The high frequency of the APM necessitates consideration of viscoelastic response by the liquid. Modeling simple liquids as Maxwellian fluids with a single relaxation time, r, gives good agreement with experimental data. When the Maxwellian fluid is driven in... 

Fig. 6. The variation of relative retardation with time for L2 at various thicknesses d ranging from 200 to 1763 pm. Solid-line curve is a calculated Maxwellian relaxation curve.

In Fig. 8 an experimental result for the sample L2 of thickness 380 ym (curve a in Fig. 8) is compared with a Maxwellian relaxation curve (curve b in Fig. 8). Both curves deviate from each other at larger t. In the first approximation, the deviation is considered to have been brought by the wall effect, i.e. the appearance of B-phase. The B-phase will grow with time under the influence of wall effect to some extent [the volume fraction of which is denoted here by Xg( ). This is schematically illustrated in Fig. 7. When the transformation into the B-phase is assumed to be analogous to one-demensional crystal-crystal transformation, the volume fraction of B-phase at t = t is given by... 

In the equation (14), f = f(x,, t)is distribution fimction of only particle, is microscopic velocity, X is the relaxation time due to collision, and g is the Boltzmann-Maxwellian distribution function (fM), is important mention that collision term has been transforming in accordance with equation (2). 

Assuming the Maxwellian hypothesis, the relaxation time spectrum can be expressed in a discrete form, leading to... 

In order to illustrate the typical nonlinear mechanical response of wormlike micelles under steady shear flow, we chose to focus on the cetylpyridinium (CPCl)/sodium saUcylate (NaSal) system. It is often considered as a model system since it follows the right scaling laws for the concentration dependence of the static viscosity and plateau modulus [32]. Moreover, for concentrations ranging from 1 to 30wt. %, the samples behave, in the linear regime, as almost perfect Maxwellian elements with a single relaxation time Tr and a plateau modulus Go- This system has been... 

Several rheological studies showed that solutions of telechelic polymers submitted to oscillatory shear exhibit a Maxwellian behavior (single relaxation time). ° ° This behavior was explained by assuming that the main stress relaxation mechanism is the exit of one hydrophobe from the... 

The relaxation equation for a Maxwellian incompressible isotropic liquid is obtained from the law of anisotropic relaxation in the following partial form of the relaxation time Xj.ju and viscosity tensors ... 

From these time-scales, it may be assumed in most circumstances that the free electrons have a Maxwellian distribution and that the dominant populations of impurities in the plasma are those of the ground and metastable states of the various ions. The dominant populations evolve on time-scales of the order of plasma diffusion time-scales and so should be modeled dynamically, that is in the particle number continuity equations, along with the momentum and energy equations of plasma transport theory. The excited populations of impurities on the other hand may be assumed relaxed with respect to the instantaneous dominant populations, that is they are in a quasi-equilibrium. The quasi-equilibrium is determined by local conditions of electron temperature and electron density. So, the atomic modeling may be partially de-coupled from the impurity transport problem into local calculations which provide quasi-equilibrium excited ion populations and effective emission coefficients (PEC coefficients) and then effective source coefficients (GCR coefficients) for dominant populations which must be entered into the transport equations. The solution of the transport equations establishes the spatial and temporal behaviour of the dominant populations which may then be re-associated with the local emissivity calculations, for matching to and analysis of observations. 

This result is purely statistical. Replacing the distribution function by particular expressions, depending on the temperature, is the last operation When a dynamical process occurs the equilibrium distribution function (maxwellian) should be modified, and the greater the reaction rate compared to the relaxation rates of both the velocities and the intramolecular states, the greater the modification Thus it is only for low reaction rates that equilibrium distribution functions can be inserted in the formulas above, and that the reaction rate depends on the temperature, but neither on the time nor on the concentrations. 

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