Fluid Maxwellian

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... 

Suppose that some constant deformation has been erected in a Maxwellian fluid and some effort is made to maintain this deformation in the course of time. Then the arising flow will gradually release the applied stress, and the effort needed to maintain the deformation will also decay. Under these conditions (r = ro, 7 = 0 at t = 0, and 7 = const for t > 0), the solution of Eq. (6.1.10) has the form... 

Viscoelastic fluids. In the monograph [181], the exact solution of Stokes first problem (6.10.1)—(6.10.3) was obtained for Maxwellian fluids with the rheological law (6.1.10), which has the following form for the problem in question ... 

Oscillating Flat-Plate Flow for Maxwellian Fluids... 

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... 

Here, is the mean streaming velocity of particles approaching the wall and (1 — a) is the fraction of fluid particles reflected at the wall, so the first term represents the distribution of particles adsorbed. The velocity distribution functions, /(v), are assumed to be Maxwellian,... 

Most fluids in engineering are Newtonian. However, fluids such as petroleum and blood are Maxwellian, that is, the stress tensor is not only a function of the strain rate but also a function of strain itself. The Maxwellian relaxation is the ratio between the fluid viscosity and stiffness. 

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. 

However, a much more tractable form can be derived, strongly reducing the number of parameters in these data tables. The first step follows from the fact, that consistent with the fluid approximation for the charged particles a we can make the assumption of a near Maxwellian distribution... 

The consequence of this organization at the molecular level is that BTC solutions in decane are viscoelastic fluids with a nearly perfect Maxwellian behavior [33]. The reason why this BTC system is viscoelastic whereas the previous UPy based system is purely viscous (Newtonian behavior) is probably related to a more rigid backbone and/or to a slower breaking of the chains. 

Using the theory developed by Chapman-Enskog (see Ref. 14), a hierarchy of continuum fluid mechanics formulations may be derived from the Boltzmann equation as perturbations to the Maxwellian velocity distribution function. The first three equation sets are well known (1) the Euler equations, in which the velocity distribution is exactly the Maxwellian form (2) the Navier-Stokes equations, which represent a small deviation from Maxwellian and rely on linear expressions for viscosity and thermal conductivity and (3) the Burnett equations, which include second order derivatives for viscosity and thermal conductivity. 

During the convective transport individual target molecules are dispersed by the presence of small eddies. The random walk motion of small particles suspended in a fluid due to bombardment by molecules obeys the Maxwellian velocity distribution. If a number of particles subject to Brownian motion are present in a given medium and there is no preferred direction for... 

To determine the energy contained in eddies of different scales, a distribution function of the kinetic energy for eddies in turbulent flows is required. A Maxwellian distribution function may be a natural and consistent choice as the eddy velocity is assumed to follow this distribution [66], but Luo and Svendsen [74] preferred an empirical energy-distribution density function for fluid particles in liquid developed by Angelidou et al [1[. The turbulent kinetic energy distribution of eddies with size A is approximated as follows ... 

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 ... 

The fact that a local equilibrium assumption is an essential part of the hydrodynamic description of fluid flow suggests that we look for solutions of the Boltzmann equation where the distribution function is close to a local Maxwellian distribution /ie(r, v, t) given by... 

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