Fluid second-order perturbation

The results of this second-order perturbation theory for a fluid whose pair potential is the Lennard-Jones 6 12 potential,... 

Figure 2. Equation of state of the 6 12 fluid. The points and curves give computer simulation and the second-order perturbation theory results for seven isotherms that are labelled with the appropriate values of kT/e. For the Lennard-Jones fluid the triple-point reduced temperature ana density are about 0.7 and 0.85, respectively, and the critical-point reduced temperature and density are about 1.30 and 0.30, respectively.

Kaloni used Oldroyd model, Schtimmer a fourth order fluid model, while Wissler a nonhnear Maxwell model Employing the perturbation method, the authors observed that the inclusion of second-order perturbation terms (which bring in the non-Newtonian effects) predicted velocity profiles with superimposed secondary circulation patterns. 

Monomer contribution As an alternative to the mentioned TPTl versions, we will attempt to describe the thermodynamics and structme of the binary reference mixture of monomers by means of a second order perturbation theory, based on an analytical solution of the Mean Spherical Approximation (MSA) of simple fluids [299,317]. In this theory, the free energy of the mixture of monomers is described perturbatively in terms of the properties of an auxiliary fluid which contains only repulsive interactions. We therefore split the full Lennard—clones potential, Vy(r) into repulsive and perturbative contributions as suggested by Barker and Henderson [300], so that the repulsive potential, contains all of the positive part of the Lennard—Jones potential, while the perturbation, wf contains all of the negative region ... 

The first and second order perturbation contributions could be evaluated by using rigorous expressions for and gjj as obtained from an integral equation theory. Such an approach has been recently undertaken with good results [301, 303], Unfortunately, the expressions are quite lengthy and somewhat inconvenient for further differentiation. For this reason, we will evaluate the perturbative contributions of the free energy by using a Van der Waals like one fluid approximation. In this approximation, one considers that the radial distribution function of the Lennard-Jones fluid mixture may be expressed in terms of the radial distribution function of a pure effective Lennard-Jones fluid with composition dependent parameters, try and ey, yet to be determined. More specifically, one assumes that gij (r) may be expressed in terms of the radial distribution function of a pure fluid as follows. 

The general aspect of this curve with an inflexion point is typical of the dynamic behaviour of a second-order overdamped system in response to a step perturbation, i.e. Heaviside function [4] A property of this curve is that the concentration reached at the plateau is equal to the concentration of the fluid entering R2. Data of these runs are summarized in table 2 and presented in Figures 3 and 4. 

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. 

It remains to determine the four sets of constants, An, Bn, Cn, and Dn, and the function fo that describes the 0(Ca) correction to the shape of the drop. For this, we still have the five independent boundary conditions, (7-207)-(7-210). It can be shown that the conditions (7-207), (7-208), and (7-209) are sufficient to completely determine the unknown coefficients in (7-213) and (7-215). Indeed, for any given (or prescribed) drop shape, the four conditions of no-slip, tangential-stress continuity and the kinematic condition are sufficient along with the far-field condition to completely determine the velocity and pressure fields in the two fluids. The normal-stress condition, (7-210), can then be used to determine the leading-order shape function /0. Specifically, we can use the now known solutions for the leading-order approximations for the velocity components and the pressure to evaluate the left-hand side of (7 210), which then becomes a second-order PDE for the function /(). The important point to note is that we can determine the 0(Ca) contribution to the unknown shape knowing only the 0(1) contributions to the velocities and the pressures. This illustrates a universal feature of the domain perturbation technique for this class of problems. If we solve for the 0(Cam) contributions to the velocity and pressure, we can... 

The method used to research stability of small perturbations of a cylindrical jet of non-viscous incompressible fluid is similar to the previously discussed method of handling the small perturbation problem. The main difference from the case of a plane surface is the axial symmetry of the problem, which consequently features the characteristic linear size a (radius of the jet). In a coordinate system moving with the jet s velocity, the jet itself is motionless. Let us neglect gravity and take into account only the force of surface tension. Then the pressure along the jet is equal to p = pa+ 2/a (since l i = oo, i 2 = a). Now proceed to linearize the equations and the boundary conditions. Assume the perturbations of the flow to be small and consider the equation of motion. After linearization, i.e. after rejecting the second order terms, one obtains Eq. (17.36) for velocity perturbations and full pressure. Since the flow is a potential one, any perturbation of the velocity potential satisfies the Laplace equation = 0, which in a cylindrical coordinate system (r, 0, z) is written as... 

Using the perturbation theory proposed by Larsen et al. Karakatsani and Economou have extended the PC-SAFT equation to account for dipole-dipole, dipole-quadrupole, quadrupole-quadrupole and dipole-induced dipole interactions. The exact second- and third-order perturbation terms in the work of Larsen are however rather complex and so a simplified version of the two terms was also proposed based on the work of Nezbeda and co-workers. While simplifying the equation, in an effort to generate a more usable, engineering-type, approach, the simplification of the dipolar term in the model introduces an additional pure-component model parameter. We refer to these as the PC-SAFT-KE and truncated PC-SAFT-KE (denoted PC-PSAFT and tPC-PSAFT by the authors respectively) approaches. In both KE approaches, the multipoles are assumed to be uniformly distributed over all segments in the molecule. The truncated and full PC-SAFT-KE approaches have been applied to study a wide range of polar fluids and their mixtures, " with the truncated equation found to be as accurate as the full PC-SAFT-KE model, and the inclusion of polar interactions found to improve the theoretical predictions in most cases. ... 

Ericksen tensors in this way leads to the so-called Rivlin-Ericksen (RE) fluid. Part of the complexity of the RE constitutive equation (say, staying with the case of A[ and A2 dependency, only) is due to keeping the full representation of the stress tensor as an isotropic tensor-valued function in A and A2. It is possible to simplify this relation by considering an alternative approach. Now one essentially looks at the problem as a perturbation expansion for slow flows. Thus, at rest, the stress tensor is given by the isotropic hydrostatic pressure only. The first order correction includes an additional term proportional to /4, which gives us the Newtonian fluid. At second order, we would include the square terms only, viz. 

The equation of the second-order fluid is so named because it contains all terms up to second order in the velocity gradient in a perturbation expansion about the rest state. The Newtonian term 2jjoD is the first-order term, and pi is the zeroth-order term. There are also equations of the third-order fluid, and so on (Bird et al., 1987). These form a series of equations in the retarded motion expansion so called because it assumes that the flow is a small perturbation from the state of test. The equations that are higher in order than second are of less practical importance because of their complexity and the restricted conditions under which they are accurate. 

What is now required are closed expressions for the first and second order contributions of a pure fluid. To this end, we employ a theory that we have recently applied successfully to study both bulk [238] and interfacial properties [99], This theory is based on a perturbation expansion proposed recently for the Lennard-Jones potential [299]. Although the expressions are obtained in a closed analytical form, they are rather lengthy and we refer the reader to the original paper for further details [238]. 

The pressures p/ and include both the initial pressures of Equation 5.22 and the perturbations given by Equation 5.23. They must be evaluated at the actual position z of the interface, but with all terms of second and higher order in small perturbation quantities neglected. The result for the pressure in fluid A is, for... 

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