Dilute solution constitutive equation

The proposed mechanisms of models to explain the drag reduction phenomenon are based on either a molecular approach or fluid dynamical continuum considerations, but these models are mainly empirical or semi-empirical in nature. Models constructed from the equations of motion (or energy) and from the constitutive equations of the dilute polymer solutions are generally not suitable for use in engineering applications due to the difficulty of placing numerical values on all the parameters. In the absence of a more generally accurate model, semi-empirical ones remain the most useful for applications. 

The same statement can be made about inelastic non-Newtonian fluids, such as the Power Law fluid, from a mathematical solution point of view. In reality, most non-Newtonian fluids are viscoelastic and exhibit normal stresses. For fluids such as those (i.e., fluids described by constitutive equations that predict normal stresses for viscometric flows), theoretical analyses have shown that secondary flows are created inside channels of nonuniform cross section (78,79). Specifically it can be shown that a zero second normal stress difference is a necessary (but not sufficient) condition to ensure the absence of secondary flow (79). Of course, the analyses of flows in noncircular channels in terms of constitutive equations—which, strictly speaking, hold only for viscometric flows—are expected to yield qualitative results only. Experimentally low Reynolds number flows in noncircular channels have not been investigated extensively. In particular, only a few studies have been conducted with fluids exhibiting normal stresses (80,81). Secondary flows, such as vortices in rectangular channels, have been observed using dyes in dilute aqueous solutions of polyacrylamide. Interestingly, these secondary flow vortices (if they exist) seem to have very little effect on the flow rate. 

The set of constitutive equations for the dilute polymer solution consists of the definition of the stress tensor (6.16), which is expressed in terms of the second-order moments of co-ordinates, and the set of relaxation equations (2.39) for the moments. The usage of a special notation for the ratio, namely... 

We can see that a set of constitutive equations for dilute polymer solutions contains a large number of relaxation equations. It is clear that the relaxation processes with the largest relaxation times are essential to describe the slowly changing motion of solutions. In the simplest approximation, we can use the only relaxation variable, which can be the gyration tensor (S i5J), defined by (4.48), or we can assume the macromolecule to be schematised by a subchain model with two particles. The last case, which is considered in Appendix F in more detail, is a particular case of equations (9.3) and (9.4), which is followed at N = 1, Ai = 2,... 

Equations (9.6) and (9.7) make up the simplest set of constitutive equations for dilute polymer solutions, which, after excluding the internal variables j, can be written in the form of a differential equation that has the form of the two-constant contra-variant equation investigated by Oldroyd (1950) (Section 8.6). 

Note once again that equations (9.6) and (9.7) determines the stresses for the completely idealised macromolecules (without internal viscosity, hydro-dynamic interaction and volume effects) in dilute solutions. To remedy the unrealistic behaviour of constitutive equations (9.6) and (9.7), some modifications were proposed (Rallison and Hinch 1988 Hinch 1994). 

The expressions for the stress tensor together with the equations for the moments considered as additional variables, the continuity equation, and the equation of motion constitute the basis of the dynamics of dilute polymer solutions. This system of equations may be used to investigate the flow of dilute solutions in various experimental situations. Certain simple cases were examined in order to demonstrate applicability of the expressions obtained to dilute solutions, to indicate the range of their applicability, and to specify the expressions for quantities which were introduced previously as phenomenological constants. 

Rabin Y, Ottinger HCh (1990) Dilute polymer solutions internal viscosity, dynamic scaling, shear thinning, and frequency-dependent viscosity. Europhys Lett 13(5) 423—428 Rallison JM, Hinch EJ (1988) Do we understand the physics in the constitutive equation J Non-Newton Fluid Mech 29(l) 37-55... 

Equations (3-32)-(3-34) are equivalent to the so-called Oldroyd-B equation. The Oldroyd-B equation is a simple, but qualitatively useful, constitutive equation for dilute solutions of macromolecules (see Section 3.6.2). Refinements to the simple elastic dumbbell model, such as the effects of the nonlinearity of the force-extension relationship at high extensions, are discussed in Section 3.6.2.2.I. 

The study of a particular adsorption process requires the knowledge of equilibrium data and adsorption kinetics [4]. Equilibrium data are obtained firom adsorption isotherms and are used to evaluate the capacity of activated carbons to adsorb a particular molecule. They constitute the first experimental information that is generally used as a tool to discriminate among different activated carbons and thereby choose the most appropriate one for a particular application. Statistically, adsorption from dilute solutions is simple because the solvent can be interpreted as primitive, that is to say as a structureless continuum [3]. Therefore, all equations derived firom monolayer gas adsorption remain vafid. Some of these equations, such as the Langmuir and Dubinin—Astakhov, are widely used to determine the adsorption capacity of activated carbons. Batch equilibrium tests are often complemented by kinetics studies, to determine the external mass transfer resistance and the effective diffusion coefficient, and by dynamic column studies. These column studies are used to determine system size requirements, contact time, and carbon usage rates. These parameters can be obtained from the breakthrough curves. In this chapter, I shall deal mainly with equilibrium data in the adsorption of organic solutes. 

Analogous equations are written for fluxes along the other faces. Equation (8) is assumed to be linear. Commonly, in electrochemical systems, the constitutive relationship may be nonlinear. For example, assuming dilute-solution theory, the flux of species j is given by... 

One of the most important applications of the theory of PS is to biomolecules. There have been numerous studies on the effect of various solutes (which may be viewed as constituting a part of a solvent mixture) on the stability of proteins, conformational changes, aggregation processes, etc., (Arakawa and Timasheff 1985 Timasheff 1998 Shulgin and Ruckenstein 2005 Shimizu 2004). In all of these, the central quantity that is affected is the Gibbs energy of solvation of the biomolecule s. Formally, equation (8.26) or equivalently (8.28), applies to a biomolecule s in dilute solution in the solvent mixture A and B. However, in contrast to the case of simple, spherical solutes, the pair correlation functions gAS and gBS depend in this case on both the location and the relative orientation of the two species involved (figure 8.5). Therefore, we write equation (8.26) in an equivalent form as ... 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

By combining Equations 10.8 and 10.9, one obtains an expression for the derivative of the activity coefficient of an infinitely dilute solute with respect to the cosolvent mole fraction in terms of characteristics of the solute-free binary solvent (y, c,°, and c°) and the parameters Aj2 and A23, which characterize the interactions of an infinitely dilute solute with the components of the mixed solvent. Even though Equation 10.8 constitutes a formal statistical thermodynamics relation in which all... 

Lumley, J. L. Applicability of Oldroyd constitutive equation to flow of dilute polymer solutions. Phys. Fluids 14 (1971) 2282. 

Unlike the case of flexible polymers the constitutive equation cannot be written in a simple closed form unless we use the decoupling approximation. Since the equation for S p is given by the same equation as that in dilute solution, eqns (8.149) and (9.57) give a closed equation for a,... 

James, D. F. (1972). Constitutive Equations for Dilute Polymer Solutions, /. Rheol. 16 175. 

This section deals with reactions in which the polymer and its low molecular reagent are in reversible equilibrium according to Equation (2) and at least one of the components is optically active. These reactions proceed under very mild conditions and in most cases it is sufficient to mix the dilute solutions of the components at room temperature. Unlike the reactions described so far, these reactions are not accompanied by any change in the constitution or configuration of the chirality elements however the interactions of the components can lead to a shift in the electrons, to ionization or polarization of dipoles, thus influencing the conformation or secondary structure of the polymers. These interactions usually cause major changes in most physical properties such as colour, solubility or viscosity. In this context, primarily, the changes in the chiroptical properties will be considered. 

Other works used, instead of spectral, lower order accuracy finite difference approximations, but they have to be noted here since they employed more suitable numerical formulations for the constitutive equations that explicitly avoided the introduction of artificial diffusivity in the numerical solution [61, 63-66]. Those works also employed the FENE-P model to simulate dilute polymer solutions [61, 63, 64] or the Giesekus model for surfactant turbulent fiow [19, 65, 66[. 

Giesekus (1982) summarized nicely a series of his papers on the formulation of a new class of constitutive equations. The origin of Eq. (3.23) comes from a modification of the upper convected Maxwell model as applied to a dilute polymer solution, namely... 

In this chapter, we have presented the fundamentals of molecular theory for the viscoelasticity of flexible homogeneous polymers, namely the Rouse/Zimm theory for dilute polymer solutions and unentangled polymer melts, and the Doi-Edwards theory for concentrated polymer solutions and entangled polymer melts. In doing so, we have shown how the constitutive equations from each theory have been derived and then have compared theoretical prediction with experiment. The material presented in this chapter is very important for understanding how the molecular parameters of polymers are related to the rheological properties of homopolymers. 

The positive force/in Eq. (10.3.7) is externally applied and is balanced by an inward-acting internal force, which, in the absence of the external force, tends to make the end-to-end distance go to zero. This, however, does not happen in practice because the spring force is not the only one acting on the chain the equilibrium end-to-end distance is given by a balance of all the forces acting on the polymer molecule. This aspect of the behavior of isolated polymer molecules will be covered in greater detail in the discussion of constitutive equations for dilute polymer solutions in Chapter 14. 

Gordon, R. J., and A. E. Everage, Jr., Bead-Spring Model of Dilute Polymer Solutions Continuum Modifications and an Explicit Constitutive Equation, J. AppL Polym. Set, 15, 1903-1909, 1971. 

We have assumed that the solution is so dilute that its volume closely approximates the volume of the solvent constituting it. Note that this volume corresponds to the STP molar volume of an ideal gas. The osmotic pressure equation also resembles the ideal gas equation. 

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