Constitutive equations Second order fluid

For almost steady flows one can expand yl1 or y about t t and obtain second-order fluid constitution equations in the co-deforming frame. When steady shear flows are considered, the CEF equation is obtained, which, in turn, reduces to the GNF equation for T i = 2 = 0 and to a Newtonian equation if, additionally, the viscosity is constant. 

The constitutive equation of second-order fluids indicates that the stress tensor is determined by — u), and consequently Gy t) is independent of t and depends only on and Af Thus the function dy reduces to a function y, and the constitutive equation for second-order fluids [Eq. (13.9)] can be expressed as... 

The simplest constitutive equation capable of predicting a first normal stress difference is the equation of the second-order fluid (Bird et al., 1987 Larson, 1988) ... 

Comparing eq. 4.3.10 with eq. 4.3.1, we see that to second order in the velocity gradient the upper-convected Maxwell equation for small strain rates reduces to a special case of the equation of the second-order fluid with V i.o = 2kr]o and V 2,o = 0. All properly formulated constitutive equations for which the stress is a smooth functional of the strain history reduce at second order in the velocity gradient to the equation of the second-order fluid. Example 4.3.3, however, illustrates that the equation of the second-order fluid cannot be trusted except for slow nearly steady flows. 

Thus for steady state uniaxial extensional flow, in contrast to steady state shearing flow, the second-order fluid result agrees with the UCM prediction only at small strain rates. The UCM, second-order fluid, and Newtonian fluid equations all differ in their predictions of the strain rate dependences of the extensional viscosity, though the strain rate dependences of the shear viscosity are the same for all three equations. This result typifies the usual finding that constitutive equations differ among themselves more strongly in their predictions of extensional viscosities than in their predictions of shear viscosities. 

Note that since m(s) and a( i, 2) are functions only of time y, then t]q, y3, and v are constants. A material that can be represented by the constitutive equation given in Eq. (3.76) is called a Coleman-Noll second-order fluid (Coleman and Markovitz 1964 Truesdell and Noll 1965). For steady-state simple shear flow, Eq. (3.76) yields... 

Equation (11) is written in the form of Newton s second law and states that the mass times acceleration of a fluid particle is equal to the sum of the forces causing that acceleration. In flow problems that are accelerationless (Dx/Dt = 0) it is sometimes possible to solve Eq. (11) for the stress distribution independently of any knowledge of the velocity field in the system. One special case where this useful feature of these equations occurs is the case of rectilinear pipe flow. In this special case the solution of complex fluid flow problems is greatly simplified because the stress distribution can be discovered before the constitutive relation must be introduced. This means that only a first-order differential equation must be solved rather than a second-order (and often nonlinear) one. The following are the components of Eq. (11) in rectangular Cartesian, cylindrical polar, and spherical polar coordinates ... 

In the general case, eqs 4 and 5 constitute a system of nonlinear coupled second-order partial differential equations. To specify the boundary conditions for this problem, it is necessary to include the external (interphase) heat and mass transfer, as both the concentration and the temperature at the external surface of the catalyst pellet may differ from the corresponding values in the bulk of the surrounding fluid phase. 

We see that application of the angular acceleration principle does reduce, somewhat, the imbalance between the number of unknowns and equations that derive from the basic principles of mass and momentum conservation. In particular, we have shown that the stress tensor must be symmetric. Complete specification of a symmetric tensor requires only six independent components rather than the full nine that would be required in general for a second-order tensor. Nevertheless, for an incompressible fluid we still have nine apparently independent unknowns and only four independent relationships between them. It is clear that the equations derived up to now - namely, the equation of continuity and Cauchy s equation of motion do not provide enough information to uniquely describe a flow system. Additional relations need to be derived or otherwise obtained. These are the so-called constitutive equations. We shall return to the problem of specifying constitutive equations shortly. First, however, we wish to consider the last available conservation principle, namely, conservation of energy. 

The first of these relations was noted by Lodge [(46), Eq. (6.43)] and by Williams and Bird (77) as a result of the study of two different empirical constitutive equations. Later Spriggs (70) obtained Eqs. (7.22) and (7.23) from the Coleman and Noll (21) theory of second-order viscoelasticity. Eqs. (7.22) and (7.23) indicate that no additional information about fluids can be obtained from normal stress oscillatory measurements than has not already been obtained by shear stress oscillatory data. Eq. (7.24) seems to be new and is probably specific to rigid dumbbell suspensions. 

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. 

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