Constitutive equation Newton

Kwon Y, Leonov AI (1995) Stability constraints in the formulation of viscoelastic constitutive equations. J Non-Newton Fluid Mech 58 25—46 Laius LA, Kuvshinskii EV (1963) Structure and mechanical properties of oriented amorphous linear polymers. Fizika Tverdogo Tela 5(11) 3113—3119 (in Russian)... 

Leonov AI (1992) Analysis of simple constitutive equations for viscoelastic liquids. J Non-Newton Fluid Mech 42(3) 323-350... 

Phan-Thien N, Tanner RI (1977) A new constitutive equation derived from network theory. J Non-Newton Fluid Mech 2(4) 353-365... 

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... 

Decoupled methods as well as direct Newton methods have been used to accoimt for the nonhnearities in the constitutive equations. Those methods exhibit limited convergence, especially when flows with singularities are considered. Newton methods are efficient but need a high storage area and powerful computational flidlities. 

In order to use these general momentum conservation equations to calculate the velocity field, it is necessary to express viscous stress terms in terms of the velocity field. The equations which relate the stress tensor to the motion of the continuous fluid are called constitutive equations or rheological equations of state. Although the governing momentum conservation equations are valid for all fluids, the constitutive equations, in general, vary from one fluid material to another and possibly also from one type of flow to another. Fluids, which follow Newton s law of viscosity (although it is referred to as a law, it is just an empirical proposition) are called Newtonian fluids. For such fluids, the viscous stress at a point is linearly dependent on the rates of strain (deformation) of the fluid. With this assumption, a general deformation law which relates stress tensor and velocity components can be written ... 

Constitutive equations, which quantitatively describe the physical properties of the fluids. The most important constitutive equations used in this book are the Newton s viscosity law, the Fourier s law of heat conduction, and the Pick s law of mass diffusion. The equation of state and more empirical relations for the physical properties of the fluid mixture also belong to this group of equations. 

Note that the coefficient matrix in (11.1.37) is symmetric. Once we have evaluated the derivatives (11.1.38)-(11.1.41), we solve (11.1.37) for AL and AVi This constitutes the Newton-Raphson method for solving sets of nonlinear algebraic equations [9]. 

Constitutive equation n. In material science, an equation that relates stress in a material to strain or strain rate. Simple examples are (1) Hooke s law, which states that, in elastic solids, strain is directly proportional to stress, and (2) Newton s law of flow, which states that, in laminar shear flow, the shear rate is equal to the shear stress divided by the viscosity. Few plastic solids and liquids obey either of these laws. 

Second, constitutive equations for the fluid flow should be formulated. These are Navier-Stokes equations, which basically are the Newton s equations of motion written for fluid movement under the action of forces of external pressure and gravity and taking into account shear forces inside the fluid ... 

If we replace the term lack of slip by viscosity, Newton s constitutive equation emerges, which, in its simplest one-dimensional form, can be written as... 

It is obvious that the simple non-Newtonian cannot be treated by Newton s law of viscosity. As such, other approaches must be taken that lead to often quite complicated rheological constitutive equations. The simplest of these is Eq. (3-3), the apparent viscosity expression, or the Ostwald-De Waele power... 

Algebraic equations (14.3) correspond to constitutive equations, which are generally based on physical and chemical laws. They include basic definitions of mass, energy, and momentum in terms of physical properties, like density and temperature thermodynamic equations, through equations of state and chonical and phase equilibria transport rate equations, such as Pick s law for mass transfer, Fourier s law for heat conduction, and Newton s law of viscosity for momentum transfer chemical kinetic expressions and hydraulic equations. 

The question arises how do we identify the variables for a Dimensional Analysis study The best way to identify the variables for use in a Dimensional Analysis is to write the conservation laws and constitutive equations underpinning the process being studied. Constitutive equations describe a specific response of a given variable to an external force. The most familiar constitutive equations are Newton s law of viscosity, Fourier s law of heat conduction, and Pick s law of diffusion. 

We could summarize Eqs. 1-1 through 1-3 by sa3dng that they introduced the concepts of kinematics and stress. More than half a century would elapse before the concept of stress would be presented in a modem framework by Cauchy, and it would require a slightly longer period of time before a constitutive equation would be developed leading to the Navler-Stokes equations. In the century between Euler and Stokes, the basic ideas associated with kinematics, stress and constitutive relations were formulated. Two centuries later, these same concepts represent the building blocks of fluid mechanics. Before we comment on the development of these concepts, we need to examine how Eqs. 1-1 through 1-3 compare with Newton s three laws of mechanics. 

Only a few years after Hooke expressed the concept that eventually led to the constitutive equation for the ideal elastic solid, Newton (Figure 2.1.1) wrote his famous Principia Mathematica. Here Newton expressed, among many other things, the basic idea for a viscous fluid. His resistance means local stress velocity by which the parts of the fluid are being separated means velocity... 

In this chapter we have developed the general constitutive equation for a viscous liquid. We found that by using the rate of deformation or strain rate tensor 2D, we can write Newton s viscosity law properly in three dimensions. By making the coefficient of 2D dependent on invariants of 2D, we can derive models like the power law. Cross, and Carreau. We also showed how to introduce a three-dimensional yield stress to describe plastic materials with models like those Bingham and Casson. We saw two ways to describe the temperature dependence of viscosity and the importance of shear heating. 

Because of their complex structure the mechanical behavior of polymeric materials is not well described by the classical constitutive equations Hooke s law (for elastic solids) or Newton s law (for viscous liquids). Polymeric materials are said to be viscoelastic inasmuch as they exhibit both viscous and elastic responses. This viscoelastic behavior has played a key role in the development of the understanding of polymer structure. Viscoelasticity is also important in the understanding of various measuring devices needed for rheometric measurements. In the fluid dynamics of polymeric liquids, viscoelasticity also plays a crucial role. " Also in the polymer-processing industry it is necessary to include the role of viscoelastic behavior in careful analysis and design. Finally there are important connections between viscoelasticity and flow birefringence. ... 

Our next goal is to find the velocity field. To do this the type of fluid and an appropriate constitutive equation must be specified. In the first case the fluid is considered to be Newtonian, is replaced with Newton s law of viscosity to obtain a differential equation for v ... 

Hooke s law, the direct proportionality between stress and strain in tension or shear, is often assumed such that the constitutive equations for a purely elastic solid are o = fjs for unidirectional extension and x = qy in simple shear flow. The latter expression is recognized from Chapter 7 as the constitutive relationship for a Newtonian fluid and, in analogy to Hooke s law for elastic solids, is sometimes termed Newton s law of viscosity. For cross-linked, amorphous polymers above 7, a nonlinear relationship can be derived theoretically. For such materials v = 0.5. When v is not 0.5, it is an indication that voids are forming in the sample or that crystallization is taking place. In either case, neither the theoretical equation nor Hooke s law generally applies. Before turning to one of the simplest mathematical models of viscoelasticity, it is important to recall that the constitutive equations of a purely viscous fluid are a = fj for elongational flow and x = qy for shear flow. 

The Gauss-Newton method is directly related to Newton s method. The main difference between the two is that Newton s method requires the computation of second order derivatives as they arise from the direct differentiation of the objective function with respect to k. These second order terms are avoided when the Gauss-Newton method is used since the model equations are first linearized and then substituted into the objective function. The latter constitutes a key advantage of the Gauss-Newton method compared to Newton s method, which also exhibits quadratic convergence. 

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