Newtonian fluid, stress-strain rate relation

Rheology relates the stress components to the respective rate of strain of the fluid. For example, the rheological equation for a Newtonian fluid shows a linear relation between stress, s, and... 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

The apparent viscosity, defined as du/dj) drops with increased rate of strain. Dilatant fluids foUow a constitutive relation similar to that for pseudoplastics except that the viscosities increase with increased rate of strain, ie, n > 1 in equation 22. Dilatancy is observed in highly concentrated suspensions of very small particles such as titanium oxide in a sucrose solution. Bingham fluids display a linear stress—strain curve similar to Newtonian fluids, but have a nonzero intercept termed the yield stress (eq. 23) ... 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

In the previous section we discussed the nature and some properties of the stress tensor t and the rate of strain tensor y. They are related to each other via a constitutive equation, namely, a generally empirical relationship between the two entities, which depends on the nature and constitution of the fluid being deformed. Clearly, imposing a given stress field on a body of water, on the one hand, and a body of molasses, on the other hand, will yield different rates of strain. The simplest form that these equations assume, as pointed out earlier, is a linear relationship representing a very important class of fluids called Newtonian fluids. 

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 ... 

To characterize Newtonian and non-Newtonian food properties, several approaches can be used, and the whole stress-strain curve can be obtained. One of the most important textural and rheological properties of foods is viscosity (or consistency). The evaluation of viscosity can be demonstrated by reference to the evaluation of creaminess, spreadability, and pourability characteristics. All of these depend largely on shear rate and are affected by viscosity and different flow conditions. If it is related to steady flow, then at any point the velocity of successive fluid particles is the same at successive periods of time for the whole food system. Thus, the velocity is constant with respect to time, but it may vary at different points with... 

To proceed formulating the momentum equation we need a relation defining the total stress tensor in terms of the known dependent variables, a constitutive relationship. In contrast to solids, a fluid tends to deform when subjected to a shear stress. Proper constitutive laws have therefore traditionally been obtained by establishing the stress-strain relationships (e.g., [11] [12] [13] [89] [184] [104]), relating the total stress tensor T to the rate of deformation (sometimes called rate of strain, i.e., giving the name of this relation) of a fluid element. However, the resistance to deformation is a property of the fluid. For some fluids, Newtonian fluids, the viscosity is independent both of time and the rate of deformation. For non-Newtonian fluids, on the other hand the viscosity may be a function of the prehistory of the flow (i.e., a function both of time and the rate of deformation). 

In these cases the relative velocity of the shearing plates is not constant but varies in a sinusoidal manner so that the shear strain and the rate of shear strain are both cyclic, and the shear stress is also sinusoidal. For non-Newtonian fluids, the stress is out of phase with the rate of strain. In this situation a measured complex viscosity (rf) contains both the shear viscosity, or dynamic viscosity (t] ), related to the ordinary steady-state viscosity that measures the rate of energy dissipation, and an elastic component (the imaginary viscosity ij" that measures an elasticity or stored energy) ... 

For most fluids the shear stress is a unique function of the strain rate. The constitutive relation of Newton assumes the shear stress to be linear in the strain rate. In the Couette problem there is only the single strain-rate component duldy and single stress component so the Newtonian viscosity law may be written... 

The stress a in an ideally elastic materials is uniquely related to a specific strain y, a = Gy (where G is the shear modulus), whereas the stress in a Newtonian fluid is uniquely related to the rate of strain, y = dy/dt,... 

The Maxwell Model. In the above development, discussion moves from elastic behavior to viscoelastic descriptions of material behavior. In a simple sense, viscoelasticity is the behavior exhibited by a material that has both viscous and elastic elements in its response to a deformation or load. In early days, this was often represented by elastic or viscous mechanical elements combined in different ways (9-12). The simplest models are two element models that contain a viscous element (dashpot) and an elastic element (spring). The dashpot is assumed to follow a Newtonian fluid constitutive law in which the stress is related directly to the strain rate by the following expression ... 

First, when finite strains are imposed on solids (especially those soft enough to be deformed substantially without breaking), the stress-strain relations are much more complicated (non-Hookean deformation) similarly, in steady flow with finite strain rates, many fluids (especially polymeric solutions and undiluted uncross-linked polymers) exhibit marked deviations from Newton s law (non-Newtonian flow). The dividing line between infinitesimal and finite depends, of course, on the level of precision under consideration, and it varies greatly from one material to another. 

To describe this nonviscometric deformation, a new material function the elongational or Trouton viscosity, is needed to relate tensile stress to the rate of tensile strain, ctu = T7e(su) iis or equivalently the function (t — 0 11 (an). For incompressible Newtonian fluids, it may be shown that = Sjj,... 

The constitutive equation for the Newtonian fluid is linear in both stress and velocity gradients. If it is desired to retain a linear relation between the stress and rate of strain tensors, but allow for time derivatives, then one can write a relation of the form... 

A flowing fluid is acted upon by many forces that result in changes in pressure, temperature, stress, and strain. A fluid is said to be isotropic when the relations between the components of stress and those of the rate of strain are the same in all directions. The fluid is said to be Newtonian when this relationship is linear. These pressures and temperatures must be fully understood so that the entire flow picture can be described. 

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