**A fluid that departs from the classic linear Newtonian relation between stress and shear rate.** In a strict sense, a fluid is any state of matter that is not a solid, and a solid is a state of matter that has a unique stress-free state. A conceptually simpler definition is that a fluid is capable of attaining the shape of its container and retaining that shape for all time in the absence of external forces. Therefore, fluids encompass a wide variety of states of matter including gases and liquids as well as many more esoteric states (for example, plasmas, liquid crystals, and foams). * See also: ***Fluids**; **Foam**; **Gas**; **Liquid**; **Liquid crystals**; **Plasma (physics)**

A newtonian fluid is one whose mechanical behavior is characterized by a single function of temperature, the viscosity, a measure of the “slipperiness” of the fluid. For the example of **Fig. 1**, where a fluid is sheared between a fixed plate and a moving plate, the viscosity is given by Eq. (1).

Thus, as the viscosity of a fluid increases, it requires a larger force to move the top plate at a given velocity. For simple, Newtonian fluids, the viscosity is a constant dependent on only temperature; but for non-Newtonian fluids, the viscosity can change by many orders of magnitude as the shear rate (velocity/height in Fig. 1) changes. * See also: ***Newtonian fluid**; **Viscosity**

Many of the fluids encountered in everyday life (such as water, air, gasoline, and honey) are adequately described as being Newtonian, but there are even more that are not. Common examples include mayonnaise, peanut butter, toothpaste, egg whites, liquid soaps, and multigrade engine oils. Other examples such as molten polymers and slurries are of considerable technological importance. A distinguishing feature of many non-Newtonian fluids is that they have microscopic or molecular-level structures that can be rearranged substantially in flow. * See also: ***Particle flow**; **Polymer**

#### Counterintuitive behavior

Our intuitive understanding of how fluids behave and flow is built primarily from observations and experiences with Newtonian fluids. However, non-Newtonian fluids display a rich variety of behavior that is often in dramatic contrast to these expectations. For example, an intuitive feel for the slipperiness of fluids can be gained from rubbing them between the fingers. Furthermore, the slipperiness of water, experienced in this way, is expected to be the same as the slipperiness of automobile tires on a wet road. However, the slipperiness (viscosity) of many non-Newtonian fluids changes a great deal depending on how fast they move or the forces applied to them.

Intuitive expectations for how the surface of a fluid will deform when the fluid is stirred (with the fluid bunching up at the wall of the container) are also in marked contrast to the behavior of non-Newtonian fluids. When a cylindrical rod is rotated inside a container of a Newtonian fluid, centrifugal forces cause the fluid to be higher at the wall. However, for non-Newtonian fluids, the normal stress differences cause the fluid to climb the rod; this is called the Weissenberg effect (**Fig. 2 a**). Intuitive understanding about the motion of material when the flow of a fluid is suddenly stopped, for example by turning off a water tap, is also notably at odds with the behavior of non-Newtonian fluids.

*See also:*

**Centrifugal force**

A non-Newtonian fluid also displays counterintuitive behavior when it is extruded from an opening. A Newtonian fluid tapers to a smaller cross section as it leaves the opening, but the cross section for a non-Newtonian fluid first increases before it eventually tapers. This phenomenon is called die swell (Fig. 2*b*). * See also: ***Nozzle**

When a Newtonian fluid is siphoned and the fluid level goes below the entrance to the siphon tube, the siphoning action stops. For many non-Newtonian fluids, however, the siphoning action continues as the fluid climbs from the surface and continues to enter the tube. This phenomenon is called the tubeless siphon (Fig. 2*c*).

#### Non-newtonian viscosity

For Newtonian fluids, the viscosity is independent of the speed at which the fluid is caused to move. However, for non-Newtonian fluids, the viscosity can change by many orders of magnitude as the speed (velocity gradient) changes. Typically, the viscosity (η) of these fluids is given as a function of the shear rate (˙γ), which is a generalization of the ratio velocity/height in Fig. 1. A common dependence for this function is given in **Fig. 3**. In this particular model (called the Carreau model), the viscosity as a function of shear rate is given by

Eq. (2). This model involves four parameters: η_{0}, the viscosity at very small shear rates; η_{∞}, the viscosity at very large shear rates; λ, which describes the shear rate at which the curve begins to decrease; and *n*, which determines the rate of viscosity decrease. Figure 3 shows the viscosity for a fluid that shear-thins (that is, the viscosity decreases as the shear rate increases, *n* is less than one, and η_{0} is greater than η_{∞}). For other non-Newtonian fluids, the viscosity might increase as the shear rate increases (shear-thickening fluids, for which *n* is greater than one and η_{0} is less than η_{∞}).

#### Nonlinear effects

Although shear-rate-dependent viscosity is the most important non-Newtonian effect for many engineering applications of these fluids, it is not by itself capable of describing any of the phenomena depicted in Fig. 2. Both the Weissenberg effect (rod climbing) and die swell are generally attributed to nonlinear effects, by which is meant stresses that are present in non-Newtonian fluids and not present in Newtonian fluids. The stress is a generalization of the concept of pressure; at any point in the fluid there are nine components of the stress, τ_{ij}, where *i* and *j* take on the three coordinate directions. Then τ_{ij} is the force per unit area in the *j* direction on a surface of constant *i* (for example, τ_{xy} is the force per unit area in the *y* direction on a surface of constant *x*). Most significant among the anomalous stresses in non-Newtonian fluids is the first normal stress difference, which causes a tension in the direction of fluid flow. (There is also a second normal stress difference, which tends to be smaller and less significant than the first.) These stresses are generally caused by the flow trying to deform the structure of the material and the microstructure of the material resisting this deformation. The most common and pronounced result in a shear flow is a tension in the direction of the flow trying to resist flow. The Weissenberg effect is easily understood in terms of the first normal stress difference. Here the flow direction is circular around the rod. The tension pulls the fluid inward toward the rod, and the fluid must rise because it cannot go downward through the bottom of the container. * See also: ***Stress and strain**

#### Viscoelasticity

Perhaps the most striking behavior of non-Newtonian fluids is a consequence of their viscoelasticity. Solids can be thought of as having perfect memory. If they are deformed through the action of a force, they return to their original shape when the force is removed. This happens when a rubber ball bounces; the ball is deformed as it hits a surface, but the rubber remembers its undeformed spherical shape. Recovery of the shape causes the ball to bounce back. In contrast, Newtonian fluids have no memory; when a force is removed, they retain their condition at the time the force is removed (or continue moving as the result of inertia). When a newtonian fluid is dropped onto a surface, it does not bounce. Non-Newtonian fluids are viscoelastic in the sense that they have fading memory. If a force is removed shortly after it is applied, the fluid will remember its undeformed shape and return toward it. However, if the force is applied on the fluid for a long time, the fluid will eventually forget its undeformed shape. If a sample of a non-Newtonian fluid is dropped onto a surface, it will bounce like a ball. However, if the fluid is simply placed on the surface, it will flow smoothly. Viscoelasticity is frequently the cause of many of the secondary flows that are observed for non-Newtonian fluids. These are fluid motions that are small for Newtonian fluids (for example, swirling motions) but can become dominant for non-Newtonian fluids. * See also: ***Elasticity**

#### Constitutive relations

Analysis of fluid flow operations is typically performed by examining local conservation relations—conservation of mass, momentum (Newton's second law), and energy. This analysis requires material-specific information (for example, the relation between density, pressure, and temperature) that is collectively known as constitutive relations. The science devoted to obtaining suitable constitutive equations for description of the behavior of non-newtonian fluids is called rheology. The most important constitutive equation for fluid mechanics is that relating the stress in the fluid to the kinematics of the motion (that is, the velocity, the derivatives of the velocity with respect to position, and the time history of the velocity).

Two general methods exist for formulation of appropriate constitutive equations: (1) empiricism combined with flow measurements, and (2) molecular or structural theories. A wide variety of standard experiments have been designed to probe the non-Newtonian behavior of fluids and to test the predictions of constitutive equations. Examples include steady shear flow as in Fig. 1, with the upper plate moving at constant velocity; small-amplitude oscillatory shear flow, where the velocity of the upper plate depicted in Fig. 1 is caused to oscillate sinusoidally in time; and the sudden inception of shear flow, where the upper plate in Fig. 1 instantaneously begins moving at constant velocity. In the latter two experiments, the stress (force) is recorded as a function of time. These experiments provide useful probes of the viscoelasticity of the fluid. Another category of flow experiment involves elongation, wherein the fluid is caused to move so that fluid elements are stretched. In addition, other experiments have been used where the fluid is caused to move in more complicated ways. All of these experiments are used to define material functions that characterize the response of the fluid (for example, the viscosity of Fig. 3).

##### Generalized newtonian fluid

The simplest constitutive equation (obtained empirically from examination of experimental viscosity data) for a non-Newtonian fluid is that for the generalized Newtonian fluid, given by Eq. (3),

where γ_{ij}, given by Eq. (4), is called the rate-of-strain tensor, and η (˙γ) is the viscosity as a function of shear rate. Here, *v*_{i} is the *i*th component of the velocity vector, and the *x*_{i} are the position coordinates. For the Carreau fluid, η (˙γ) is given by Eq. (2).

The generalized Newtonian fluid constitutive equation is not capable of describing viscoelasticity or any effects of normal stress differences. The simplest constitutive equation that is capable of describing viscoelasticity is the Maxwell model (named after James C. Maxwell). In the Maxwell model,

Eq. (5) expresses the stresses as time integrals over the history of the velocity derivatives, where *G* (the shear modulus) and λ (the relaxation time) are material-dependent constants, and the time integral extends over all past time. It is straightforward to see how the Maxwell model incorporates the memory of the fluid; the stresses at the current time depend on the velocity derivatives at past times, and the memory fades because of the exponential that becomes smaller at more distant past times. For smaller values of λ the memory fades more rapidly, and for larger values of λ the fluid behaves more elastically (remembers better). The Maxwell model has been modified to describe fluid elasticity more accurately by writing the stress as a sum of integrals each with different values for *G* and λ, but the model still suffers from the serious deficit in predicting that the viscosity is constant (that is, independent of shear rate).

##### Requirements and generalizations

Constitutive equations, whether formulated empirically or as the result of a structural theory, are generally required to obey several fundamental postulates, including frame invariance and objectivity. Frame invariance specifies that the stress in a material should not depend on the measuring system (for example, the coordinate system) that is used to quantify it. From a practical point of view, this means that the mathematical language of tensor calculus must be used for the formulation. Objectivity is a more confusing concept, but it means loosely that the stress in a material must be independent of any rigid-body motion of the material. * See also: ***Tensor analysis**

Unfortunately, the Maxwell model as formulated above does not satisfy the objectivity postulate. However, it has been modified in a variety of ways by using more complicated descriptions of the fluid motion to make it objective. It has also been modified in other ways to incorporate shear-rate-dependent viscosity and normal stress effects. In addition, a large class of constitutive equations has been developed that expresses the stress as the solution of a differential equation (as opposed to a time integral), and these are often easier to use in analyzing specific flows.

Although the non-Newtonian behavior of many fluids has been recognized for a long time, the science of rheology is, in many respects, still in its infancy, and new phenomena are constantly being discovered and new theories proposed. Advancements in computational techniques are making possible much more detailed analyses of complex flows and more sophisticated simulations of the structural and molecular behavior that gives rise to non-Newtonian behavior. Engineers, chemists, physicists, and mathematicians are actively pursuing research in rheology, particularly as more technologically important materials are found to display non-Newtonian behavior. * See also: ***Fluid-flow principles**; **Rheology**