# Non-Newtonian flow

## What are Newtonian and non-Newtonian flows?

A Newtonian Fluid is one where there is a linear relationship between stress and strain rate: the ratio of stress to strain rate is the fluid’s viscosity. A Hookean solid is one where there is a linear relationship between stress and strain: the ratio of stress to strain is the modulus of the solid. Many materials have intermediate properties between a Newtonian fluid and a Hookean solid. If the properties are predominantly solid-like, the materials are called non-Hookean, which are described as viscoelastic. If they are predominantly fluid-like, they are called non-Newtonian, and the materials are defined as elastic viscous.

A non-Newtonian fluid has, therefore, a simultaneously elastic and viscous nature. All fluids are non-Newtonian on an appropriate time scale, though the time scale is concise for many common fluids such as air and water. When the time-scale of a flow is much less than the relaxation time  Of a viscous elastic material, elastic effects dominate. This typically happens when there are abrupt changes in flow geometry. When on the other hand is much greater than  Elastic effects relax sufficiently for viscous effects to dominate. This typically happens when there are no abrupt changes in flow geometry. The ratio of to  Is a dimensionless number of particular significance in the study of non-Newtonian fluids. This number is called the Deborah Number or the Weissenberg number. If the Deborah or Weissenberg number is small, elastic effects can be neglected and the non-Newtonian fluid treated as a purely viscous material, albeit with a non-constant viscosity. Non-Newtonian fluids are divided into time-dependent and time-independent, or purely viscous.

Time-Independent Fluids are those fluids for which the shear rate at a given point is only dependent upon the instantaneous shear stress at that point. These materials are sometimes referred to as “non-Newtonian viscous fluids” or “purely viscous fluids.”

In some fluids, the viscosity varies with time when the shear rate is constant. Two different types of behavior can be distinguished, thixotropic and rheopectic. Viscosity decreases with increasing time; for a rheopectic fluid, viscosity increases with increasing time. The variation of viscosity with time is associated with changes in the fluid structure. When straining the fluid breaks down that structure, for example, by destroying local linkages within the fluid, viscosity decreases with increasing time, and the fluid is thixotropic. A similar argument can be advanced for the decrease in viscosity with increasing shear rate for a pseudoplastic fluid.

In contrast, when straining builds up the structure, for example, by causing local alignment in the fluid, viscosity increases with increasing time, and the fluid is rheopectic. Again, a similar argument can be advanced for increasing viscosity with a dilatant fluid’s increasing shear rate. Therefore, an analogy exists between thixotropy and pseudo-plasticity on the one hand and between rheopexy and dilatancy on the other. Nevertheless, variation of viscosity with time is different from, and should not be confused with, variation with shear rate.

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Time-Independent Fluids are those fluids for which the shear rate at a given point is only dependent upon the instantaneous shear stress at that point. These materials are sometimes referred to as “non-Newtonian viscous fluids” or “purely viscous fluids.”