Newtonian and Non-Newtonian Fluids

Fluids in which shear stress is directly proportional to the rate of deformation are “Newtonian fluids. Most common fluids such as water, air and gasoline are Newtonian under normal conditions. If the fluid is Newtonian, then

The constant of proportionality in Eq. (1) is the absolute (or dynamic) viscosity, μ. The Newton's law of viscosity is given for one-dimensional flow by

Note that, since the dimensions of ′τ′ are [F/L²] and the dimensions of dv/dy are [1/t], μ has dimensions [Ft/L²]. Since the dimensions of force, F, mass, m, length, L, and time, t, are related by Newton's second law of motion, the dimensions of μ, can also be expressed as [M/Lt]. In the British Gravitational system, the units of viscosity are lbf.s/ft² or slug/(ft .s). In the Absolute Metric system, the basic unit of viscosity is called a poise [1 poise = 1 g/(cm .s)]; in the SI system the units of viscosity are kg/(m. s) or Pa. s (1 Pa. s = 1 N. s/m²).

In fluid mechanics the ratio of absolute viscosity, μ, to density, 𝜌, often arises. This ratio is given the name kinematic viscosity and is represented by the symbol ′𝜗′. Since density has dimensions [M/L³], the dimensions of 𝜗 are [L²/t]. In the Absolute Metric system of units, the unit for 𝜗 is a stoke (1 stoke = 1 cm²/s). For gases, viscosity increases with temperature, whereas for liquids, viscosity decreases with increasing temperature. If one considers the deformation of two different Newtonian fluids, say Glycerine and water, one recognizes that they will deform at different rates under the action of same applied stress. Glycerine exhibits much more resistance to deformation than water. Thus we say it is more viscous.

Non-Newtonian Fluids

Fluids in which shear stress is not directly proportional to deformation rate are non- Newtonian. Many common fluids exhibit non-Newtonian behaviour. The familiar example is toothpaste. Toothpaste behaves as a "fluid" when squeezed from the tube. However, it does not run out by itself when the cap is removed. There is a threshold or yield stress below which toothpaste behaves as a solid. Non-Newtonian fluids commonly are classified as having time-independent or time-dependent behaviour.

Numerous empirical equations have been proposed to model the observed relations between τ and dv/dy for time-independent fluids. They may be adequately represented for many engineering applications by the power law model, which for one-dimensional flow becomes

where the exponent, n, is called the flow behaviour index and the coefficient, k, the consistency index. This equation reduces to Newton's law of viscosity for n = 1 with k = μ. To ensure that τ has the same sign as dv/dy, Eq. (2) is rewritten in the form

The idea behind Eq. (3) is that we end up with a viscosity 𝜂 that is used in a formula that is the same form as Eq. (2), in which the Newtonian viscosity μ is used. The big difference is that while μ is constant (except for temperature effects), 𝜂 depends on the shear rate. Most non-Newtonian fluids have apparent viscosities that are relatively high compared with the viscosity of water.

Fluids in which the apparent viscosity decreases with increasing deformation rate (n < 1) are called pseudo plastic (or shear thinning) fluids. Most non-Newtonian fluids fall into this group; examples include polymer solutions, colloidal suspensions, and paper pulp in water. If the apparent viscosity increases with increasing deformation rate (n > 1) the fluid is termed dilatant (or shear thickening). Suspensions of starch and of sand are examples of dilatant fluids.

A "fluid" that behaves as a solid until a minimum yield stress, τ, is exceeded and subsequently exhibits a linear relation between stress and rate of deformation is referred to as an ideal or Bingham plastic. Clay suspensions, drilling muds, and toothpaste are examples of substances exhibiting this behaviour. The study of non-Newtonian fluids is further complicated by the fact that the apparent viscosity may be time-dependent. Thixotropic fluids show a decrease in 𝜂 with time under a constant applied shear stress; many paints are thixotropic. Rheopectic fluids show an increase in 𝜂 with time. After deformation some fluids partially return to their original shape when the applied stress is released; such fluids are called viscoelastic.

Fig. 1 Variation of Shear Stress with Velocity Gradient

Non-Newtonian Liquids

For most fluids the dynamic viscosity is independent of the velocity gradient in straight and parallel flow, so Newton’s hypothesis is fulfilled. A graph of stress against rate of shear is a straight line through the origin with slope equal to μ. There is a fairly large category of liquids for which the viscosity is not independent of the rate of shear, and these liquids are referred to as non-Newtonian. Solutions (particularly of colloids) often have a reduced viscosity when the rate of shear is large, and such liquids are said to be pseudo-plastic. Gelatine, clay, milk, blood and liquid cement come in this category.

A few liquids exhibit the converse property of dilatancy; that is, their effective viscosity increases with increasing rate of shear. Concentrated solutions of sugar in water and aqueous suspensions of rice starch (in certain concentrations) are examples. Additional types of non-Newtonian behaviour may arise if the apparent viscosity changes with the time for which the shearing forces are applied. Liquids for which the apparent viscosity increases with the duration of the stress are termed rheopectic; those for which the apparent viscosity decreases with the duration are termed thixotropic.

A number of materials have the property of plasticity. Metals when strained beyond their elastic limit or when close to their melting points can deform continuously under the action of a constant force and thus in some degree behave like liquids of high viscosity. Their behaviour, is non-Newtonian, and most of the methods of mechanics of fluids are therefore inapplicable to them.

Viscoelastic materials possess both viscous and elastic properties; bitumen, nylon and flour dough are examples. In steady flow, that is, flow not changing with time, the   of shear is constant and may well be given by τ/μ where μ represents a constant dynamic viscosity as in a Newtonian fluid. Elasticity becomes evident when the shear stress is changed. A rapid increaseof stress from τ to τ +δτ causes the material to be sheared through an additional angle δτ /G where G represents an elastic modulus; the corresponding rate of shear is (1/G)∂τ/∂t so the total rate of shear in the material is (τ/μ) + (1/G)∂τ/∂t.

The fluids with which engineers most often have to deal are Newtonian, that is, their viscosity is not dependent on either the rate of shear or its duration, and the term mechanics of fluids is generally regarded as referring only to Newtonian fluids. The study of non-Newtonian liquids is termed rheology.

Inviscid Fluid

An important field of theoretical fluid mechanics involves the investigation of the motion of a hypothetical fluid having zero viscosity. Such a fluid is sometimes referred to as an ideal fluid. Although commonly adopted in the past, the use of this term is now discouraged as imprecise. A more meaningful term for a fluid of zero viscosity is inviscid fluid.