and Non-Newtonian Fluids
If two parallel plane
elements in a fluid are moving relative to one another, it is found that
a steady force must be applied to maintain a constant relative speed.
This force is called the viscous drag because it arises from the action
of viscous forces.
If the plane elements are at a distance Z apart, and if their relative velocity is v, then the force F required to maintain the motion has been found, experimentally, to be proportional to v and inversely proportional to Z for many fluids. The coefficient of proportionality is called the viscosity of the fluid, and it is denoted by the symbol m (mu).
From the definition of viscosity we can write
is the force applied, A is the area over which force is applied,
Z is the distance between planes, v is the velocity
of the planes relative to one another, and m
is the viscosity.
There is some ambivalence
about the writing and the naming of the unit of viscosity; there is no
doubt about the unit itself which is the N s m-2, which is
also the Pascal second, Pa s, and it can be converted to mass units using
the basic mass/force equation. The older units, the poise and its sub-unit
the centipoise, seem to be obsolete, although the conversion is simple
with 10 poises or 1000 centipoises being equal to 1 N s m-2,
and to 1 Pa s.
The viscous properties of many of the fluids and plastic materials that must be handled in food processing operations are more complex than can be expressed in terms of one simple number such as a coefficient of viscosity.
From the fundamental definition of viscosity in eqn. (3.14) we can write:
where t (tau) is called the shear stress in the fluid. This is an equation originally proposed by Newton and which is obeyed by fluids such as water. However, for many of the actual fluids encountered in the food industry, measurements show deviations from this simple relationship, and lead towards a more general equation:
which can be called the power-law equation, and where k is a constant of proportionality.
Where n = 1 the fluids are called Newtonian because they conform to Newton's equation (3.14) and k = m; and all other fluids may therefore be called non-Newtonian. Non-Newtonian fluids are varied and are studied under the heading of rheology, which is a substantial subject in itself and the subject of many books. Broadly, the non-Newtonian fluids can be divided into:
(1) Those in which n < 1. As shown in Fig. 3.6 these produce a concave downward curve and for them the viscosity is apparently high under low shear forces decreasing as the shear force increases. Such fluids are called pseudoplastic, an example being tomato puree. In more extreme cases where the shear forces are low there may be no flow at all until a yield stress is reached after which flow occurs, and these fluids are called thixotropic.
(2) Those in which n > 1. With a low apparent viscosity under low shear stresses, they become more viscous as the shear rate rises. This is called dilatancy and examples are gritty slurries such as crystallized sugar solutions. Again there is a more extreme condition with a zero apparent viscosity under low shear and such materials are called rheopectic. Bingham fluids have to exceed a particular shear stress level (a yield stress) before they start to move.
In many instances in practice non-Newtonian characteristics are important, and they become obvious when materials that it is thought ought to pump quite easily just do not. They get stuck in the pipes, or overload the pumps, or need specially designed fittings before they can be moved. Sometimes it is sufficient just to be aware of the general classes of behaviour of such materials. In other cases it may be necessary to determine experimentally the rheological properties of the material so that equipment and processes can be adequately designed.
For further details see, for example, Charm (1971), Steffe (2000).
Fluid-flow theory > STREAMLINE & TURBULENT FLOW
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