Consider the motion of fluid illustrated in Fig. 1.2. All particles are moving in the same direction, but different layers of the fluid move with differ- ent velocities (as indicated here by the lengths of the arrows). Thus one layer moves relative to another. We assume for the moment that the paral- lel movements of the layers are in straight lines. A particular small portion of the fluid will be deformed from its original rectangular shape PQRS to
PQRS as it moves along. However, it is not the displacement of PQ
22 Fundamental concepts
Fig. 1.2
Fig. 1.3
diagram of Fig. 1.3 represents a smaller degree of deformation than does the left-hand diagram, although the relative movement between top and bottom of the portion considered is the same in each case. The linear displacement is a matter of the difference of velocity between the two planes PQ and SR but the angular displacement depends also on the distance between the planes. Thus the important factor is the velocity gradient, that is, the rate at which the velocity changes with the distance across the flow.
Fig. 1.4
Suppose that, within a flowing fluid, the velocity u of the fluid varies with distance y measured from some fixed reference plane, in such a man- ner as in Fig. 1.4. Such a curve is termed the velocity profile. The velocity gradient is given by δu/δy or, in the limit as δy → 0, by ∂u/∂y. The partial derivative∂u/∂y is used because in general the velocity varies also in other directions. Only the velocity gradient in the y direction concerns us here.
Figure 1.5 represents two adjoining layers of the fluid, although they are shown slightly separated for the sake of clarity. The upper layer, supposed the faster of the two, tends to draw the lower one along with it by means of a force F on the lower layer. At the same time, the lower layer (by Newton’s Third Law) tends to retard the faster, upper, one by an equal and opposite force acting on that. If the force F acts over an area of contact A the shear stressτ is given by F/A.
Newton (1642–1727) postulated that, for the straight and parallel motion of a given fluid, the tangential stress between two adjoining layers is pro- portional to the velocity gradient in a direction perpendicular to the layers. That is
Viscosity 23
Fig. 1.5
or
τ = µ∂u∂y (1.9)
where µ is a constant for a particular fluid at a particular temperature. This coefficient of proportionalityµ is now known by a number of names. The preferred term is dynamic viscosity – to distinguish it from kinematic viscosity (Section 1.6.4) – but some writers use the alternative terms absolute
viscosity or coefficient of viscosity. The symbolsµ and η are both widely used
for dynamic viscosity; in this bookµ will be used. The restriction of eqn 1.9 to straight and parallel flow is necessary because only in these circumstances does the increment of velocityδu necessarily represent the rate at which one layer of fluid slides over another.
It is important to note that eqn 1.9 strictly concerns the velocity gradient and the stress at a point: the change of velocity considered is that occurring over an infinitesimal thickness and the stress is given by the force acting over an infinitesimal area. The relationτ = µ u/ y, where u represents the change of velocity occurring over a larger, finite distance y, is only true for a velocity profile with a linear velocity gradient.
To remove the restriction to straight and parallel flow, we may substitute ‘the rate of relative movement between adjoining layers of the fluid’ forδu, and ‘rate of shear’ for ‘velocity gradient’. As will be shown in Section 6.6.4, if angular velocity is involved then the rate of shear and the velocity gradient are not necessarily identical; in general, the rate of shear represents only part of the velocity gradient. With this modification, eqn 1.9 may be used to define viscosity as the shear stress, at any point in a flow, divided by the rate of shear at the point in the direction perpendicular to the surface over which the stress acts.
The dynamic viscosityµ is a property of the fluid and a scalar quantity. The other terms in eqn 1.9, however, refer to vector quantities, and it is important to relate their directions. We have already seen that the surface over which the stressτ acts is (for straight and parallel flow) perpendicular to the direction of the velocity gradient. (With the notation of eqn 1.9 the surface is perpendicular to the y coordinate or, in other words, parallel to the
x–z plane.) We have seen too that the line of action of the force F is parallel
to the velocity component u. Yet what of the sense of this force? In Fig. 1.5, to which of the two forces each labelled F does eqn 1.9 strictly apply?
If the velocity u increases with y, then∂u/∂y is positive and eqn 1.9 gives a positive value ofτ. For simplicity the positive sense of the force or stress is defined as being the same as the positive sense of velocity. Thus, referring again to Fig. 1.5, the value ofτ given by the equation refers to the stress acting on the lower layer. In other words, both velocity and stress are considered positive in the direction of increase of the coordinate parallel to them; and
24 Fundamental concepts
the stress given by eqn 1.9 acts over the surface facing the direction in which the perpendicular coordinate (e.g. y) increases.
For many fluids the magnitude of the viscosity is independent of the rate of shear, and although it may vary considerably with temperature it may be regarded as a constant for a particular fluid and temperature. Such fluids are known as Newtonian fluids. Those fluids that behave differently are discussed in Section 1.6.5.
Equation 1.9 shows that, irrespective of the magnitude ofµ, the stress is zero when there is no relative motion between adjoining layers. Moreover, it is clear from the equation that ∂u/∂y must nowhere be infinite, since such a value would cause an infinite stress and this is physically impossible. Consequently, if the velocity varies across the flow, it must do so continu- ously and not change by abrupt steps between adjoining elements of the fluid. This condition of continuous variation must be met also at a solid boundary; the fluid immediately in contact with the boundary does not move relative to it because such motion would constitute an abrupt change. In a viscous fluid, then, a condition that must always be satisfied is that there should be no slipping at solid boundaries. This condition is commonly referred to as the no-slip condition.
It will be seen that there is a certain similarity between the dynamic viscos- ity in a fluid and the shear modulus of elasticity in a solid. Whereas, however, a solid continues to deform only until equilibrium is reached between the internal resistance to shear and the external force producing it, a fluid con- tinues to deform indefinitely, provided that the external force remains in action. In a fluid it is the rate of deformation, not the deformation itself, that provides the criterion for equilibrium of force.
To maintain relative motion between adjoining layers of a fluid, work must be done continuously against the viscous forces of resistance. In other words, energy must be continuously supplied. Whenever a fluid flows there is a loss of mechanical energy, often ascribed to fluid friction, which is used to overcome the viscous forces. The energy is dissipated as heat, and for practical purposes may usually be regarded as lost forever.