ABSTRACT
An important mechanical property of uids is viscosity. Physical systems and applications as diverse as uid ow in pipes, the ow of blood, lubrication of engine parts, the dynamics of raindrops, volcanic eruptions, and planetary and stellar magnetic eld generation, to name just a few, all involve uid ow and are controlled to some degree by uid viscosity. Viscosity is the tendency of a uid to resist ow and can be thought of as the internal friction of a uid. Microscopically, viscosity is related to molecular diffusion and depends on the interactions between molecules or, in complex uids, larger-scale ow units. Diusion tends to transfer momentum from regions of high momentum to regions of low momentum, thus smoothing out variations in ow velocity. In this sense, the internal friction of a uid is analogous to the macroscopic mechanical friction, which causes an object sliding across a planar surface to slow down. In the mechanical system, energy must be supplied to sustain the motion of the object over the plane, while in a uid, energy must be supplied to sustain a ow. Since viscosity is related to the diffusive transport of momentum, the viscous response of a uid is called a momentum transport process. e ow velocity within a uid will vary, depending on location. Consider a viscous uid at constant pressure between two closely spaced parallel plates as shown in Figure 46.1. A force, F, applied to the
46.1 Shear Viscosity ................................................................................46-1 46.2 Newtonian and Non-Newtonian Fluids .....................................46-3
46.6 Oscillation Methods .....................................................................46-17 46.7 Acoustic Methods .........................................................................46-20 46.8 Microrheology ...............................................................................46-22
High-Pressure Rheometry References ..................................................................................................46-29 Further Information .................................................................................46-31
top plate causes the uid adjacent to the upper plate to move in the direction of F ⃗. e uid adjacent to the top plate is constrained by the no-slip boundary condition to move at the same speed as the plate. Similarly, the uid next to the stationary bottom plate must be stationary. e motion of the top plate thus causes the uid to ow with a velocity prole across the liquid that decreases linearly from the upper to the lower plate, as shown in Figure 46.1. is arrangement is referred to as simple shear. e applied force is called a shear, and the force per unit area a shear stress. e resulting deformation rate of the uid, or equivalently the velocity gradient dUx/dz, is called the shear strain rate, g zx. e mathematical expression describing the viscous response of the system to the shear stress is simply
dU
dz = = (46.1)
where τzx, the shear stress, is the force per unit area exerted on the upper plate in the x-direction (and
hence is equal to the force per unit area exerted by the uid on the upper plate in the negative x-direction)
dUx/dz is the gradient of the x-velocity in the z-direction in the uid, that is, the shear strain rate η is the coefficient of viscosity
Note that in general, the shear strain rate is a more complex function of the uid velocity-gradient tensor. In this case, because one is concerned with a shear force that produces the uid motion, η is more specically called the shear dynamic viscosity. In uid mechanics, where the motion of a uid is considered without reference to force, it is common to dene the kinematic viscosity, ν, which is simply given by
n h
r = (46.2)
where ρ is the mass density of the uid. e viscosity dened by Equation 46.1 is relevant only for laminar (i.e., layered or sheetlike) or stream-
line ow as depicted in Figure 46.1, and it refers to the molecular viscosity or intrinsic viscosity. e molecular viscosity is a property of the material that depends microscopically on interactions between individual molecules and is manifested macroscopically as the uid’s resistance to ow. When the ow is turbulent, small-scale turbulent vortices can contribute to the overall diusion of momentum, resulting in an eective viscosity, sometimes called the eddy viscosity, that, depending on the Reynolds number, can be as much as 106 times larger than the intrinsic viscosity.
