ABSTRACT
An important mechanical property of žuids is viscosity. Physical systems and applications as diverse as žuid žow in pipes, the žow of blood, lubrication of engine parts, the dynamics of raindrops, volcanic eruptions, and planetary and stellar magnetic Ÿeld generation, to name just a few, all involve žuid žow and are controlled to some degree by žuid viscosity. Viscosity is the tendency of a žuid to resist žow and can be thought of as the internal friction of a žuid. Microscopically, viscosity is related to molecular diffusion and depends on the interactions between molecules or, in complex žuids, larger-scale žow units. Di¥usion tends to transfer momentum from regions of high momentum to regions of low momentum, thus smoothing out variations in žow velocity. In this sense, the internal friction of a žuid 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 žuid, energy must be supplied to sustain a žow. Since viscosity is related to the diffusive transport of momentum, the viscous response of a žuid is called a momentum transport process. œe žow velocity within a žuid will vary, depending on location. Consider a viscous žuid 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 žuid adjacent to the upper plate to move in the direction of F .⃗ œe žuid adjacent to the top plate is constrained by the no-slip boundary condition to move at the same speed as the plate. Similarly, the žuid next to the stationary bottom plate must be stationary. œe motion of the top plate thus causes the žuid to žow with a velocity proŸle across the liquid that decreases linearly from the upper tothe lower plate, as shown in Figure 46.1. œis arrangement is referred toas simple shear. œe applied force is called ashear, and the force per unit area ashear stress. œe resulting deformation rate of the žuid, or equivalently the velocity gradient dUx/dz, is called the shear strain rate, g zx. œe 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 žuid on the upper plate in the negative x-direction)
dUx/dz is the gradient of the x-velocity in the z-direction in the žuid, that is, the shear strain rate η is the coe¼cient of viscosity
Note that in general, the shear strain rate is a more complex function of the žuid velocity-gradient tensor. In this case, because one is concerned with a shear force that produces the žuid motion, η is more speciŸcally called the shear dynamic viscosity. In žuid mechanics, where the motion of a žuid is considered without reference to force, it is common to deŸne the kinematic viscosity, ν, which is simply given by
n h
r = (46.2)
where ρ is the mass density of the žuid. œe viscosity deŸned by Equation 46.1 is relevant only for laminar (i.e., layered or sheetlike) or stream-
line žow as depicted in Figure 46.1, and it refers to the molecular viscosity or intrinsic viscosity. œe molecular viscosity is a property of the material that depends microscopically on interactions between individual molecules and is manifested macroscopically as the žuid’s resistance to žow. When the žow is turbulent, small-scale turbulent vortices can contribute to the overall di¥usion of momentum, resulting in an e¥ective viscosity, sometimes called the eddy viscosity, that, depending on the Reynolds number, can be as much as 106 times larger than the intrinsic viscosity.
