VISCOSITY AND THE MECHANISMS OF MOMENTUM TRANSPORT

Consider a pair of large parallel plates, each one with area A, separated by a distance Y. In the space between them is a fluid (gas or liquid). When the final state (steady state) has been attained, a constant force F is required to maintain the motion of the lower plate.

At constant velocity, V

Fluid gains momentum

- Sheering force per unit area = Newton’s law of viscosity- Flux of x-momentum in the positive y direction, where the term "flux" means "flow per unit area."- All gases and all liquids with molecular weight of less than about 5000 (Newtonian fluids)- Polymeric liquids, suspensions, pastes, slurries, and other complex fluids (non-Newtonian fluids)

In the neighborhood of the moving solid surface at y = 0 the fluid acquires a certain amount of x-momentum. This fluid, in turn, imparts momentum to the adjacent layer of liquid, causing it to remain in motion in the x direction. Hence x-momentum is being transmitted through the fluid in the positive y direction.

Therefore τyx may also be interpreted as the flux of x-momentum in the positive y direction, where the term "flux" means "flow per unit area." This interpretation is consistent with the molecular picture of momentum transport and the kinetic theories of gases and liquids..

The idea in the preceding paragraph may be paraphrased by saying that momentum goes "downhill" from a region of high velocity to a region of low velocity-just as a sled goes downhill from a region of high elevation to a region of low elevation.

The velocity gradient can therefore be thought of as a "driving force" for momentum transport.

PROBLEMS

Two parallel plates are 10 cm apart. The bottom plate is stationary. The fluid between the plates is water which has a viscosity of 0.001 kgm-1s-1. Calculate the momentum flux or force per unit area necessary to maintain a plate in motion at a velocity of 30 cms-1.

A Newtonian fluid with a viscosity of 0.001 kgm-1s-1 is placed between two large parallel plates. The distance between the plates is 4 mm. The lower plate is pulled in the positive x-direction with a force of 0.5 N, while the upper plate is pulled in the negative x-direction with a force of 2 N. Each plate has an area of 2.5 m2. If the velocity of the lower plate is 0.1 m/s, calculate:

i) The steady state momentum flux ii) The velocity of the upper plate

GENERALIZATION OF NEWTON'S LAW OF VISCOSITY

In Fig. 1.2-1 is shown a small cube-shaped volume element within the flow field, each face having unit area. The center of the volume element is at the position x, y, z.

There will be two contributions to the force: that associated with the pressure, and that associated with the viscous forces.

The pressure force will always be perpendicular to the exposed surface.

Hence in (a) the force per unit area on the shaded surface will be a vector pδx - that is, the pressure (a scalar) multiplied by the unit vector δx in the x direction. Similarly, the force on the shaded surface in (b) will be pδy and in (c) the force will be pδz.

The pressure forces will be exerted when the fluid is stationary as well as when it is in motion.

This tabulation is a summary of the forces per unit area (stresses) exerted within a fluid, both by the thermodynamic pressure and the viscous stresses. Sometimes we will find it convenient to have a symbol that includes both types of stresses, and so we define the molecular stresses as follows:

τ as the viscous stress tensor (with components τij) as the molecular stress tensor (with components ij)

Conclusion: two coefficients characterizing the fluid: the viscosity µ and the dilatational viscosity K. Usually, in solving fluid dynamics problems, it is not necessary to know K. If the fluid is a gas, we often assume it to act as an ideal monoatomic gas, for which K is identically zero. If the fluid is a liquid, we often assume that it is incompressible, and for incompressible liquids (.v) = 0, and therefore the term containing K is discarded anyway. The dilational viscosity is important in describing sound absorption in polyatomic gases and in describing the fluid dynamics of liquids containing gas bubble.

So, generalization for Newton's law of viscosity:

CONVECTIVE MOMENTUM TRANSPORT

Momentum can be transported by the bulk flow of the fluid, and this process is called convective transport.

At the center of the cube (located at x, y, z) the fluid velocity vector is v. The volume rate of flow across the shaded unit area in (a) is vx. This fluid carries with it momentum ρv per unit volume. Hence the momentum flux across the shaded area is vxρv; note that this is the momentum flux from the region of lesser x to the region of greater x.

ρvxvy is the convective flux of y-momentum across a surface perpendicular to the x direction.

the combined momentum flux, which is the sum of the molecular momentum flux and the convective momentum flux:

Keep in mind that the contribution pδ contains no velocity, only the pressure; the combination ρvv contains the density and products of the velocity components; and the contribution τ contains the viscosity and, for a Newtonian fluid, is linear in the velocity gradients.

PRESSURE AND TEMPERATURE DEPENDENCE OF VISCOSITY

How to estimate viscosity of gas & liquid. Experimental but sometime experimental data

are lacking and there is not time to obtain them.

The viscosity can be estimated by empirical methods - a corresponding-states correlation

The plot in Fig. 1.3-1 gives a global view of the pressure and temperature dependence of viscosity. The reduced viscosity µr = µ/µc is plotted versus the reduced temperature Tr= T/Tc for various values of the reduced pressure pr = p/pc.

The viscosity of a gas at low density increases with increasing temperature, whereas the viscosity of a liquid decreases with increasing temperature.

µc may be estimated in one of the following ways:

(i) if a value of viscosity is known at a given reduced pressure and temperature, preferably at conditions near to those of interest, then µc can be calculated from µc = µ/µr

(ii) if critical p-V-T data are available, then µc may be estimated from these empirical relations:

estimation of viscosities of mixtures

EXAMPLE

PROBLEMS