VIEWS: 5 PAGES: 15 POSTED ON: 8/13/2012
viscosity The most important distinction between a solid such as steel and a viscous fluid such as water or air is that shear stress in a solid material is proportional to shear strain, and the material ceases to deform when equilibrium is reached, whereas the shear stress in a viscous fluid is proportional to the time rate of strain. The proportionality factor for the solid is the shear modulus. The proportionality factor for the viscous fluid is the dynamic, or absolute, viscosity. For example, the shear stress of a fluid near a wall is given by τ=μ(dv/dy) where τ(tau) is the shear stress, μ(mu) is the dynamic viscosity, and dv/dy is the time rate of strain, which is also the velocity gradient normal to the wall. Thus the definition of the viscosity, μ, is the ratio of the shear stress to the velocity gradient, μ= τ/(dv/dy). Consider the flow shon in fig.2.1 this velocity distribution is typical of that for laminar (nonturbulent) flow next to a solid boundary. Several observations relation to this figure will help you appreciate the interaction between viscosity and velocty disribution. Figure 2.1 Velocity distribution next to a boundary. First, the velocity gradient ant the boundary is finite. The curve of velocity variation cannot be tangent to the boundary because this would imply an infinite velocity gradient and, in turn, and infinite shear stress, whech is impossible. Second, a velocity gradient that becomes less steep (dv/dy becomes smaller ) whith distance from the boundary has a maximum shear stress at the boundary, and the shear stress decreases with distance from the boundary. Also note that the velocity of the fluid is zero at the stationary boundary. This is characteistic of all flows dealt with in this basic txt. That is, at the boundary surface the fluid has the velocity of the boundary –no slip occurs. Many of the equations of fluid mechanics include the combination μ/ρ in them .Because it occurs so frequently, this combination has been given the special name kinematic viscosity (so called because the force dimension cancels out in the combination μ/ρ ). The symbol used to identify kinematic viscosity is ν(nu). Whenever shear stress is applied to fluids, motion occurs. This is the basic difference between fluids and solids. Solids can resist shear stress in a static condition, but fluidsdeform continuously under the action of a shear stress. Another important characteristic of fluids is that the viscous resistance is independent of the normal force (pressure) acting within the fluid. In constrast, for two solids sliing relative to each other, the shearing resistance is totally dependent on the normal force between the two. The manner in which viscous forces are produced can be seen in the conveyor-belt analogy. Consider a type of transit system in which people are carried from one part of a city to another on conveyor belts(Fig.2.2a) . People ride the fast-moving belt from left to right-an equilibrium condition exists. Next visualixe the action when the people Figure 2.2 step off the fast belt onto Conveyor-belt transportation a slower-moving belt (Fig. system. 2.2b) . Before they step off the fast-moving belt, each possesses a certain amount of momentum in the x direction. But as soon sa they acquire the wpeed of the slower belt, their momentum in the x direction is reduced by a significant amount. It is knownfrom basic mechanics that a change in monentum of a body results from an external force acting on that body. In our example, it is the slower belt that exers a force in the negative z direction as each person steps on the belt. Conversely, as each person steps on the slower belt, a forc eis exerted on the belt in the positive x direction. Now, if the people step from the faster belt to the slower 퓨딧 at a rather steady rate, then a rather continyuous force is exerted on the slower belt . In effect, by the action of the people moving in the negative y direction, they produe a force on the slow belt in the positive zx direction. One may think of this as shear force in the x direction. In a sililar manner, it can be visualixed that if people stepped from the slow-moving belt to the faster one, a “ shear force” in the negative x direction vwould be imposed on the faster belt. If the people were continuously going both ways (back and forth) from one the direction of motion ) on the slow belt and a like retarding force on the fast belt. Furthermore, as the relative speed between the belts changes (analogous to a change in velocity gradient ) , the shear force is increased or decreased in direct proportion to the increase or decrease in relative velocity. Thus if both belts were mad to have the same speed, the shear force would be zero. In fluid flow we can think of streams of fluid traveling in a given general direction, such as in a pipe, with the fluid nearer the pipe center traveling faster9analogous to the faster belt) while the fluid nearer the wall is traveling more slowly. The interaction between streams, in the case of gas flow, occurs when the molecules of gas travel back and forth between adjacent streams, thus creating a shear stress in the fluid. Because the rate of activity (back-and- forth motion)of the gas molecules increases with an increase with the temperature of the gas, such is indeed the case, as can be seen in gig.2.3 For liquids, the shear stress is involved with the cohesive forces between molecules. Thes eforce s decrease with temprature, which resuts in a decrease in viscosity with an increase in temperature (see Fig.2.3) The variation of viscosity (dynamic and kinematic ) for other fluids is given in figs,A.2 and A.3 in the Appendix. Figure 2.3 Kinematic viscosity for air and crude oil. Units of Viscosity From Eq. (2.6) tit can be seen that thee units of μ are N.s/m2 μ= τ/(dv/dy)=N/m2/((m/s)/m)=N.s/m2 ν are m2/s ν= μ/ ρ=N.s/m2/(N. s2 /m4)=m2/s