Viscosity, $\mu$

We all have a feel for viscosity. More viscous a fluid more difficult for it to flow. Oils flow at a slower rate than water. We understand viscosity as a property that tends to retard fluid motion.But we do have a more rigorous definition of viscosity, which can be developed from the thought experiment described before.

Figure 1.4: Flow between parallel plates

It was seen that when a shear force is applied to the top plate the fluid undergoes a continuous deformation ( What is a Fluid? Fig.1.4). As a result the block of fluid abcd deforms to ab'c'd after a time $\delta$t. Let the speed of the top plate be U. It is an important property of fluids that the layer of fluid adjacent to a solid surface moves with the same velocity as the solid surface. This is called the "No Slip" condition. Accordingly fluid layer closer to the top plate moves with a speed U while that closer to the lower plate is at rest. Thus the velocity of the fluid varies continuously from zero on the lower plate to U at the upper plate. In other words a velocity gradient develops in the fluid. In the simple case of the flow between parallel plates this is a linear profile. The velocity gradient is given by

where h is the distance between the two plates.

In a small instant of time $\delta t$ we find that the upper plate has moved by a distance bb' which is equal to $U \delta t$.

Now

Noting that for solids the shear stress $\tau$ is proportional to strain $\delta\alpha$ while for fluids it is proportional to rate of strain, $\dot{\gamma}$, which in turn is defined as

$$\dot{\gamma} = \lim_{\delta t \rightarrow 0} {{\delta\alpha} \over {\delta t}}$$ (1.4)

. Substituting for $\delta\alpha$ we have

$$\dot{\gamma}= {{U} \over {h}} = {{du} \over {dy}}$$ (1.5)

Since $\tau$ is proportional to $\dot{\gamma}$, we have

$$\tau \propto \dot{\gamma},$$

or

$$\tau \propto {{du} \over {dy}}$$ (1.6)

It is found that for common fluids such as air, water and oil the relationship between shear stress and velocity gradient can be expressed as,

$$\tau = \mu {{du} \over {dy}}$$ (1.7)

The constant of proportionality $\mu$ is an important property of fluids in determining the flow behaviour and is called Dynamic Viscosity or Absolute Viscosity. It is usual to refer to it as just Viscosity. It has the dimensions $FTL^{-2}$ and units of $Ns/m^2$ in the SI system.

Fluids for which the shear stress varies linearly as rate of strain are called Newtonian Fluids. Many of the common fluids belong to this category- air, water. When the relationship between shear stress and rate of strain is not linear, the fluid is designated Non-Newtonian. Examples of this category are some of the industrial fluids such as plastics, sludge and biological fluids such as blood. Typical plots of shear stress vs rate of strain are shown in Fig.1.5. Rheology is the branch of fluid mechanics which specialises in these fluids. We consider primarily common fluids such as water and air and hence restrict ourselves to Newtonian fluids.

Figure 1.5 : Flow between parallel plates

Viscosity of a fluid is strongly dependent on temperature and is a weak function of pressure. For example, when the pressure of air is increased from 1 atm to 50 atm, its viscosity increases only by about 10 percent allowing one to ignore its dependence on pressure. Fig.1.6 shows the manner of dependence of viscosity on temperature for some of the common fluids. It is seen that the viscosity of liquids deceases with temperature while that for the gases increases with temperature. This difference in behaviour is explained by the cohesive and intermolecular forces within the fluid. Liquids are characterized by strong cohesive forces and close packing of molecules. When temperature increases cohesive forces are weakened and there is less resistance to motion. Hence viscosity decreases. With gases, the cohesive forces are very weak and the molecules are spaced apart. Viscosity is due to the exchange of momentum between molecules as a result of random motion. As the temperature increases the molecular activity increases giving rise to an increased resistance to motion or in other words viscosity increases.

Figure 1.6 : Viscosity of Air and Water plotted against temperature

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney