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Viscosity

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Viscosity represents the internal resistance of a fluid to motion. Thicker fluids have a higher viscosity. For example, the viscosity of oil is higher than that of water, the oil is more viscous than the water. Viscosity may be known as either dynamic (absolute) viscosity or kinematic viscosity. This discussion is limited to dynamic viscosity. The symbol μ is used to represent the coefficient of viscosity, also called the viscous coefficient, dynamic viscosity of a fluid, and absolute viscosity of a fluid. The coefficient of viscosity is a proportionality constant that is unique for each material.

Nomenclature

μ	coefficient of viscosity
τ	shear stress acting on fluid layer
F	force
A	area
u( y)	velocity profile

Details
The magnitude of the drag force depends, in part, on viscosity.

To obtain a relation for viscosity, consider a fluid layer between two very large parallel plates (or equivalently, two parallel plates immersed in a large body of a fluid) separated by a distance L. Now a constant parallel force F is applied to the upper plate while the lower plate is held fixed. After the initial transients, it is observed that the upper plate moves continuously under the influence of this force at a constant velocity V. The fluid in contact with the upper plate sticks to the plate surface and moves with it at the same velocity, and the shear stress τ acting on this fluid layer is:

τ =

where A is the contact area between the plate and the fluid. Note that the fluid layer deforms continuously under the influence of shear stress.

The fluid in contact with the lower plate assumes the velocity of that plate, which is zero (because of the no-slip condition). In steady laminar flow, the fluid velocity between the plates varies linearly between 0 and V, and thus the velocity profile and the velocity gradient are:

u( y) =

and

where y is the vertical distance from the lower plate.

During a differential time interval dt, the sides of fluid particles along a vertical line MN rotate through a differential angle dβ while the upper plate moves a differential distance da = V dt. The angular displacement or deformation (or shear strain) can be expressed as:

dβ ≈ tan β =

V dt

Rearranging, the rate of deformation under the influence of shear stress τ becomes:

dβ

Thus it is concluded that the rate of deformation of a fluid element is equivalent to the velocity gradient du/dy. Further, it can be verified experimentally that for most fluids the rate of deformation (and thus the velocity gradient) is directly proportional to the shear stress τ,

τ ∝

dβ

or:

τ ∝

In one-dimensional shear flow of Newtonian fluids, shear stress can be expressed by the linear relationship:

τ = μ

where the constant of proportionality μ is called the coefficient of viscosity or the dynamic (or absolute) viscosity of a fluid, whose unit is kg/(ms), or equivalently, (Ns)/m² (or Pa*s where Pa is the pressure unit pascal). A common viscosity unit is poise, which is equivalent to 0.1 Pa*s (or centipoise, which is one-hundredth of a poise). The viscosity of water at 20°C is 1 centipoise, and thus the unit centipoise serves as a useful reference. A plot of shear stress versus the rate of deformation (velocity gradient) for a Newtonian fluid is a straight line whose slope is the viscosity of the fluid, as shown. Note that viscosity is independent of the rate of deformation.

The shear force acting on a Newtonian fluid layer (or, by Newton's third law, the force acting on the plate) is:

F = τA = μA

where again A is the contact area between the plate and the fluid. Then the force F required to move the upper plate in the figure at a constant velocity of V while the lower plate remains stationary is:

F = μA

This relation can alternately be used to calculate μ when the force F is measured. Therefore, the experimental setup just described can be used to measure the viscosity of fluids. Note that under identical conditions, the force F will be very different for different fluids.

For non-Newtonian fluids, the relationship between shear stress and rate of deformation is not linear, as shown. The slope of the curve on the τ versus du/dy chart is referred to as the apparent viscosity of the fluid. Fluids for which the apparent viscosity increases with the rate of deformation (such as solutions with suspended starch or sand) are referred to as dilatant or shear thickening fluids, and those that exhibit the opposite behavior (the fluid becoming less viscous as it is sheared harder, such as some paints, polymer solutions, and fluids with suspended particles) are referred to as pseudoplastic or shear thinning fluids. Some materials such as toothpaste can resist a finite shear stress and thus behave as a solid, but deform continuously when the shear stress exceeds the yield stress and thus behave as a fluid. Such materials are referred to as Bingham plastics.

In fluid mechanics and heat transfer, the ratio of dynamic viscosity to density appears frequently. For convenience, this ratio is given the name kinematic viscosity.

The viscosity of a fluid is a measure of its "resistance to deformation." Viscosity is due to the internal frictional force that develops between different layers of fluids as they are forced to move relative to each other. Viscosity is caused by the cohesive forces between the molecules in liquids and by the molecular collisions in gases, and it varies greatly with temperature. The viscosity of liquids decreases with temperature, whereas the viscosity of gases increases with temperature, as shown. This is because in a liquid the molecules possess more energy at higher temperatures, and they can oppose the large cohesive intermolecular forces more strongly. As a result, the energized liquid molecules can move more freely.

In a gas, on the other hand, the intermolecular forces are negligible, and the gas molecules at high temperatures move rrandomly at higher velocities. This results in more molecular collisions per unit volume per unit time and therefore in greater resistance to flow. The viscosity of a fluid is directly related to the pumping power needed to transport a fluid in a pipe or to move a body (such as a car in air or in a submarine in the sea) through a fluid.

The kinetic theory of gases predicts the viscosity of gases to be proportional to the square root of temperature. That is, μ_gas ∝ T^1/2. This prediction is confirmed by practical observations, but deviations for different gases need to be accounted for by incorporating some correction factors. The viscosity of gases is expressed as a function of temperature by the Sutherland correlation (from the U.S. Standard Atmosphere) as:

For gases:

μ =

aT^1/2

1 + b/T

where T is absolute temperature and a and b are experimentally determined constants. Note that measuring viscosities at two different temperatures is sufficient to determine these constants. For air, the values of these constants are a = 1.458 × 10^-6 kg/(m*s*K^1/2) and b = 110.4 K at atmospheric conditions. The viscosity of gases is independent of pressure at low to moderate pressures (from a few percent of 1 atm to several atm). But viscosity increases at high pressures due to the increase in density.

For liquids, the viscosity is approximated as:

μ = a10^{b/(T − c)}

where again T is absolute temperature and a, b, and c are experimentally determined constants. For water, using the values a = 2.414 × 10^-5 Ns/m², b = 247.8 K, and c = 140 K results in less than 2.5 percent error in viscosity in the temperature range of 0°C to 370°C.

Consider a fluid layer of thickness L within a small gap between two concentric cylinders, such as the thin layer of oil in a journal bearing. The gap between the cylinders can be modeled as two parallel flat plates separated by a fliud. Noting that torque is T = FR (force times the moment arm, which is the radius R of the inner cylinder in this case), the tangential velocity is V = ωR (angular velocity times the radius), and taking the wetted surface area of the inner cylinder to be A = 2πRL by disregarding the shear stress acting on the two ends of the inner cylinder, torque can be expressed as:

T = FR = μ

2πR³ωL

= μ

4π²R³nL

where L is the length of the cylinder and n is the number of revolutions per unit time, which is usually expressed in rpm (revolutions per minute). Note that the angular distance traveled during one rotation is 2π rad, and thus the relation between the angular velocity in rad/min and the rpm is μ = 2πn. The above equation can be used to calculate the viscosity of a fluid by measuring torque at a specified angular velocity. Therefore, two concentric cylinders can be used as a viscometer, a device that measures viscosity.

The viscosities of different fluids may differ by several orders of magnitude. Also note that it is more difficult to move an object in a higher-viscosity fluid such as engine oil than it is in a lower-viscosity fluid such as water. Liquids, in general, are much more viscous than gases.

Viscosity is a quantitative measure of a fluid's resistance to flow. More specifically, it determines the fluid strain rate that is generated by a given applied shear stress. Air can be easily moved through—it has a low viscosity. Movement is more difficult in water, which has 50 times higher viscosity. Still more resistance is found in SAE 30 oil, which is 300 times more viscous than water. Glycerin is five times more viscous than SAE 30 oil, or blackstrap molasses, another factor of five higher than glycerin. Fluids may have a vast range of viscosities.

Consider a fluid element sheared in one plane by a single shear stress τ, as in the figure. The shear strain angle δθ will continuously grow with time as long as the stress τ is maintained, the upper surface moving at speed δu larger than the lower. Such common fluids as water, oil, and air show a linear relation between the applied shear and resulting strain rate:

τ ∝

δθ

δt

From the geometry of the figure:

(Eq1)

tan δθ =

δu δt

δy

In the limit of infinitesimal changes, this becomes a relation between shear strain rate and velocity gradient:

dθ

From Eq1, then, the applied shear is also proportional to the velocity gradient for the common linear fluids. The constant of proportionality is the viscous coefficient μ:

(Eq2)

τ = μ

dθ

= μ

Eq2 is dimensionally consistent; therefore μ has dimensions of stress–time. The British unit is slugs per foot-second, and the SI unit is kilograms per meter-second. The linear fluids that follow Eq2 are called newtonian fluids.

The strain angle θ(t) is not of much concern in fluid mechanics, concentration is instead focused on the velocity distribution u( y), as in the figure. Eq2 can be used to derive a differential equation for finding the velocity distribution u( y)—and, more generally, V(x, y, z, t)—in a viscous fluid. The figure illustrates a shear layer, or boundary layer, near a solid wall. The shear stress is proportional to the slope of the velocity profile and is greatest at the wall. Further, at the wall, the velocity u is zero relative to the wall: This is called the no-slip condition and is characteristic of all viscous fluid flows.

The viscosity of newtonian fluids is a true thermodynamic property and varies with temperature and pressure. At a given state ( p, T) there is a vast range of values among the common fluids.

Generally speaking, the viscosity of a fluid increases only weakly with pressure. Temperature, however, has a strong effect, with μ increasing with T for gases and decreasing for liquids. It is customary in most engineering work to neglect the pressure variation.

Variation of Viscosity with Temperature

Temperature has a strong effect and pressure a moderate effect on viscosity. The viscosity of gases and most liquids increases slowly with pressure. Water is anomalous in showing a very slight decrease below 30°C. Since the change in viscosity is only a few percent up to 100 atm, pressure effects may be neglected, depending on the application.

Gas viscosity increases with temperature. Two common approximations are the power law and the Sutherland law. The power law is:

μ₀

≈

(

T₀

)

and the Sutherland law is:

μ₀

(T/T₀)^3/2(T₀ + S)

T + S

where μ₀ is a known viscosity at a known absolute temperature T₀ (usually 273 K). The constants n and S are fit to the data, and both formulas are adequate over a wide range of temperatures. For air, n ≈ 0.7 and S ≈ 110 K = 199°R.

Liquid viscosity decreases with temperature and is roughly exponential, μ ≈ ae^−bT; but a better fit is the empirical result that ln μ is quadratic in 1/T, where T is absolute temperature:

μ₀

≈ a + b

(

T₀

)

+ c

(

T₀

)

For water, with T₀ = 273.16 K, μ₀ = 0.001792 kg/(m*s), suggested values are a = –1.94, b = –4.80, and c = 6.74, with accuracy about ±1%.