§1.3 Major physical properties of fluid

Important physical properties of fluid include inertia, compressibility, expansibility, viscosity and surface tension.

1.3.1 Inertia

Inertia is the resistance of any physical obj ect to any change in its state of motion (this includes changes to its speed, direction or state of rest).It is the tendency of obj ects to keep moving in a straight line at constant velocity.Mass is the quantitative measure of inertia of a body.The greater the mass of a body, the greater is its inertia. Mass density of a substance is its mass per unit volume.The symbol most often used for density isρwith a unit of kg/m3.For homogeneous fluid, mass density is defined as mass divided by volume as described in equation(1-1 a):

where m is mass and V is volume.

If the fluid is not homogeneous, then its density varies between different regions of the fluid.In that case the density around any given location is determined by calculating the density of a small volume (ΔV)around that location.In the limit of an infinitesimal volume, the density of an inhomogeneous fluid at a point becomes:

where dV is an elementary volume at position r.

The density of a material varies with temperature and pressure.This variation is typically small for liquids.The density of liquid can be considered as a constant.For example, water density is 1000 kg/m3and mercury density is 13600 kg/m3.

The density of gas varies with temperature and pressure greatly.Air density is ·7·1.29 kg/m3at 0 ℃ and under a standard atmospheric pressure.

1.3.2 Compressibility

1．Compressibility of liquid

In fluid mechanics, compressibility is a measure of the relative volume change of a liquid as a response to a pressure (or mean stress)change.The compressibility of liquid can be quantified by coefficient of compressibility, which can be denoted by k with a unit of Pa-1.The compressibility of liquid can be determined by equation(1-2):

where V is volume and p is pressure.

With the enhancement of pressure, the volume of liquid become smaller and the mass remains unchanged (dm=0), leading to the increase of density.The relationship between relative increments can be determined by d m=d(ρV)=ρd V+V dρ=0, and then we can write:

which was substituted into equation(1-2), then equation(1-2)becomes:

The smaller the value of k, the more difficult is the liquid to be compressed.

The bulk modulus K is used to represent the compressibility of the liquid in engineering.Bulk modulus is the reciprocal of coefficient of compressibility (k).

The unit of bulk modulus is Pa (N/m2), which is same as that of pressure.The greater the value of K, the more difficult is the liquid to be compressed.When K tends to infinity, the liquid is impossible to be compressed.

K and k values vary with the type of liquid.For example, the compressibility of mercury is around 8% of that of water and the compressibility of nitric acid is around 6 times greater than that of water.For the same liquid, the values of k and K vary with temperature and pressure.However, this variation is small.

Bulk modulus variation is not very large at fixed temperature and moderate pressure.Bulk modulus can be approximatively described as:

or

In the field of engineering design, bulk modulus of water is approximated to be 2.1 ·8·×109Pa, indicating the relative change of the volume is about when the variation of the pressure is an atmospheric pressure.Thus, when Δp is not large, the compressibility of water can be ignored and water density can be considered as a constant.However, the compressibility of water is beyond neglect when water-hammer problems of water flow in the pipeline are discussed.

Expansion coefficient αv can be used to represent expansion of liquid, indicating the relative variation rate of the volume when pressure remains unchanged and variation of liquid temperature is 1 K, which can be described as equation(1-6):

where d T is temperature variation and the unit of αv is K-1 or ℃-1.

Expansion coefficient of liquid varies with pressure and temperature.At an atmospheric pressure and 20℃, αv is 2.1×10-4 K-1, which can be ignored.However, expansion of water must be taken into account when the variation of working temperature is large.

2．Compressibility of gas

Gas density is determined by the equation of state.

An ideal gas is a theoretical gas composed of many randomly moving point particles that do not interact except when they collide elastically.Its equation of state can be described as equation(1-7):

where p is absolution pressure of gas, R is ideal gas constant with the value of 287 N·m/(kg·K), and T is absolute temperature.

Ideal gas does not exist.However, at normal conditions such as standard temperature and pressure, most real gases behave qualitatively like an ideal gas.Many gases such as nitrogen, oxygen, hydrogen, noble gases, and some heavier gases like carbon dioxide can be treated like ideal gases within reasonable tolerances.

Example 1-1:The volume of water is 2.5 m3at 20 ℃.How much is its volume increase when the temperature rises to 80℃?

Solution:Water densityρ1is 998.23 kg/m3at 20 ℃ while water densityρ2is 971.83 kg/m3at 80 ℃.Due to conservation of mass, the volume of water increases with the decrease of density when the temperature rises.Then, we can write

Data were substituted into the equation above, then

Then,

The volume of water increases by 2.64 percent.

Example 1-2:How much is pressure increase in order that the relative reduction of water volume can reach 0.1% and 1%? Bulk modulus K of water is 2000 MPa.

Solution:Equation(1-5)was rearranged to become

Pressure increase is 2.0 MPa and 20 MPa, respectively in order that the relative reduction of water volume can reach 0.1 % and 1 %.

Example 1-3:The length and diameter of delivery pipeline is 200 m and 0.4 m, respectively.During hydraulic testing, stop pressurizing after pressure in pipe reaches 5.39×106Pa.The pressure in pipe drops to 4.90×106Pa one hour after stopping pressurizing.Without considering pipe deformation, how much is the volume of water leaking out of pipe crack per second? Water compressibility k is 4.83×10-10Pa-1.

Solution:After water leak out of pipe crack, the pressure in the pipe drops, leading to water expansion.The volume of expanding water is

The expansion volume of water is 5.95× 10-3m3, which is the volume of water leaking out of pipe crack in one hour.Then the volume of water leaking out of pipe crack per second is

1.3.3 Viscosity

1．Viscosity and internal friction of fluid

Fluid is flowable.Even a very small shearing force will deform a fluid body at rest while fluid can resist deformation by shear stress in motion state.The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress.

To illustrate this viscosity of fluid, consider fluid between two parallel plates and adhering to them acted on by a constant shearing force F.The distance between two parallel plates is Y.The plates are assumed to be very large, so that one need not consider what happens near their edges.When the lower plate keeps fixed, the upper plate moves parallel to the lower one at constant speed U under the action of external force (see Fig.1-1).If the speed of the upper plate is small enough, the fluid particles will move parallel to it, and their speed will vary linearly from zero at the bottom to U at the top.If the distanceY between the plates is not big or the velocityU is not high, the velocity distribution of the fluid in the direction normal to the plate is linear.Then, equation(1-8)can be obtained as:

or

The flowing fluid can be considered to be composed of a series of thin layers.Each layer of fluid will move faster than the one j ust below it, and friction between them will give rise to a force resisting their relative motion, i.e.internal friction, which is characterization of viscosity.The fast moving fluid layer exerts on the slow moving layer an internal friction in the direction same as its motion.The internal friction accelerates the movement.On the other hand, the slower fluid moving layer exerts on the fast moving layer an internal friction in the direction opposite to its motion.The internal friction slows down the movement.Thus, the flow pattern of the fluid can be changed.Due to the presence of viscosity, work was done and mechanical energy was consumed in order to overcome the frictional force in the process of fluid movement. Therefore, fluid viscosity is the reason why fluid mechanical energy was lost.

2．Law of Newton internal friction

In 1686, the law proposed by Isaac Newton and proved by the descendants through experiments is that for most fluids, the shearing force is in direct proportion to the contact area A and the velocity gradient .Thus, we can write

The proportional factorμis introduced, then equation(1-10)can be obtained:

According to equation(1-10), viscous shearing force τ is a constant when flow velocity distribution is linear as shown in Fig.1-1.

The velocity gradient is the change rate of flow velocity in the direction perpendicular to the flow layer.In order to explain the physical implication of the velocity gradient, a rectangular fluid micelle (abcd)was considered between the two plates and the thickness of rectangle is dy.After dt, this fluid micelle moves from abcd to a＇b＇c＇d＇, as shown in Fig.1-2.Due to the presence of velocity difference du between the two moving layers, fluid micelle has angular deformation dθ besides translation movement.Because dt is very small, dθis also very small.Thus, we can write

According to equation(1-11), is local shear velocity of fluid micelle.Because shearing force leads to the generation of angular deformation, angular deformation is also called the rate of shear deformation.

Therefore, Law of Newton internal friction can be rewritten as:

indicating the shear force of the fluid is proportional to the rate of shear deformation, which is a very important characteristic of a fluid that is different from the solid.The shearing force of solid is proportional to the angular deformation, and is dependent of the deformation rate.

The proportional factorμis known as dynamic viscosity (abbreviated as viscosity), which reflects the measurement of fluid viscosity.Its unit is Pa·s.τis viscous shearing force, indicating internal friction per area.The unit ofτis Pa.The viscosity of fluid varies with the type of fluid.Under same conditions, the viscosity of the liquid is greater than that of gas and varies with temperature and pressure.And basically, the viscosity of the fluid is caused by the cohesive force of the flowing fluid and the molecular momentum transfer.The viscosity of liquids decreases with increasing temperature while the viscosity of gases increases with temperature.In liquids, where the molecules are packed as closely together as the repulsive forces will allow, each molecule is in the range of attraction of several others.The exchange of sites among molecules, responsible for the deformability, is impeded by the force of attraction from neighboring molecules, indicating intermolecular attraction (cohesive force)is large. The contribution from these intermolecular forces to the force on surface elements of fluid particles having different macroscopic velocities is greater than the contribution from the molecular momentum transfer.For gases, the mean distance between the molecules is great (about ten effective molecular diameters)and cohesive force is small. The viscosity of gases, where the momentum transfer is basically its only source, increases with temperature, since increasing the temperature increases the thermal velocity of the molecules, and thus the momentum exchange is favored.Although the viscosity of both gases and liquids are affected by temperature, the influence is different.The effect of temperature on the viscosity of liquid is greater than that on the gas viscosity.For example, the viscosity of water is reduced by about half and the viscosity of the air increase by about 9% when the temperature rises from 15 to 50℃. The pressure has little effect on the viscosity of the fluid.For common fluids such as water and air, the effect of pressure can be ignored.

The viscosity of fluid can also be represented by kinematic viscosity.The kinematic viscosity (also called“momentum diffusivity”)is the ratio of the dynamic viscosityμto the density of the fluidρ.It is usually denoted by the Greek letter nu (ν)with a unit of m2/s.Kinematic viscosity also varies with the temperature.The kinematic viscosityνof water can be calculated by the empirical equation:

where t is centigrade temperature.

Example 1-4:Try to plot distribution curve of liquid flow velocity and shearing forces.The flow velocity distribution of the upper and lower layers of the liquid between two horizontal plates is linear.The viscosity of the upper layer of liquid is μ1and the height of this liquid layer is h1.The viscosity of the lower layer of liquid is μ2and the height is h2.The movement speed of plate is U, as shown in Fig.1-3a.

Solution:The flow rate on the interface of the liquid layers was set as u.

The flow velocity distribution of liquid is linear.Thus, according to the law of Newton's internal friction, shearing force on the upper and lower layers can be described as:

Upper:

Lower:

On the interface of liquid layers, it is known:τ1=τ2=τ.

Then, we can write

which was rearranged as:

Thus, the flow velocity distribution is:

Upper:

Lower:

Because the flow velocity distribution of each layer of fluid is linear, distribution curve of liquid flow velocity can be obtained and the shear stress is uniformly distributed as long as the flow velocity on the interface was plotted qualitatively, as shown in Fig. 1-3b.

Example 1-5:A wood board with the base area of 0.4× 0.45 m2, the height of 0.01 m and the mass of 5 kg moves downward along a inclined plane coated with lubricant at a constant velocity, as shown in Fig.1-4.The movement speed u of this board is 1 m/s.The thicknessδof lubricant is 0.1 mm.The velocity distribution of oil movement driven by the wood board is linear.Calculate the viscosity of the lubricant.

Solution:Analyze the forces exerted on the wood board in the direction of motion.

The component force of gravity in the direction of motion is mg sinθ, and the shearing force is:

According to Newton's laws of motion, we can write

Then the equation below can be derived:

Thenμcan be calculated as:

Example 1-6:A disc (the diameter is 0.1 m), which is driven by a rotating shaft, rotates on a fixed plate.A layer of oil film with the thicknessδof 1.5 mm is trapped between disc and plate.When the disc rotated with the rate of 50 r/min, the torque M was measured to be 2.94×10-4N·m.The velocity distribution in the oil film is linear in the direction perpendicular to the disc.Determine the viscosity of oil and shearing forceτ0at the edge of disc.

Solution:For a microelement, the flow rate is:

The shearing force is:

Viscous friction torque exerted on the microelement of this disc is:

Total friction torque is:, τdistribution is linear, as shown in Fig.1-5.

Thus, when r=0.05 m, τ0=1.50 Pa.

1.3.4 Surface tension

Surface tension is the elastic tendency of liquids that makes them acquire the least surface area possible.At liquid-air interfaces, surface tension results from the greater attraction of liquid molecules to each other (due to cohesion)than to the molecules in the air (due to adhesion).The net effect is an inward force at its surface that causes the liquid to behave as if its surface were covered with a stretched elastic membrane.Thus, the surface becomes under tension from the imbalanced forces, which is probably where the term“surface tension”came from.Surface tension can make the hemispherical water drop suspending at the outlet of the water tap without dropping.When tube is inserted into the liquid, surface tension can draw liquid up the tube in a phenomenon known as capillary action if the tube is sufficiently narrow and the liquid adhesion to its walls is sufficiently strong.

Surface tension, usually represented by the symbolσ, is measured in force per unit length.Its SI unit is newton per meter.Surface tensionσvaries with the type of fluid and temperature.At 20℃, surface tension of water and mercury is 0.074 N/m and 0.51 N/m, respectively when exposure to air.

Surface tension is not great.Thus its influence can be ignored in practical engineering.For experiments regarding capillary rise, water droplet and bubble formation, the dispersion of liquid j et and small hydraulic model, surface tension is important.

The values of main physical parameters related to water, air and other fluids are included in CD.