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The engineering science of fluid mechanics: fluid properties, the application of the basic laws of mechanics and thermodynamics, and orderly experimentation.

- The properties of density and viscosity play principal roles in open- and closed-channel flow and in flow around immersed objects.
- Surface-tension effects are important in the formation of droplets, in the flow of small jets, and in situations where liquid-gas-solid or liquid-liquid-solid interfaces occur, as well as in the formation of capillary waves.
- The property of vapor pressure, which accounts for changes of phase from liquid to gas, becomes important when reduced pressures are encountered.
- This chapter: liquid is defined and the International System of Units (SI) of force, mass, length, and time units are discussed before the discussion of properties and definition of terms is taken up.

1.1 DEFINITION OF A FLUID

- Fluid: substance that deforms continuously when subjected to a shear stress, no matter how small that shear stress may be.
- Shear force is the force component tangent to a surface, and this force divided by the area of the surface is the average shear stress over the area. Shear stress at a point is the limiting value of shear force to area as the area is reduced to the point.
- Fig. 1.1

Figure 1.1 Deformation resulting from application of constant shear force.

A substance is placed between two closely spaced parallel plates so large that conditions at their edges may be neglected. The lower plate is fixed, and a force F is applied to the upper plate, which exerts a shear stress F/A on any substance between the plates. A is the area of the upper plate. When the force F causes the upper plate to move with a steady (nonzero) velocity, no matter how small the magnitude of F, one may conclude that the substance between the two plates is a fluid.

The fluid in immediate contact with a solid boundary has the same velocity as the boundary (no slip at the boundary)

- Fig.1.1: fluid in the area abcd flows to the new position ab'c'd, each fluid particle moving parallel to the plate and the velocity u varying uniformly from zero at the stationary plate to U at the upper plate.
- Experiments: other quantities being held constant, F is directly proportional to A and to U and is inversely proportional to thickness t. In equation form

μ is the proportionality factor and includes the effect of the particular fluid

If τ = F/A for the shear stress,

- The ratio U/t: angular velocity of line ab, or it is the rate of angular deformation of the fluid (rate of decrease of angle bad)
- The angular velocity may also be written du/dy – more general
- The velocity gradient du/dy may also be visualized as the rate at which one layer moves relative to an adjacent layer, in differential form,

(1.1.1)

- Newton's law of viscosity

- proportionality factor μ: viscosity of the fluid

Materials other than fluids cannot satisfy the definition of a fluid

- A plastic substance will deform a certain amount proportional to the force, but not continuously when the stress applied is below its yield shear stress.
- A complete vacuum between the plates would cause deformation at an ever-increasing rate.
- If sand were placed between the two plates, Coulomb friction would require a finite force to cause a continuous motion.
- plastics and solids are excluded from the classification of fluids.

Fluids:

- Newtonian
- non-Newtonian
- Newtonian fluid: linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq. (1.1.1)] (Fig. 1.2)
- Non-Newtonian fluid: nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation
- An ideal plastic has a definite yield stress and a constant linear relation of τ to du/dy.
- A thixotropic substance, such as printer's ink, has a viscosity that is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest.
- Gases and thin liquids tend to be Newtonian fluids, while thick, long-chained hydrocarbons may be non-Newtonian.

For purposes of analysis, the assumption is frequently made that a fluid is nonviscous

- With zero viscosity the shear stress is always zero, regardless of the motion of the fluid.
- If the fluid is also considered to be incompressible, it is then called an ideal fluid and plots as the ordinate in Fig. 1.2.

1.2 FORCE, MASS, LENGTH, AND TIME UNITS

- Force, mass, length, and time: consistent units
- greatly simplify problem solutions in mechanics
- derivations may be carried out without reference to any particular consistent system
- A system of mechanics units: consistentwhen unit force causes unit mass to undergo unit acceleration
- The International System (SI)
- newton (N) as unit or force
- kilogram (kg) as unit of mass
- metre (m) as unit of length
- the second (s) as unit of time

With the kilogram, metre, and second as defined units, the newton is derived to exactly satisfy Newton's second law of motion

(1.2.1)

- The force exerted on a body by gravitation is called the force of gravity or the gravity force. The mass m of a body does not change with location; the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g:

(1.2.2)

- For example, where g = 9.876 m/s2, a body with gravity force of 10 N has a mass m = 10/9.806 kg. At the location where g = 9.7 m/s2, the force of gravity is
- Standard gravity is 9.806 m/s2. Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg.

1.3 VISCOSITY

- Viscosity requires the greatest consideration in the study of fluid flow.
- Viscosity is that property of a fluid by virtue of which it offers resistance to shear.
- Newton's law of viscosity [Eq. (1.1.1)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity.
- Molasses and tar are examples of highly viscous liquids; water and air have very small viscosities.

The viscosity of a gas increases with temperature, but the viscosity of a liquid decreases with temperature – it can be explained by examining the causes of viscosity.

- The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum.
- A liquid, with molecules much more closely spaced than a gas, has cohesive forces much larger than a gas. Cohesion - predominant cause of viscosity in a liquid; and since cohesion decreases with temperature, the viscosity does likewise.
- A gas, on the other hand, has very small cohesive forces. Most of its resistance to shear stress is the result of the transfer of molecular momentum.

Fig.1.3: rough model of the way in which momentum transfer gives rise to an apparent shear stress, considering two idealized railroad cars loaded with sponges and on parallel tracks

- Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track. First, consider A stationary and B moving to the right, with the water from its nozzles striking A and being absorbed by the sponges. Car A will be set in motion owing to the component of the momentum of the jets which is parallel to the tracks, giving rise to an apparent shear stress between A and B. Now if A is pumping water back into B at the same rate, its action tends to slow down B and equal and opposite apparent shear forces result. When both A and B are stationary or have the same velocity, the pumping does not exert an apparent shear stress on either car.

Model illustrating transfer of momentum.

- Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it. When one layer moves relative to an adjacent layer, the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig. 1.3. The measure of the motion of one layer relative to an adjacent layer is du/dy.

Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces, and since molecular activity increases with temperature, the viscosity of a gas also increases with temperature.

- For ordinary pressures viscosity is independent of pressure and depends upon temperature only. For very great pressures, gases and most liquids have shown erratic variations of viscosity with pressure.
- A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up, regardless of the viscosity, because du/dy is zero throughout the fluid
- fluid statistics - no shear forces considered, and the only stresses remaining are normal stresses, or pressures greatly simplifies the study of fluid statics, since any free body of fluid can have only gravity forces and normal surface forces acting on it

Dimensions of viscosity: from Newton's law of viscosity – solving for the viscosity μ

and inserting dimensions F, L, T for force, length, and time,

shows that μ has the dimensions FL-2T.

- With the force dimension expressed in terms of mass by use of Newton's second law of motion, F = MLT-2, the dimensions of viscosity may be expressed as ML-1T-1.
- The SI unit of viscosity which is the pascal second (symbol Pa·s) has no name.

Kinematic Viscosity

- μ - absolute viscosity or the dynamic viscosity
- ν - kinematic viscosity (the ratio of viscosity to mass density):

(1.3.1)

- occurs in many applications (e.g., in the dimensionless Reynolds number for motion of a body through a fluid, Vl/ν, in which V is the body velocity and l is a representative linear measure or the body size)

- The dimensions of ν are L2T-1.
- SI unit: 1 m2/s, has no name.
- Viscosity is practically independent of pressure and depends upon temperature only.
- The kinematic viscosity of liquids, and of gases at a given pressure, is substantially a function of temperature.

Example 1.1

A liquid has a viscosity or 0.005 Pa·s and a density or 850 kg/m3. Calculate the kinematic viscosity:

Example 1.2

In Fig. 1.4 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank. The clearance is δ and the viscosity μ. Write a program in BASIC to determine the average energy loss per unit time in the sleeve. D = 0.8 in, L = 8.0 in, δ = 0.001 in, R = 2 ft, r = 0.5 ft, μ = 0.0001 lb s/ft2, and the rotation speed is 1200 rpm.

The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion. The period T is 2π/w, where w = dθ/dt. The sleeve force depends upon the velocity. The force Fi and position xi are found for 2n equal increments of the period. Then by the trapezoidal rule the work done over the half period is found

Using the law of sines to eliminate φ, we get

Figure 1.5 lists the program, in which the variable RR represents the crank radius r.

1.4 CONTINUUM

- In dealing with fluid-flow relations on a mathematical or analytical basis: consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum.
- Example: velocity at a point in space is indefinite in a molecular medium, as it would be zero at all times except when a molecule occupied this exact point, and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood.
- This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point. With n molecules per cubic centimetre, the mean distance between molecules is of the order n-1/3 cm.

Molecular theory, however, must be used to calculate fluid properties (e.g., viscosity) which are associated with molecular motions, but continuum equations can be employed with the results of molecular calculations.

- Rarefied gases (the atmosphere at 80 km above sea level): the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow.
- The flow regime is called gas dynamicsfor very small values of the ratio; the next regime is called slip flow; and for large values of the ratio it is free molecular flow.
- In this text only the gas-dynamics regime is studied.

1.5 DENSITY, SPECIFIC VOLUME, UNIT GRAVITY FORCE, RELATIVE DENSITY, PRESSURE

- The densityρ: its mass per unit volume.
- Density at a point: the mass Δm of fluid in a small volume ΔV surrounding the point

(1.5.1)

- For water at standard pressure (760 mm Hg) and 4oC, ρ = 1000 kg/m3.
- The specific volumevs: the volume occupied by unit mass of fluid

(1.5.2)

The unit gravity force, γ: the force of gravity per unit volume. It changes with location depending upon gravity

(1.5.1)

- Water: γ = 9806 N/m3 at 5oC, at sea level.
- The relative densityS of a substance: the ratio of its mass to the mass of an equal volume of water at standard conditions. (may also be expressed as a ratio or its density to that of water).
- The average pressure : the normal force pushing against a plane area divided by the area.
- The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point.
- If a fluid exerts a pressure against the walls or a container, the container will exert a reaction on the fluid which will be compressive.
- Liquids can sustain very high compressive pressures, but are very weak in tension absolute pressures in this book are never negative (otherwise fluid would be sustaining a tensile stress)
- Units force per area, which is newtons per square metre, called pascals (Pa).
- Absolute pressure : P, gage pressures : p.

1.6 PERFECT GAS

- This treatment: thermodynamic relations and compressible-fluid-flow cases are limited generally to perfect gases (defined in this section)
- The perfect gas : substance that satisfies the perfect-gas-law

(1.6.1)

and that has constant specific heats. P is the absolute pressure; vs is the specific volume; R is the gas constant; T is the absolute temperature.

The perfect gas must be carefully distinguished from the ideal fluid. An ideal fluid frictionless and incompressible. The perfect gas has viscosity and can therefore develop shear stresses, and it is compressible according to Eq. (1.6.1).

- Eq.(1.6.1): the equation of state for a perfect gas; may be written

(1.6.2)

- The units of R can be determined from the equation

Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law. As the pressure increases, the discrepance increases and becomes serious near the critical point.

- The perfect-gas law encompasses both Charles' law and Boyle's law.
- Charles' law: for constant pressure the volume of a given mass of gas varies as its absolute temperature.
- Boyle's law (isothermal law): for constant temperature the density varies directly as the absolute pressure.

The volume v of m mass units of gas is mvs

(1.6.3)

- With being the volume per mole

(1.6.4)

- If n is the number of moles of the gas in volume ϑ

(1.6.5)

- The product MR, called the universal gas constant, has a value depending only upon the units employed

(1.6.6)

- The gas constant R can then be determined from

(1.6.7)

knowledge of relative molecular mass leads to the value of R

The specific heat cv of a gas : number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant.

- The specific heat cp : the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant.
- The specific heat ratio k = cp/cv.
- The intrinsic energy u (dependent upon P, ρ and T) : the energy per unit mass due to molecular spacing and forces.
- The enthalpy h : important property of a gas given by h=u+P/ρ.
- cv and cp : units joule per kilogram per kelvin (J/kg·K)
- 4187 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
- R is related to cv and cp by

Example 1.2

A gas with relative molecular mass of 44 is at a pressure or 0.9 MPa and a temperature of 20oC. Determine its density.

From Eq.(1.6.7),

Then, from Eq.(1.6.2)

1.7 BULK MODULUS OF ELASTICITY

- For most purposes a liquid may be considered as incompressible, but for situations involving either sudden or great changes in pressure, its compressibility becomes important; also when temperature changes are involved, e.g., free convection.
- The compressibility of a liquid is expressed by its bulk modulus of elasticity.
- If the pressure of a unit volume of liquid is increased by dp , it will cause a volume decrease -dV ; the ratio dp/dV is the bulk modulus of elasticity K
- For any volume V

(1.7.1)

- Expressed in units of p. For water at 20oC K = 2.2. GPa.

Example 1.3

A liquid compressed in a cylinder has a volume of 1 L (1000cm3) at 1 MN/m2 and volume of 995 cm3 at 2 MN/n2. What is its bulk modulus of elasticity?

1.8 VAPOR PRESSURE

- Liquids evaporate because or molecules escaping from the liquid surface : vapor molecules exert partial pressure in the space - vapor pressure.
- If the space above the liquid is confined, after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time, and equilibrium exists.
- Depends upon temperature and increases with it. Boiling: when the pressure above a liquid equals the vapor pressure of the liquid
- 20oC water: 2.447 kPa, mercury: 0.173 Pa
- When very low pressures are produced at certain locations in the system, pressures may be equal to or less than the vapor pressure the liquid flashes into vapor - cavitation.

1.9 SURFACE TENSION

Capillarity

- At the interface between a liquid and a gas, or two immiscible liquids, a film or special layer seems to form on the liquid, apparently owing to attraction of liquid molecules below the surface
- The formation or this film may be visualized on the basis of surface energyor work per unit area required to bring the molecules to the surface. The surface tension is then the stretching force required to form the film, obtained by dividing the surface-energy term by unit length of the film in equilibrium.
- The surface tension of water varies from about 0.074 N/m at 20oC to 0.059 N/m at 100oC (Table 1.2)

Table 1.2 Approximate properties of common liquids at 20oC and standard atmospheric pressure

The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet.

- For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body (see Sec. 2.6),
- For the cylindrical liquid jet of radius r, the pipe-tension equation applies:
- Both equations: the pressure becomes large for a very small radius of droplet or cylinder

Capillary attraction is caused by surface tension and by the relative value of adhesion between liquld and solid to cohesion of the liquid.

- A liquid that wets the solid has a greater adhesion than cohesion. Surface tension in this case: causes the liquid to rise within a small vertical tube that is partially immersed in it
- For liquids that do not wet the solid, surface tension tends to depress the meniscus in a small vertical tube. When the contact angle between liquid and solid is known, the capillary rise can be computed for an assumed shape of the meniscus.
- Figure 1.4: the capillary rise for water and mercury in circular glass tubes in air

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