Fluid Mechanics Concepts

Fundamental Concepts in Fluid Mechanics

1. DEFINITION OF FLUID MECHANICS 

Fluid mechanics is that branch of applied mechanics that is concerned with the statics and dynamics of liquids and gases. The analysis of the behaviour of fluids is based upon the fundamental laws of applied mechanics that relate to the conservation of mass, energy and momentum. The subject branches out into sub-disciplines such as aerodynamics, hydraulics, geophysical fluid dynamics and bio-fluid mechanics.

2. FLUIDS 

A fluid is a substance that may flow. That is, the particles making up the fluid continuously change their positions relative to one another. Fluids do not offer any lasting resistance to the displacement of one layer over another when a shear force is applied. This means that if a fluid is at rest, then no shear forces can exist in it, which is different from solids; solids can resist shear forces while at rest. To summarize, if a shear force is applied to a fluid it will cause flow. Recall the example in class when a book was placed between my hands that were previously moving parallel to one another, even in the presence of the fluid, air. The book was somewhat distorted by the shear forces exerted on it by my hands, but eventually adopted a deformed position that resisted the force.

A further difference between solids and fluids is that a solid has a fixed shape whereas a fluid owes its shape at any particular time to that of the vessel containing it.

3. CONTINUUM CONCEPT 

The behaviour of individual molecules comprising a fluid determines the observed properties of the fluid and for an absolutely complete analysis, the fluid should be studied at the molecular scale. The behaviour of any one molecule is highly complex, continuously varying and may indeed be very different from neighbouring molecules at any instant of time. The problems normally encountered by engineers do not require knowledge and prediction of behaviour at the molecular level but on the properties of the fluid mass that may result. Thus the interest is more on the average rather than the individual responses of the molecules comprising the fluid. At a microscopic level, a fluid consists of molecules with a lot of space in between. For our analysis, we do not consider the actual conglomeration of separate molecules, but instead assume that the fluid is a continuum, that is a continuous distribution of matter with no empty space. The sketch below illustrates this. Note that the fluid particle consists of an assembly of molecules each having properties such as pressure, temperature, density etc. However, we are interested in the property of the fluid particle at P and therefore we regard P as being a “smear” of matter (represented as a solid filled circle in the figure) with no space.

Recall the example of a crowd in a stadium given in class.

4. DIMENSIONS AND UNITS
Physical quantities require quantitative descriptions when solving engineering problems. Density, which is one such physical quantity, is a measure of the mass contained in unit volume. Density, however, does not represent a fundamental magnitude. There are nine quantities considered to be fundamental magnitudes, and they are: length, mass, time, temperature, amount of a substance, electric current, luminous intensity, plane angle, and solid angle. The magnitudes of all the quantities can be expressed in terms of the fundamental magnitudes.

To give the magnitude of a quantity a numerical value, a set of units must be selected. Two primary systems of units are commonly used in Fluid Mechanics, namely, the Imperial System (sometimes called the English units) and the International System, which is referred to as SI (Systeme International) units.

The fundamental magnitudes and their units and the factors for conversion from the English unit system to the SI are shown in the two tables on the following pages.

Refer also to the discussion in class on the Bernoulli Equation.

5. FLUID PROPERTIES

i. Density Density is the ratio of the mass of a given amount of the substance to the volume it occupies.

Mean density is defined as the ratio of a given amount of a substance to the volume that this amount occupies. The density is said to be uniform if the mean density in all parts of the substance is the same. ( See sketch in class)

Table 1. Fundamental Magnitudes and Their Units Table 2. Derived Magnitudes

Quantity	Magnitude	SI unit	English unit
Area A	L2	m2	ft2
Volume V	L3	M3; L (litre)	ft3
Veloctiy v	LT-1	m/s	ft/sec
Acceleration a	LT-2	m/s2	ft/sec2
Angular velocity	T-1	s-1	sec-1
Force F	MLT-2	kg m/s2; (newton)	N slug-ft/sec2 lb (pound)
Density ñ	ML-3	kg/m3	slug/ft3
Specific weight ã	ML-2T-2	N/m3	lb/ft3
Frequency f	T-1	s-1	sec-1
Pressure p	ML-1T-2	Pa (pascal) N/m2	lb/ft2
Stress ô	ML-1T-2	N/m2	lb/ft2
Surface tension	MT-2	N/m	lb/ft
Work W	ML2T-2	J (joule) N m	ft-lb
Energy E	ML2T2	J (joule) N m	ft-lb
Heat rate Q&	ML2T-3	J/s	Btu/sec
Torque T	ML2T-2	N m	ft-lb
Power P	ML2T-3	J/s W (watt)	ft-lb/sec
Viscosity ì	ML-1T-1	N s/m2	lb-sec/ft2
Mass flux m&	MT-1	kg/s	Slug/sec
Flow rate Q	L3T-1	m3/s	ft3/sec
Specific heat c	L2T2È-1	J/(kg K)	Btu/slug-°R
Conductivity K	MLT-3È-1	W/(m K)	lb-sec-°R

Density at a point is the limit to which the mean density tends as the volume considered is indefinitely reduced. Expressed mathematically, it is:

limV →ε m

V

where ε is taken as the minimum volume of a fluid particle below which the continuum assumption fails.

This is illustrated in the sketch below (as completed in class).

ρ

V

ii. Compressibility The degree of compressibility of a substance is characterized by the bulk modulus of elasticity, K, defined as:

δp

K =−

δV

V

where δp represents the small increased in pressure applied to the substance that causes a decrease of the volume by δV from its original volume of V.

Note the negative sign in the definition to ensure that the value of K is always positive.

K has the same dimensional formula as pressure, which is: [ML-1T-2]

K can also be expressed as a function of the accompanying change in density caused by the pressure increase. Using the definition of density as mass/volume, it can be shown that:

The reciprocal of the bulk modulus is compressibility.

Note that the value of K depends on the relation between pressure and density under which the compression occurs. The isothermal bulk modulus is the value when compression occurs while the temperature is held constant. The isentropic bulk modulus is the value when compression occurs under adiabatic conditions.

For liquids, K is very high (2.05 GPa for water at moderate pressure) and so there is very little change of density with pressure. For this reason, the density of liquids can be assumed to be constant without any serious loss in accuracy.

On the other hand, gases are very compressible.

iii. Surface Tension Surface tension is the surface force that develops at the interface between two immiscible liquids or between liquid and gas or at the interface between a liquid and a solid surface. Because of surface tension, small water droplets, gas bubbles and drops of mercury tend to maintain spherical shapes.

The presence of surface tension and its dynamics are due to complex interactions at the molecular level along interfaces. Away from interfaces, molecules are surrounded by like molecules on all sides and so intermolecular force interactions result in a zero net force. At interfaces, molecules interact with molecules of the same fluid on only one side. The molecules at the interfaces experience a net force that puts the interface under tension.

The ultimate magnitude and direction of this tension force is determined not only by what happens on either side of the interface, but by the way molecules of the two fluids interact with each other. Surface tension, therefore, is specific to the participating fluids. Surface tension forces are also sensitive to the physical and chemical condition of the solid surface in contact, such as its roughness, cleanliness, or temperature.

If a line is imagined drawn in a liquid surface, then the liquid on one side of the line pulls that on the other side. The magnitude of surface tension is defined as that of the tensile force acting across and perpendicular to a short, straight element of the line drawn in the surface divided by the length of that line.

Dimensional Formula: [MLT-2]/[L] = [MT-2]

A common symbol for surface tension is σ.

The forces of attraction binding molecules to one another give rise to cohesion, the tendency of the liquid to remain as one assemblage of particles rather than to behave as a gas and fill the entire space within which it is confined. On the other hand, forces between the molecules of a fluid and the molecules of a solid boundary give rise to adhesion between the fluid and the boundary. It is the interplay of these two forces that determine whether the liquid will “wet” the solid surface of the container. If the adhesive forces are greater than the cohesive forces, then the liquid will wet the surface; if vice versa, then the liquid will not. It is rare that the attraction between molecules of the liquid exactly equals that between molecules of the liquid and molecules of the solid and so the liquid surface near the boundary becomes curved.

For a curved surface, the resultant surface tension forces is towards the concave side. For equilibrium, the pressure on the concave side must be greater than that on the convex side by an amount equal to

⎛ 11 ⎞

σ⎜⎜+ ⎟⎟

RR

⎝ 12 ⎠ where, R1 and R2 are the surface radii of curvature in two perpendicular directions.

The capillarity phenomenon is due to the rise or depression of the meniscus of the liquid due to the action of surface tension forces.

The water column in the sketch below rises to a height h such that the weight of the column is balanced by the resultant surface tension forces acting at θ to the vertical at the contact with the tube.

Hydraulic Systems

The Basic Concept of a Hydraulic System

Hydraulic power transmission is a drive or transmission system that uses hydraulic fluid under pressure to drive machinery. The system is concerned with the generation, modulation, and control of pressure and flow, and in general such systems include:

·         Pumps which convert available power from the prime mover to hydraulic power at the actuator.

·         Valves which control the direction of pump-flow, the level of power produced, and the amount of fluid flow to the actuators. The power level is determined by controlling both the flow and pressure level.

·         Actuators which convert hydraulic power to usable mechanical power output at the point required.

·         The medium, which is a liquid, provides rigid transmission and control as well as lubrication of components, sealing in valves, and cooling of the system.

·         Connectors which link the various system components provide power conductors for the fluid under pressure, and fluid flow return to tank (reservoir).

·         Fluid storage and conditioning equipment which ensure sufficient quality and quantity as well as cooling of the fluid.

Advantages of hydraulic power transmission

Some of the major advantages of hydraulic power transmission are as follows:

·         Great efficiency and economy due to low friction losses and high system reliability (efficiency is approx. 70 to 80 percent)

·         Freedom of location of input and output power converters such as prime movers, pumps, and actuators.

·         Safety and overload protection by means of relief valves.

·         Emergency power stored in an accumulator.

·         Infinitely variable control of output force, output torque, output speed, and actuator position.

·         Extremely high output forces and force multiplication by means of the "hydraulic lever".

·         Low inertia and ease of shock absorption during actuator motion, reversal, start, and stop.

·         Hydraulic systems are self-lubricating and power can be diverted to alternative actuators.

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Milan Tomic

Hi. I’m Designer of Blog Magic. I’m CEO/Founder of Stuftech. I’m Creative App Developer, Web Designer, UI/UX Designer, Interaction Designer, Industrial Designer, Web Developer, Business Enthusiast, StartUp Enthusiast, Speaker, Writer and Photographer. Inspired to make things looks better.

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