Fluid Mechanics
- Fluid mechanics is the branch of physics which involves the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; and fluid dynamics, the study of the effect of forces on fluid motion.
- It is branch of continuum mechanics, a subject which models matter without using the information that it is made out of atoms; that is, it models matter from a macroscopic viewpoint rather than from microscopic.
- Fluid mechanics, especially fluid dynamics, is an active field research with many problems that are partly or wholly unsolved.
- Fluid mechanics can be mathematically complex, and can be solved by numerical methods, typically using computers. A modem discipline, called computational fluid dynamics (CFD), is devoted to this approach to solving fluid mechanics problems.
- Particle image velocimetry, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow.
- The study of fluid mechanics goes back at least to the days of an ancient Greece, when Archimedes investigated fluid statics and buoyancy and formulated his famous law known now as the Archimedes' principle, which was published in his work On Floating Bodies - generally considered to be the first major work on fluid mechanics.
- Fluid statics or hydrostatics is the branch of fluid mechanics that studies fluids at rest. It embraces the study the conditions under which fluids are at rest in stable equilibrium; and is contrasted with fluid dynamics, the study of fluids in motion.
- Hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids. It is also relevant to geophysics and astrophysics to meteorology, to medicine and many other fields.
- Hydrostatics offers physical explanations for many phenomena of everyday life, such as why atmospheric pressure changes with altitude, why wood and oil float on water, and why the surface of water is always flat and horizontal whatever the shape of its container.
- Fluid dynamics is a sub discipline of fluid mechanics that deal with fluid flow-the natural science of fluids in motion.
- It has several sub disciplines itself, including aerodynamics and hydrodynamics. Fluid dynamics has a wide range of applications, including calculating force and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation.
- Some of its principles are even used in traffic engineering, where traffic is treated as a continuous fluid, and crowd dynamics.
- Fluid dynamics offers a systematic structure-which underlies these practical disciplines-that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems.
- The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such as velocity, pressure, density, and temperature, as functions of space and time.
- Like any mathematical model of the real world, fluid mechanics makes some basic assumptions about the materials being studied. These assumptions are turned into equations that must be satisfied if the assumptions are to be held true.
- Fluid mechanics assumes that every fluid obeys the following:
\[-\] Conservation of mass
\[-\] Conservation of energy
\[-\] Conservation of momentum
- Further, it is often useful (at subsonic conditions) to assume a fluid is incompressible - that is, the density of the fluid does not change.
- Similarly, it can sometimes be assumed that the viscosity of the fluid is zero (the fluid is in viscid). Gases can often be assumed to be in viscid. If a fluid is viscous, and its flow contained in some way (e.g. in a pipe), then the flow at the boundary must have zero velocity.
- Fluids are composed of molecules that collide with one; another and solid objects. The continuum assumption, however, considers fluids to be continuous. That is, properties such as density, pressure, temperature, and velocity are taken to be well-defined at "infinitely" small points, defining a REV (Reference Element of Volume), at the geometric order of the distance between two. Adjacent molecules of fluid.
- Properties are assumed to vary continuously from one point to another, and are averaged values in the REV. The fact that the fluid is made up of discrete molecules is ignored.
- The continuum hypothesis is basically an approximation, in the same way planets are approximated by point particles when dealing with celestial mechanics, and therefore results in approximate solutions.
- Consequently, assumption of the continuum hypothesis can lead to results which are not of desired accuracy. However, under the right circumstances, the continuum hypothesis produces extremely accurate results.
- Those problems for which the continuum hypothesis does not allow solutions of desired accuracy are solved using statistical mechanics. To determine whether or not to use conventional fluid dynamics or statistical mechanics, the Knudsen number is evaluated for the problem.
- The Knudsen number is defined as the ratio of the molecular mean free path length to a certain representative physical length scale.
- This length scale could be, for example, the radius of a body in a fluid. Problems with Knudsen numbers at or above one are best evaluated using statistical mechanics for reliable solutions.
- The Navier-Stokes equations are the set of equations that describe the motion of fluid substances such as liquids and gases.
- These equations state that changes in momentum (force) of fluid particles depend only on the external pressure and internal viscous forces (similar to friction) acting on the fluid.
- Thus, the Navier-Stokes equations describe the balance of forces acting at any given region of the fluid.
- The Navier-Stokes equations are differential equations which describe the motion of a fluid. Such equations establish relations among the rates of change of the variables of interest.
- This means that solutions of the Navier-Stokes equations for a given physical problem must be sought with the help of calculus. In practical terms only the simplest cases can be solved exactly in this way.
- These cases generally involve non-turbulent, steady flow (flow does not change with time) in which the Reynolds number is small.
- A wattmeter is essentially an inherent combination of an ammeter and a voltmeter and, therefore, consists of two coils known as current coil (CC) and potential coil (PC).
- Power in ac circuit is given by \[P=VI\,\,\cos \,\phi \] where V and I are rms values of voltage and current respectively and \[f\] is the phase angle between V and I. The fact that the power factor \[\left( \cos \,\phi \right)\] is involved in the expression for power, we cannot measure power in ac circuits by using an ammeter and a voltmeter.
- Since deflecting torque in a dynamometer type wattmeter is directly proportional to true power in both cases dc and ac and the instrument is spring controlled, the scale of the instrument is uniform.
- In order to cause the resultant flux in shunt magnet to lag in phase by exactly \[90{}^\circ \] behind the applied voltage, one or more copper rings, known as copper shading bands, are provided on one limb of the shunt magnet.
- In the two wattmeter method of measurement of power in a 3-phase circuit the readings of two wattmeters depends upon the pf of load. One of the wattmeter gives negative reading when the phase angle f of the load varies from \[60{}^\circ \] to \[90{}^\circ \]
- One of the two wattmeters, in a two wattmeter method of measuring power in a balanced 3-phase circuit, will keep showing zero as the load is varied if and only if the power factor of the load remains constant at 0.5. The reading of other wattmeter will vary with the variation in load.
- Wattmeter reading for inductive loads are given as
\[{{W}_{1}}={{V}_{L}}\,{{I}_{L}}\cos \,\,\left( 30{}^\circ -\phi \right)\]
\[{{W}_{2}}={{V}_{L}}\,{{I}_{L}}\cos \,\,\left( 30{}^\circ +\phi \right)\]
For unity pf load, both wattmeters give equal readings, each equal to half the total power of the circuit.
For pf of load 0.5 (lagging). Wattmeter \[{{W}_{1}}\] will give total power of load and wattmeter \[{{W}_{2}}\] gives zero reading.
For zero pf load. The readings of two wattmeters are equal but of opposite sign.
So wattmeter \[{{W}_{2}}\] gives \[-\,ve\] reading when phase angle \[\phi \] varies from \[60{}^\circ \] to \[90{}^\circ .\] for obtaining the reading of wattmeter \[{{W}_{2}}\] either the connection of current coil or pressure coil should be changed and readings obtained after the reversal of connections should be subtracted from the other wattmeter reading in order to get the total power.
- The essential difference between an energy meter and a wattmeter is that the former is fitted with some type of registration mechanism whereby all the instantaneous readings of power are summed over a definite period of time whereas the latter indicates the value of power at particular instant when it is read.
- The rotating system of the energy meter is made as small as possible so that frictional torque is reduced to the minimum.
- An aluminium disc is usually preferred to a copper disc in order to have resistance per unit weight smaller.
- The number of revolutions made by the aluminium disc for 1 kWh of energy consumption is called the meter constant of an energy meter.
- In some energy meters, the disc continues rotating when the potential coils are excited but with no load current flow. This defect is known as creeping.
- Creeping error can be avoided by either cutting two holes (or slots) in the disc on the opposite sides of the spindle or attaching a small piece of iron wire to the edge of the disc.
- The disc rotates continuously in the series magnetic field under load conditions and so there is a dynamically induced emf in the disc which induces eddy currents. Due to interaction of eddy currents with the field of series magnet a braking torque proportional to the square of the energizing current is developed. The braking torque so developed is called the self-braking torque.
