Basic Laws of Fluid Dynamics

8/10/2019 Basic Laws of Fluid Dynamics

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One of the most important variable of the fluid is velocity . Theposition of a fluid particle is given by the vector position.

is a function of time ( if the particle is in motion) . The timederivative of position gives the speed of particles

= . Calculating

the speed ( x, y, z, t ) of all particles gives the velocity field.


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There are two basic methods for describing the flow. In the first,Euler's method, is observed a fixed area, called control volume,i.e. analyzing the situation in the fixed points of area .


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The second , Lagrange's method follows the motion of individualparticles and analyze how the characteristics of fluids related to theparticle, are changing as a function of time.


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As an illustration of the differences between these two methods canbe used an example in biology, ornithological observing the migrationof birds. Euler method would require setting up observation stations

and measuring the number of birds fly over at a given time, while theLagrange's method meant placing radio transmitters on some birdsand tracking their movements.

Generally, in fluid mechanics is used Euler method description flow ,while the Lagrange's method is used in particular cases . An

important fact is that the basic laws of physics are defined forparticles, and they should be for the purposes of fluid mechanicsconverted into the corresponding field description (condition,distribution) certain physical quantities in space ( part of space -volume control ).


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Material particles, material volume and control volume

The simplest form of universal physical laws relate to the materialparticle that is so small (infinitesimal small),that the velocity v,

density and other immanent properties are uniform inside it.

The basic conservation laws for material particle can be expressed:

- The law of conservation of mass: =

The law of conservation of mass, or principle of mass conservation,states that for any system closed to all transfers of matter

and energy (both of which have mass), the mass of the system mustremain constant over time, as system mass cannot change quantity ifit is not added or removed. Hence, the quantity of mass is"conserved" over time. The law implies that mass can neither becreated nor destroyed, although it may be rearranged in space, or

the entities associated with it may be changed in form.


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- The law of conservation of momentum: =

The lawofmomentum conservationcan be stated as follows. For acollision occurring between object 1 and object 2 in an isolatedsystem, the total momentumof the two objects before the collision isequal to the total momentumof the two objects after the collision.

- The law of conservation of angular momentum: =

Thelaw of conservation of angular momentumstates that when noexternal torque acts on an object or a closed system of objects, nochange of angular momentumcan occur.


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- The law of conservation of energy(first law of thermodynamics):

The law of conservation of energy states that the total energy of anisolated system is constant; energy can be transformed from oneform to another, but cannot be created or destroyed. The first law

is often formulated by stating that the change in the internalenergy of a closed system is equal to the amount of heat supplied tothe system, minus the amount of work done by the system on itssurroundings.

= +

where e is the total energy ( kinetic + internal potential ) per unitmass , Wwork done on the border of material particles and heatQbrought material particle on its border.


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In the previous given expressions for conservation, indicatesinfinitesimally small size, while d/dt refers to time derivative .

It is necessary to clarify that the concept of the law ofconservation is indeed the preservation of quantity just in case ofthe law of conservation of mass, while for the other three laws it ismore appropriate to term them as equilibrium law. A term, the law ofconservation is still being used by pointing to certain analogy to the

law of conservation of mass .

In fluid dynamics as a continuum, it is necessary to generalise thelaws of conservation of the material particles on material volume .Physical volume is the system of an infinite number of infinitely

small of particles and always ( at each point in time ) consists of thesame particles .


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The law of conservation ofmass, expressed in the materialvolume is:

=

The law of conservationof momentum is:

=

The law of conservation ofangular momentum is:

=

The law of conservation ofenergy (i.e. the first law ofthermodynamics)

= +


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Explanation of the material derivatives

Material derivatives also referred as substantial, individual or fullderivation. Operator =

+(.)

is the operator of the material derivative . Application of materialderivative on a scalar of size gives

=

+(.)


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The first member on the right side /t represents the rate ofchange of the quantity at a given point in space and is called localderivation. The second member of the right side () representschange of quantity due to movement of the particles in space and iscalled the convection (advection) derivative . In a certain sensematerial derivative D/dt indicates a time-based change felt by an

observer who moves together with fluid (fluid particle) , while /tindicates the change that an observer would feel if he is stationarywithin control volume .

=

+(.)

In the Cartesian coordinate system material derivate can be written

=

+

+

+


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In all four previous terms of the conservation of the material volumewe used forms

where represents a quantity per unit volume (mass, momentum,angular momentum, energy ) which is the first integrated by thematerial volume MV and then the result derived on material

derivative D/Dt by time . When fluid flow, the material volume movesand deforms and with it the material surface that is unknownfunction in time, until the problem itself has not been resolved .Since the form of the material volume is unsuitable for use inengineering, there is a need to define the system where conservation

laws can be applied. Therefore, we define the notion of controlvolume.


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The relationship between material and control volume

Control volume ( CV ) is arbitrary,theoretically defined volume limited

by control surface (CS) which definesthe dynamic and thermodynamiceffects of fluid. Through controlvolume over time, is passing differentmaterial volumes. The control surface

encloses part of the space and in acertain coordinate system can bestatic or movable, can expand or tocontract, depending on the preferenceof the observer. Control volume and

control surfaces are analogous toEuler description of fluid flow, whilethe concept of material volumecorresponds to Lagrange descriptionof the flow .


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Often is taken that the border of control volume (labeled CS) coincide in partwith fixed limits (inner wall S3) , while others are oriented perpendicular tothe flow direction (A1, A2) in order to simplify the process.


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The transformation of the material in the control volume,Reynolds transport theorem

Reynolds transformation theorem enables the transformation of the

fundamental laws of material volume to control volume. The basicpremise of the transformation is that material volume and a controlvolume coincide at a certain point in time. After a moment ofcoincidence of two volumes, with passage of time they will notoverlap because the material volume will be dragged away together

with particles that it form, while the control volume is at rest ormove at the discretion of the viewer. For the implementation oftransformation it is of interest only the moment when the twovolumes overlap. Written mathematical Reynolds transformation is:

=

+

.

where represents a general quantity


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The law of conservation of mass

By applying the Reynolds transport theorem, from formulation of thematerial volume, it comes to the formulation of the conservation ofmass in the control volume, related to the Euler fluid flow description.General quantity is replaced by the density i.e. = , it follows:

=

+

.

+

. = 0

=

As the physical interpretation, it could be said that the increasein mass within the control volume plus the mass outflow from thecontrol volume through its border is equal to zero at all times.


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Solving previous integral calculusand assume for above pipe that and are average values throughthe cross sectional area:

111 = 222 = The term represents mass flow [kg/s] . By introducing a newvariable flow rate or less flow:

= [3 ]

previous relation can be written: 11 = 22 =

or for stationary flow with constant density of the fluid inside the

control volume: = 11 = 22In addition to the concept of mass flow is used weight flow

= = [ ]

Basic Laws of Fluid Dynamics

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