FLUIDS IN MOTION;CONCEPT OF FLOW AND LINE OF CURRENT.

Flow is defined as a fluid in motion. We will describe the flow of a fluid as a function of certain physical variables such as pressure, density, and velocity at all points in the fluid. We are going to describe the movement of a fluid concentrating on what happens at a certain point in space (x, y, z) at a certain instant of time t. Thus, the density of a flow, for example, will be given by , and the speed of the flow at time t at that same point will be .

Particles within a flow can follow defined paths called “streamlines”. A stream line is a continuous line drawn through a fluid following the direction of the velocity vector at each point. Thus, the velocity vector is tangent to the current line at all points in the flow. There is no flow through a streamline, but along it and indicates the direction the moving fluid takes at each point.

In order to observe the flow of a fluid, different substances , such as shiny particles, dye or smoke, can be injected into it , thus tracking the movement of the particles. The traces left by these substances are called “emission lines”.

A “stream tube” is defined as a portion of the flow formed by all streamlines that cross a certain small area transversely.

 

 

 

TYPES OF FLOW

Let’s see the different types of flows that we can find:

Ø      STATIONARY FLOW -> This type of flow occurs when the variables that characterize it are constant over time. These variables will no longer depend on time, such as the speed which can have a certain constant value at the point (x , y 1 , z 1 ), but could change its value at another point (x 2 , y 2 , z 2 ). Thus it is true that:

A flow is non-stationary if the physical variables that characterize it depend on time at all points of the fluid , then:

Since in a steady flow the velocity at one point is constant over time, all fluid particles reaching a certain point will continue to move along the streamline that passes through that point. Therefore, in this type of flow the path of the particles is the current line itself and there cannot be two current lines that pass through the same point, that is, the current lines cannot cross. In a steady flow the pattern of the current lines is constant in time.

If the flow is not stationary, streamlines can change direction from one instant to the next, so a particle can follow one streamline at one instant and follow a different streamline the next.

Ø      UNIFORM FLOW -> We have this type of flow when the physical variable is the same at all points of the flow. For example, in a uniform flow the speed of all the particles is the same at any instant of time, therefore, the speed will not depend on the position of the fluid particle, although it can vary in time :

When the physical variables vary from point to point, the flow is said to be non-uniform.

Ø      INCOMPRESSIBLE FLOW -> When a fluid flow is compressed, if the density remains constant, the flow is said to be incompressible. Otherwise, the flow is said to be compressible.

Ø      VISCOSE FLOW -> We already know that the viscosity of a fluid is the resistance it presents to tangential stresses. It could be considered the equivalent of friction in the movement of solid bodies. The higher the viscosity in a flow, the greater the external forces must be applied to maintain the flow. When the effect of the viscosity on the flow is negligible, it is considered that we are dealing with a non-viscous flow.

Ø       IRROTATIONAL FLOW ->  When you have a fluid that moves in a circular current, but the fluid particles do not rotate around the axis that passes through its center of mass, the flow is said to be irrotational . Otherwise we are facing a rotational flow.

Ø      LAMINAR FLOW AND TURBULENT FLOW ->  A flow is laminar when its particles move along smooth paths in sheets or layers, so that one layer smoothly glides over another adjacent layer. This type of flow complies with Newton’s Law of Viscosity.

A flow is turbulent when its particles move in very irregular trajectories that cause collisions between the particles, producing a significant amount of movement between them. Turbulence establishes significant shear stresses and causes energy losses throughout the flow.

The action of the viscosity dampens the turbulence in a flow. Therefore, if we have a fluid with low viscosity, high speed and great extension, moving   with a laminar flow, it would very quickly become a turbulent flow.

The laminar or turbulent nature of a flow is indicated by the ” Reynolds number “.

NUMBER OF REYNOLDS

The Reynolds number is the relationship between the inertia present in the flow due to its movement and the viscosity of the fluid.

For a circular pipe of diameter Φ, through which a fluid of density ρ and viscosity η flows, with a speed v, the Reynolds number can be calculated by the expression:

A turbulent flow flowing through a glass tube becomes laminar when the velocity is reduced to a Reynold number equal to 2000. This value is called the “lower critical Reynolds number “. All flows for which , are laminar flows.

In a pipe installation a laminar flow will change to turbulent in the range . Above 4000 the flow is considered turbulent. Experimentally it has been verified that certain very special flows continue to have a laminar behavior with a Reynolds number greater than 12000.

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