Elements of Hydrodynamics, Physics tutorial

Introduction:

Hydrodynamics is a study of liquids in motion. Specially, it looks at the ways different forces influence movement of liquids. The series of equations describe how conservation laws of mass, energy, and momentum apply to liquids, mainly those which are not compressed.

Hydrodynamics is part of the larger field known as fluid mechanics which studies how energy and forces interrelate with fluids, comprising gases and liquids.

Hydrodynamics is mainly utilized in engineering. Studies focus mainly on flow through pipes, and over different obstacles. This is very helpful information for building structures which attempt to control or divert water flow in the controlled manner.

Concept of Streamlines and velocity:

Streamline is a path of the fluid flowing gradually and without considerable turbulence. The body is said to be streamlined if shape offers least possible resistance to the current of air, water, or other fluid. Current that streamlined body breaks just reunites in wake, as contrasted with retarding eddies and turbulence made by incomplete vacuum in wake of the non-streamlined body.

Knowledge of laws governing fluid flow is very significant in distribution of water, gas and oil in pipelines and for effective transmission of energy in hydraulic machines. Rate of flow of the fluid is estimated by volume of fluid which passes through the cross-section per unit time.

Consider the pipe with some liquid flowing through it. Consider the section S of cross-sectional area A and length (l) moving with the average velocity V. Volume of liquid V moving per second is provided as:

V/t = Al/t...........Eq.1

As volume = area x length

But its average velocity = length/time

Therefore length (l) = velocity (v)

Therefore V/t = Al/t = Avt/t = Av

That is the volume per time is equivalent to area times the average velocity of fluid. When the fluid is at rest, pressure is the same at all points at same elevation of container. But once in motion, pressure is no longer the same. There is a decrease in pressure as fluid flows. Any thorough explanation of fluids in motion is fairly complicated. Flow of fluids can be explained as being the steady flow or stationary flow and turbulent. When it is a steady flow we have streamlines. Absence of streamlines makes flow turbulent.

Velocity of the fluid doesn't change at the given point, as time elapses, but it may be different at different places. When flow is stable, the paths really followed by moving particles of the fluid are known as streamlines. Streamlines of flow represent the directions of the velocities of the particles of the fluid. The flow is uniform or laminar.

Pressure and Velocity:

Consider a fluid at some initial time t lying between two cross-sections. At a time interval Δt, the fluid moves from one section to another through a distance of Δs1

Therefore Δs1 = V1Δt

Where, v1 is the speed at one end of the tube. Similarly, at the same time interval Δt, the fluid moves from c to d through a distance of Δs2

Δs1 = V2Δt

Where, v2 is the velocity of the fluid at the other end of the tube.

Let A1 be the cross-sectional area at a and A2 is the cross-sectional area at c. Change in volume of the fluid = A1Δs1 = A2Δs2. Now, we will determine the work done during this time interval Δt by using the relation,

FORCE(F)/AREA(A) = PRESSURE(P)

F = PA

But the fore at a = P1A1 the force at c = P2A2
net work done = (P1 - P2) X change in volume
Therefore P1 - P2 = Network done/Change in volume
Change in pressure = Network done/Change in volume

Bernoulli's Principle Definition:

This principle generally provides relation between velocity, pressure and height of flowing non viscous fluid in the horizontal flow.

According to it, speed and pressure of the flowing fluid are inversely proportional to each other, i.e., if velocity increases, it will lead to the automatic decrease in pressure of fluid.

Or

Theorem defines that for streamline flow of the ideal liquid, the total energy (sum of pressure energy, potential energy and kinetic energy) per unit mass remains constant at every cross-section, all through the flow.

P + 1/2ρv2 + ρgh =constant

Where p is the pressure, ρ is the density, V is the velocity, h is elevation, and g is the gravitational acceleration.

Applications of Bernoulli's Principle:

Different applications of Bernoulli's principle in terms of:

  • Suction effect of the person at railway station;
  • Venture (filter) pump sometimes known as suction pump;
  • Aerofoil lift in airplanes; and
  • Flow of liquids from wide tanks.

i) At Railway Station

Person standing close to platform at the railway station experiences the suction effect when the fast train passes by. Fast moving air between person and train generates reduction in pressure and excess air pressure on other side of person pushes him or her towards train.

ii) Filter Pump (Venturi Pump)

Filter pump is utilized in many laboratories for fast effective filtration of precipitates in solutions. Pump has the narrow section in middle so that the jet of water from tap flows fast here. This causes the drop in pressure near it and air thus flows in from side tube to which the vessel is joined. Air and water together are excluded through bottom of the filter pump.

iii) Aerofoil Lift

Curved shape of the aerofoil or wings of areoplanes produces fast flow of over its top surface than lower surface. This is described by closeness of streamlines above aerofoil compared with those below. According to Bernoulli's principle, pressure of air below is greater than that above. This generates the lift on aerofoil.

The Change in Kinetic Energy of fluid:

At the initial stage, initial kinetic energy the fluid (KE1) is provided as

KE1 = 1/2 x mass of fluid x (velocity)2 = 1/2m1v12

From the definition of density, the mass can be expressed as

m1 = density x volume = PA1Δs1

Therefore, expression for kinetic energy is as:

KE1 = 1/2P(A1Δs1) x v12

Where, the symbols

P = density of fluid

v1= velocity of fluid at ab

A1= area

Δs1 = displacement

At the end of the time interval the final kinetic energy is where is given as

KE2 = 1/2m2v22 = 1/2P(A2Δs2)xv22

But A2Δs2 = change in volume. These expressions of kinetic energies can be interpreted in terms of change in volume as

KE1 = 1/2P x change in volume x v12

and KE2 = 1/2 P x change in volume x v22

The net change in kinetic energy

ΔKE = KE2 -KE1

= 1/2P x change in volume x(v22 - v12)

Therefore Change in KE/ Change in volume = 1/2P(v22 - v12)

The Change in Potential Energy of the Fluid:

To find out change in potential energy of fluid, initial potential energy of fluid at a, at height y1 is provided as PE1

PE1 = M1gy1

where, M1 = PA1Δs1

∴ PE1 = PA1Δs1y1g

PE1 = Py1 x (change in volume) x g

Similarly, at the other end of the time interval Δt, the potential energy at d, height y2 is given as PE2 where,

PE2 = M2gy2

Where, M2 = PA2Δs2

∴ PE2 = PA2Δs2y2g

PE2 = Py2 x (change in volume) x g

The change in potential energy ΔU is given as

ΔU = Pg(y2 - y1) x change in volume

Change in PE/Change in volume = Pg(y2 - y1)

As the work done by pressure difference per unit volume of the fluid flowing along the pipe is equal to kinetic energy per unit volume plus gain in potential energy per unit volume, we can define that:

W/ΔV = ΔKE/ΔV + ΔU/ΔV

Where, ΔV - change in volume of fluid

Principle of conservation of energy may also be utilized to arrive at this conclusion

Consequently,

P1 - P1 = 1/2P (v22 - v12) + Pg (y2 - y1)

On rearranging the terms in equations

We get ∴ P1 - P1 = 1/2 Pv22 - Pv12 + Pgy2 - Pgy1

 P1 + 1/2 = Pv12 + Pgy1 = + P2 + 1/2 = Pv22 + Pgy2 G

Generally, we can write that

P + 1/2 = Pv2 + Pgy = CONSTANT

Consequently, for streamline motion of the incompressible non-viscous fluid, sum of pressure at any point plus kinetic energy per unit volume plus the potential energy per unit volume is always a constant. Statement given above describes Bernoulli's principle.

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