Fluid Concept of Fluid Density Pressure: Pressure in a Fluid. Pascal´s principle Buoyancy. Archimede´s Principle. Forces on submerged surfaces Fluids in.

Fluid• Concept of Fluid

• Density

• Pressure: Pressure in a Fluid. Pascal´s principle

• Buoyancy. Archimede´s Principle.

• Forces on submerged surfaces

Fluids in motion

• Continuity equation.

• Bernoulli´s Equation

• Viscous Flow. Viscosity. Poiseuille´s Law. Skin drag

• turbulent flow. References: Tipler; wikipedia,…

Fluids in Motion . Description of the Fluid Flow

FLUID. Fluids in Motion

A fluid flow may be described in two different ways: the Lagrangian approach (named after the French mathematician Joseph Louis Lagrange), and the Eulerian approach (named after Leonhard Euler, a famous Swiss mathematician). In the Lagrangian approach, one particle is chosen and is followed as it moves through space with time. The line traced out by that one particle is called a particle pathline.

A Eulerian approach is used to obtain a clearer idea of the flow at one particular instant. The entire flow field is easily visualized. The lines comprising this flow field are called streamlines. A streamline is a line drawn through a fluid in such a manner that it has the direction of the fluid velocity at every point.

Thus, a pathline refers to the trace of a single particle in time and space whereas a streamline presents the line of motion of many particles at a fixed time. Both coincide in a steady flow.

m, V,ρ, v, .. streamlineStream tube

path line

Fluids in Motion . Description of the Fluid Flow

FLUID. Fluids in Motion

Terminology: Compressible vs incompressible flow Viscous vs No-Viscous flow Steady vs unsteady flow Laminar vs turbulent flow Newtonian vs non-Newtonian fluids

Volume flow rate… Mass flow rate… through a cross-section

Cross-sectional Area A1

Cross-sectional Area A2

flowing from left to right, inside of a stream tube ….

What is the volume flow rate going in [leaving from] the tube? What is the volume flow rate that travels through the tube?

FLUID. Basic Equations of Fluids in Motion. Continuity Equation

Continuity equation

1v2v

A1

A2

The mass flow rate going in the tube through the cross section A1 : ρ1 A1 v1

The mass flow rate leaving from the tube through the cross-section A2 ρ2 A2 v2

For a steady flow, from the general law of conservation of mass,

v1; v2 : Average velocity over the cross-sections A1; A2. Then, no variation in velocity in the cross-section will be considered

222111 AvAv Mass flow rate has to be the same through any cross-section along the stream tube

For the incompressible flow case

21

2211 AvAv Volume flow rate [discharge rate] passing any cross-section along the stream tube will be the same

Blood flows in an aorta of radius 1 cm at 30 cm/s. What is the volume flow rate?. What is the pumping rate of the heart?. Give this result in liters per minute. If the aorta reduce its radius to 0,5 cm because of thickening of the arterial walls (arteriosclerosis), what is the speed of the blood in the narrower region?.

FLUID. Fluids in Motion. Steady and Incompressible flow

Compressible vs incompressible flowAll fluids are compressible to some extent, that is changes in pressure or temperature will result in changes in density. However, in many situations the changes in pressure and temperature are sufficiently small that the changes in density are negligible. In this case the flow can be modeled as an incompressible flow. In a incompressible flow the density is constant throughout the fluid. Per example, to assume it for liquids in most situations is an excellent approximation

Steady vs unsteady flowWhen all the magnitudes describing the flow in one point of space no vary with time, the flow is considered to be steady. Otherwise, it is called unsteady.

FLUID. Fluids in Motion. Viscous and non viscous flow

Viscous vs inviscid flowViscous problems are those in which fluid friction has significant effects on the fluid motion. Non viscous flow implies no dissipation of mechanical energy. For fluids having relatively small viscosity- water, air,..-, the effect of internal friction is appreciable only in a narrow region surrounding the fluid boundaries –boundary-layer hypothesis- The no-slip condition for viscous fluid states that at a solid boundary, the fluid will have zero velocity relative to the boundary. The fluid velocity at all liquid-solid boundaries is equal to that of the solid boundary. Conceptually, one can think of the outermost molecules of fluid stick to the surfaces past which it flows.

Drag: A object submerged in a flowing fluid is subjected to a fluid force component on it in the direction of the approach velocity, called the drag. The drag force is caused by skin friction drag, which is caused by shear stress component (viscous effect) in the flow direction, and by form, or profile, drag, caused by the lower pressure on the downstream side of the object.

FLUID. Fluids in Motion. Turbulent flow

Newtonian vs non-Newtonian fluidsSir Isaac Newton showed how stress and the rate of strain are very close to linearly related for many familiar fluids, such as water and air. These Newtonian fluids are modeled by a coefficient called viscosity, which depends on the specific fluid.However, some of the other materials, such as emulsions and slurries and some visco-elastic materials (eg. blood, some polymers), have more complicated non-Newtonian stress-strain behaviours. These materials include sticky liquids such as latex, honey, and lubricants which are studied in the sub-discipline of rheology.

Laminar vs turbulent flowTurbulence is flow dominated by recirculation, eddies, and apparent randomness. Flow in which turbulence is not exhibited is called laminar.

FLUID. Basic Equations of Fluid in Motion: Bernoulli´s Equation

Bernoulli´s EquationConsider a fluid flowing in a tube as shown in the figure in a steady, incompressible, no viscous flow. We apply the work-energy theorem to a sample of fluid initially contained between point 1 and 2. During time Δt this sample moves along the tube to the region between points 1´and 2´. So,

Wall forces = ΔK [Change of kinetic Energy]

Wall forces include gravitational and pressure forces. We neglecting internal frictional forces. Non-viscous flow. No mechanical dissipation energy is considered.

Work of gravitational forces can be computed as the variation of potential energy of sample. Considering the continuity equation we can obtain

Change of kinetic energy of sample will be

Fluid moving in a pipe that varies in both height and cross-sectional area. The net effect on the sample during a time Δt is that a mass initially at height h1 and speed v1 is transferred to a height h2 with speed v2

)()()( 2121 hhVghhgmU

)())(( 21

222

121

222

1 vvVvvmK

FLUID. Basic Equations of Fluid in Motion: Bernoulli´s Equation

The work done by the pressure forces exerted by the fluid behind the sample and by the fluid in front of the sample will be

Applying the work-energy theorem we can obtain

)()()( 21

222

11221 vvVhhgVVPP

VPtvAPW

VPtvAPW

F

F

2222

1111

2

1

Units: Joule (energy)

If we divide both sides by ΔV

)()()( 21

222

11221 vvhhgPP

Units: Joule (energy) per unit of volume

We can rearrange the terms

constvhgPorvhgPvhgP 22

1222

122

212

111

The Bernoulli Equation states that the energy is the same at any two points along a streamline in steady, incompressible and frictionless flow.

FLUID. Basic Equations of Fluid in Motion: Bernoulli Equation

constvhgP

vhgPvhgP

2

2

1

222

122

212

111

Remarks about Bernoulli Equation This combination of quantities has the

same value at any point along of the stream tube.

The Bernoulli equation aids in solving problem in which the losses due to internal friction (viscous flow) can be neglected or corrected for application of an experimentally determined coefficient.

Bernoulli´s equation can also be obtained by integrating the Euler´s equation, which is derived applying directly the Newton´s laws to a particle of the fluid

g

v

h

g

P

constg

vh

g

P

2

2

22

2

Flow work or pressure energy; Is the portion of the potential energy term that the fluid is capable of yielding of its sustained pressure. IS Unit: Joule per Newton (meter). Dimension : Length

(1)

(2)

Potential energy, due to the gravitational field. IS Unit: meter

Each term having the same units (1) [Energy per unit of volume] (2) [Energy per unit of weight]. Expression (1) can also be considered as energy per volume flow rate, and (2) as energy per weight flow rate

Kinetic energy. IS Unit: meter (Joule per Newton)

FLUID. Basic Equations of Fluid in Motion: Bernoulli Equation. Applications

hgvb 2

The Bernoulli Equation states that the energy is the same at any two points along a streamline in steady, incompressible and frictionless flow.

Torricelli´s Law In the case of a tank of liquid having a nozzle on the side at a distance Δh below the surface of water, we can consider that a streamline connects points a and b, and applying Bernoulli

equation to each point

Torricelli´s law: The water emerges from the nozzle with a speed equal it would have if it dropped in free-fall distance Δh

222

122

212

111 vhgPvhgP

22

122

1bbbaaa vhgPvhgP

Considering same pressure in each point (free atmosphere) and v1, velocity at the surface, almost nil (steady flow)

Exercise. A large reservoir of water, open at the top has a hole of diameter 10 cm, placed a distance of 10 m below of surface of the water. (a) Calculate the speed of the water as it flows out of the nozzle. (b) What is the emerging volume flow rate? © What is the emerging mass flow rate. If the emerging water moves a turbine (d) What is the maximum power can we obtain if the efficiency of turbine were 100%.

FLUID. Basic Equations of Fluid in Motion: Bernoulli Equation. Applications

The Bernoulli Equation states that the energy is the same at any two points along a streamline in steady, incompressible and frictionless flow.

The venturi meter The venturi meter shown in the figure is used to measure the rate of flow of a fluid in a close conduit (pipe). It consists of a converging portion from 1, to a throat section at 2. The Bernoulli equation for point 1 and 2 can be written as above, and considering that h1 is equal to h2, we can obtain

222

122

212

111 vhgPvhgP

1 2

222

12

212

11 vPvP

By use of continuity equation

2211 vAvA

When the speed of a fluid increase, the pressure drops (in a horizontal close conduit). This effect is called “Venturi effect”

The speed in the throat section increase, then the pressure at section 2 drops.

The discharge rate, Q, (volume flow rate), will be )(

)(222

21

21212211 AA

PPAAvAvAQ

To obtain the discharge rate we need to measure the pressure difference between the section 1 and the section 2

FLUID. Basic Equations of Fluid in Motion: Bernoulli Equation. Applications

The Venturi meter

)(

)(222

21

21212211 AA

PPAAvAvAQ

Measuring the pressure difference P1-P2

1.-Inserting a vertical pipe open at the top in each section we can measure the pressure at section 1 and at section 2.

Applying hydrostatic equation, since the column is at rest

1 2h1h2

22

11

hgPP

hgPP

atm

atm

hghhgPP )( 2121

2.- We can insert a U-tube manometer partially filled with a liquid of density ρL, connecting the section 1 and section 2, as shown in figure.

In this case the difference of pressure will be determined from the difference of height at the manometer

hgPP L 21

FLUID. Basic Equations of Fluid in Motion: Bernoulli Equation

21

222

122

212

111

lossesvhgPvhgP

Energy equation

The Bernoulli equation aids in solving problem in which the losses due to internal friction (viscous flow) can be accounted experimentally by a determined coefficient. So, we can write

The available energy at 1 is equal to the available energy at 2, plus all the losses between the two sections.

Pumps, Turbines If a pump were in action between the sections 1 and 2, its total dynamic head would appear in the loss term as a negative quantity. Similarly a turbine would be treated as a loss.

So, having in account that Power/Volume flow rate = Energy per unit of volume

Energy per unit of volume

Q

PowervhgPvhgP 222

122

212

111

FLUID. Basic Equations of Fluid in Motion: Bernoulli Equation. Applications

siphon

A siphon is a device for transferring a liquid from one container to another . The tube shown in figure must be filled to start the siphon, but once this has been done, fluid will flow through the tube. (a) Using Bernoulli´s equation, show that the speed of water in the tube is v = √(2gd). (b) What is the pressure at the highest part of the tube?. (c) When the flow in the siphon will stop?

Pitot-static tubeFigure shows a Pitot-static tube, a device used for measuring the speed o a gas. The inner pipe faces the incoming fluid, while the ring of holes in the outer tube is parallel to the gas flow. Show that the speed of the gas is given by v2= 2gh(ρL-ρg)/ ρg ρL

density of the liquid used in the manometer ρg

density of the gas

FLUID. Basic Equations of Fluid in Motion: Bernoulli Equation. Applications

Water flows through the pipe in figure and exits to the atmosphere at the right end of section C. The diameter of the pipe is 2.0 cm at A; 1.00 cm at B and 0.800 cm at C. The gauge pressure in the pipe at the center of section A is 1.22 atm and the flow rate is 0.8 liter/s. The vertical pipes are open to the air at the top. Find the level of the liquid-air interfaces in the two vertical pipes. Assume laminar non-viscous flow.

FLUID. Basic Equations of Fluid in Motion: Bernoulli Equation. Applications.

Airplane in flight

http://www.inta.es/descubreAprende/Hechos/Hechos05.htm

(a) Field velocity of the flowing air in a wing (b) Field pressure

With the airplane in flight, an air flow passes over the wing, and the shape of the wing's cross section--curved above, flat or nearly flat below--reduces the pressure on top, causing the extra pressure from below to exert a lifting force ("lift"). The lift increases if the front of the wing is raised slightly, biting into the moving air at a small angle ("angle of attack"), and for a given lifting force, this kind of wing produces much less air resistance ("drag") than a flat kite.

FLUID. Basic Equations of Fluid in Motion: Bernoulli Equation. Applications

Atomizer

FLUID. Basic Equations of Fluid in Motion: Viscous flow. ViscosityShear Stress Fs/A Shear Strain

∆X/L

Shear stress on a solid is proportional to shear strain

Boundary layer

dz

dv

A

Fz

v

A

F

swayrigourousain

s

In a laminar flow, the coefficient of (dynamic) viscosity is the proportionally constant between the shear stress and the relative velocity of deformation.

IS Units: Pascal . Second; 1 Pa. s = 10 poise

Velocity profile. Turbulent flow

For fluids having relatively small

viscosity- water, air,..-, the effect of internal friction is

appreciable only in a narrow region

surrounding the fluid boundaries

Internal friction between layer is explained as a consequence of molecular agitation (case of laminar flow). There is a constant interchange of molecules between adjacent horizontal layers. F Adjacent layer

interchange “parcels” of fluid (eddies)

FLUID. Basic Equations of Fluid in Motion: Viscous flow. Viscosity

z

v

A

Fs

Typically the coefficient of viscosity of a liquid increase as the temperature decrease. In the case of gases the variation is in opposite way.

Exercises

Calculate the kinematic viscosity of the water at 20 ºC.

The viscosity of olive oil is 80 cP -10-2 Poise- at 20ºC and its specific gravity at the same temperature is 0.915. Express the viscosity of olive oil in Pa.s and calculate its kinematic viscosity.

Calculate the shear stress required to maintain a relative speed of deformation of 1 m/s through a distance of z = 1 cm (a) in water; (b) in engine oil

Kinematic viscosity

FLUID. Basic Equations of Fluid in Motion: Viscous Flow. Poiseuille´s Law

Horizontal pipe, steady flow, constant cross section, laminar flow

P1

According Bernoulli´s equation the pressure along the pipe would have to be constant. In practice, we observe a pressure drop ΔP = P1-P2.

ΔP = P1-P2 = Q R

R: resistance to flow; it depends on the length L, the radius r, and the viscosity of the fluid. Q: volume flow rate

P2

As a result of viscous forces, the velocity of the fluid is not constant across the diameter of the pipe The velocity is a maximum at the centre line and varies (parabolically) to zero at the wall (non slip condition). 4

4

8

8

r

LR

Qr

LP

Poiseuille´s Law

Laminar, steady flow, horizontal pipe

FLUID. Basic Equations of Fluid in Motion: Turbulence. Reynolds number

Exercise: Crude oil has a viscosity of about 0.200 Pa.s at normal temperature. You are the engineering in charge of constructing a 50.0 km horizontal pipeline to deliver oil at a rate of 500 liter/s. The flow through the pipeline must be laminar. Assuming that the density of crude oil is 700 kg/m3 (a) estimate the diameter of pipeline that should be used. (b) What the power of the pumps should be to maintain constant the flow ?

Turbulence: Reynolds number When the flow speed becomes sufficiently great, laminar flow breaks down and turbulence sets in. Turbulence is flow dominated by recirculation, eddies, and apparent randomness. Turbulence implies a increasing of internal friction because adjacent layer interchange “parcel” of fluid by eddies and so, a greater pressure drop per unit of length. The flow (laminar or tubulent) can be characterized by a dimensionless number called the Reynolds number , Re, which is defined by

vr2

Re In the case of flow through a pipe: Re < 2000 Laminar Re > 3000 Turbulentr radius of the pipe; ρ Density;

v: speed (average across the section); η viscosity

Water flows through a pipe of diameter D =1 inch. It is required maintain laminar flow. What the maximum volume flow rate will be?. Calculate the average velocity in a cross section. 1 inch = 2.54 cm

FLUID. Basic Equations of Fluid in Motion:

Linear momentum equation. Not considered in this text

Olive oil flows through a horizontal pipe of radius r = 10 cm in a laminar, steady flow. The volume flow rate is 20 liter per second. (a) What is the average velocity in a cross-section?. (b) What is the pressure drop over a length L = 10 m?. Estimate the maximum volume flow rate that is possible carry out in the pipe maintaining the flow as laminar. In this last case, estimate the pressure drop over a length L= 10 m. Physical properties of the olive oil at 20º: Viscosity: 80 cP; density: 915 kg/m3.

L= 10m

A firefighter holds a hose with a bend in it as in Figure. Water is expelled from the hose in a stream of radius 1.5 cm at a speed of 30 m/s. (a) What mass of water emerges from the hose in 1 s?. (b) What is the horizontal momentum of this water; (c) Find the force exerted on the water by the hose.

http://www.it.iitb.ac.in/vweb/engr/civil/fluid_mech/course.html

FLUIDS: Summary

http://hyperphysics.phy-astr.gsu.edu/hbase/fluid.html