Chapter 9

Solids and Fluids (c)

Lead has a greater density than iron, and both are denser than water. Is the buoyant force on a solid lead object (a) greater than, (b) less than, or (c) equal to the buoyant force on a solid iron object of the same dimensions?

Quiz 15 (QUICK QUIZ 9.6)

Fluids in Motion: Streamline FlowStreamline flow: every particle that passes a particular point moves exactly along the smooth path followed by particles that passed the point earlier. Also called laminar flow;

Different streamlines do not cross each other;

The streamline at any point coincides with the direction of fluid velocity at that point.

Fluids in Motion: Turbulent and Viscocity

Turbulence : the flow becomes irregular when It exceeds a certain velocityThere are any conditions that causes abrupt

changes in velocity

Eddy currents are a characteristic of turbulent flow

Viscosity: is the degree of internal friction in the fluid;The internal friction is associated with the resistance between two adjacent layers of the fluid moving relative to each other

Ideal Fluid (the main focus of our lectures)

The fluid is nonviscousThere is no internal friction between adjacent

layersThe fluid is incompressible

Its density is constantThe flow is in steady state

Its velocity, density and pressure do not change in time

The flow is without turbulenceNo eddy currents are present

Equation of Continuity

Thus, the speed is high where the pipe is narrow and speed is low where the pipe has a large diameterAv is called the flow rate

The fluid is taken to be incompressible;

The amount of liquid is conserved: what goes in at one end must come out the other end (per unit time).

These considerations imply: A1v1 = A2v2

Bernoulli’s Equation

• Relates pressure to fluid speed and elevation• Bernoulli’s equation is a consequence of the

work-energy relation, applied to an ideal fluid

constant gyv2

1P 2

Its physical content is: the sum of the pressure, kinetic energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline.

How can we see that this is true? …… --->

Bernoulli’s Equation: derivationPhysical basis: Work-energy relation

All together now:

WithWe get:

Applications of Bernoulli’s Principle: Venturi Meter

Shows fluid flowing through a horizontal constricted pipeSpeed changes as diameter changesCan be used to measure the speed of the fluid flowSwiftly moving fluids exert less pressure than do slowly moving fluidsSwiftly moving fluids exert less pressure than do slowly moving fluids

Example: Venturi Meter (Problem 9.47)

The inside diameters of the larger portions of the horizontal pipe in the figure are 2.50 cm. Water flows to the right at a rate of 1.80 x 10–4 m3/s. Determine the inside diameter of the constriction.

Solution:

1. The velo-city from the left:

4 3

1 221

1.80 10 m s0.367 m s

2.50 10 m 4

flowratev

A

12

Example: Venturi Meter

2. Difference in pressures:

, which yields;

1

2

1 2 1 2 5.00 cmP P g h h g

3. Bernoulli’s principle:

2 2 22 1 1 2 1 1 2

22v v P P v g h h

2 22 0.367 m s 2 9.80 m s 5.00 10 m 1.06 m sv

4. Xsec. area at 2:

4 34 2

22

1.80 10 m s1.71 10 m

1.06 m sflowrate

Av

1.47 cm 4 2

222

4 1.71 10 m41.47 10 m

Ad

5. Diameter:

Surface Tension

Net force on molecule A is zeroPulled equally in all directions

Net force on B is not zeroNo molecules above to act on itPulled toward the center of the fluid

Surface Tension, cont

The net effect of this pull on all the surface molecules is to make the surface of the liquid contractMakes the surface area of the liquid as small as possibleExample: Water droplets take on a spherical

shape since a sphere has the smallest surface area for a given volume

Surface Tension on a Needle

Surface tension allows the needle to float, even though the density of the steel in the needle is much higher than the density of the waterThe needle actually rests in a small depression in the liquid surfaceThe vertical components of the force balance the weight

The surface tension is defined as the ratio of the magnitude of the surface tension force to the length along which the force acts:

SI units are N/mIn terms of energy, any equilibrium configuration of an object is one in which the energy is a minimum

Surface Tension

L

F

Notes About Surface Tension

The surface tension of liquids decreases with increasing temperatureSurface tension can be decreased by adding ingredients called surfactants to a liquid

A Closer Look at the Surface of Liquids

Cohesive forces are forces between like moleculesAdhesive forces are forces between unlike moleculesThe shape of the surface depends upon the relative size of the cohesive and adhesive forces

Liquids in Contact with a Solid Surface – Case 1

The adhesive forces are greater than the cohesive forcesThe liquid clings to the walls of the containerThe liquid “wets” the surface

Liquids in Contact with a Solid Surface – Case 2

Cohesive forces are greater than the adhesive forcesThe liquid curves downwardThe liquid does not “wet” the surface

Angle of Contact

In a, Φ > 90° and cohesive forces are greater than adhesive forcesIn b, Φ < 90° and adhesive forces are greater than cohesive forces

Capillary Action

Capillary action is the result of surface tension and adhesive forcesThe liquid rises in the tube when adhesive forces are greater than cohesive forcesAt the point of contact between the liquid and the solid, the upward forces are as shown in the diagram

Capillary Action, cont.

Here, the cohesive forces are greater than the adhesive forcesThe level of the fluid in the tube will be below the surface of the surrounding fluid

Capillary Action, final

The height at which the fluid is drawn above or depressed below the surface of the surrounding liquid is given by:

cosgr

2h

Viscous Fluid Flow

Viscosity refers to friction between the layersLayers in a viscous fluid have different velocitiesThe velocity is greatest at the centerCohesive forces between the fluid and the walls slow down the fluid on the outside

Coefficient of Viscosity

Assume a fluid between two solid surfacesA force is required to move the upper surface

η is the coefficientSI units are Ns/m2

cgs units are Poise1 Poise = 0.1 Ns/m2

d

AvF

Poiseuille’s Law

Gives the rate of flow of a fluid in a tube with pressure differences

L8

)PP(R

t

V

flowofRate

214

At sufficiently high velocity, a fluid flow can change from streamline to turbulent flowThe onset of turbulence can be found by a

factor called the Reynold’s Number, RN

If RN = 2000 or below, flow is streamlineIf 2000 <RN<3000, the flow is unstableIf RN = 3000 or above, the flow is turbulent

Reynold’s Number

vd

RN

Transport Phenomena

Movement of a fluid may be due to differences in concentrationThe fluid will flow from an area of high concentration to an area of low concentrationThe processes are called diffusion and osmosis

Diffusion and Fick’s Law

Molecules move from a region of high concentration to a region of low concentrationBasic equation for diffusion is given by Fick’s Law

D is the diffusion coefficient

L

CCDA

time

MassrateDiffusion 12

Diffusion

Concentration on the left is higher than on the right of the imaginary barrierMany of the molecules on the left can pass to the right, but few can pass from right to leftThere is a net movement from the higher concentration to the lower concentration

Osmosis

Osmosis is the movement of water from a region where its concentration is high, across a selectively permeable membrane, into a region where its concentration is lowerA selectively permeable membrane is one that

allows passage of some molecules, but not others

Motion Through a Viscous Medium

When an object falls through a fluid, a viscous drag acts on itThe resistive force on a small, spherical object of radius r falling through a viscous fluid is given by Stoke’s Law:

rv6Fr

Motion in a ViscousMedium

As the object falls, three forces act on the objectAs its speed increases, so does the resistive forceAt a particular speed, called the terminal speed, the net force is zero

)(9

gr2v f

2

t