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**preview**shows pages 1-3. to view the full**55 pages of the document.**CHAPTER 12: STATIC FLUIDS JUNE 17, 2014

Introduction

•Term fluid includes liquids and gases

oAre deformable – can evolve towards a mechanical equilibrium

•Intermolecular forces dominate = fluid is found in liquid state

•Thermal energy of system dominates = fluid found in gaseous state

12.1 Model System: The Ideal Stationary Fluid

•Fluids are systems that yield to any force that attempts to alter their shape, causing

system to flow until it reaches mechanical equilibrium

oFluid conforms to shape of the container

oRefers to liquid and gases (not solids b/c remain unchanged when placed in

containers of different sizes)

•Molecules in a liquid maintain a fixed intermolecular distance

•Liquid put into container with volume larger than volume of liquid, liquid forms a

surface

•Gases have no surface b/c gases adjust intermolecular distance and fill any space

•Fluids in mechanical equilibrium are called stationary

•Properties essential for an ideal stationary fluid

oThe ideal stationary fluid is incompressible

Volume and density (ρ = m/V) are constant and doesn’t depend on P

Does not apply to gases (use ideal gas model)

oThe ideal stationary fluid is deformable under influence of forces and seeks

mechanical equilibrium

Mechanical equilibrium needs to be established for fluid to be

stationary

Applies to both liquids and gases

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12.2 Pressure in an Ideal Stationary Fluid

•Pressure variations in liquids are more profound

12.2.1 Pascal’s Law

•A small rectangular prism of fluid is identified in a beaker with three forces acting on

it:

oTwo contact forces due to remaining fluid on top and below the object (Fup

and Fdown) and weight of fluid element (W)

oElement is at rest in an ideal stationary fluid

FNet,y = 0 = Fup – W – Fdown

•Rectangular prism has horizontal surface area A and height of Δy = yup – ydown

oThis is a small length b/c pressure in the fluid expected to vary in vertical

direction

•Volume is given by V = AΔy

• All three forces are rewritten

oMass is rewritten as m = ρV = ρAΔy

oWeight of fluid element is W = mg = ρgAΔy

•Fluid element stretches from y0 to y0 + Δy

oAt y0 the pressure is labeled p and at y0 + Δy it is labeled p + Δp

oFup = pA

oFdown = (p + Δp)A

•Rewrite Newton’s law in the form of

opA – (p + Δp)A – ρgAΔy = 0

oCombine first two terms and then divide by A, we get

oΔp = -ρgΔy

•This equation applies whether the density is constant or varies with depth

oRestrict further discussion to cases with constant density b/c

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Ideal stationary fluid is incompressible

General case requires calculus

•When ρ = constant, previous equation applies for any depth difference Δy

•Pascal’s Law: p2 – p1 = -ρg(y2 – y1)

•Used when surface of fluid can’t be identified and can’t be used as reference point

•Pascal’s law states that difference b/w pressures at two different positions in a fluid

of constant density is proportional to vertical distance b/w two positions

oProportionality factor is product of the ρ of the fluid and gravitational

acceleration

12.2.2 Pressure in Liquids with a Visible Surface

•In systems with an identifiable surface of the liquid, index 1 in above equation is

referred to the surface of the liquid

•y1 = 0 and p1 = patm

•patm is proper value for pressure of the fluid surface since in mechanical equilibrium

•Force pushing surface upwards equals force caused by air pressure pushing

downwards

•More useful to define depth below the surface as a positive distance (y2 = d)

op = patm + ρgd

•Pascal’s law doesn’t describe pressure variations in the atmosphere

oIllustrated by calculating height of upper end of atmosphere by using y2 = ymax

o(py,max – pground) = -ρg(ymax – yground)

At max height pressure drops to 0 atm b/c ymax is where vacuum of

outer space would begin, yground = 0 m and pground = patm, and ρ = 1.2

kg/m^3

ymax = 8614 m

Therefore, assumption of constant density throughout atmosphere is

inadequate b/c gases are compressible and density depends on

pressure

•Pascal’s law doesn’t contain any info about shape of container & regardless of

shape of container, pressure increases below surface and results in fixed value at

given depth

•Pressure data are often given in non-standard units convert

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