Fluid Mechanics 3502 Day 2

Fluid Properties Part 2

Solid-Liquid-Gas

Solid under shear vs Fluid under Shear

Density / Specific Weight

Compressible vs. Non-Compressible (cont’d)

Viscosity

Newtonian vs. Non-Newtonian Fluids

Surface Tension

Vapor Pressure

The mass density of a liquid is the mass per unit volume, or ρ = 𝑀/𝑉 (SI units: kg/m3)

If the fluid is incompressible , then the density will only be a function of temperature ; it may depend on pressure but only in the average sense (gas law)

The specific weight of a liquid is the weight per unit volume, or

𝛾 = 𝑊/V (SI units: N/m3)

Density and the Specifics

On the earth’s surface, the mass of fluid M will impose a weight force W, that is W = Mg (in SI units, N)

𝛾 = 𝜌𝑔 think: it does not depend on the volume so it is truly a property of the fluid

Response to pressure (normal force / A): common wisdom and qualitative assessment...

• A mass of (incompressible) liquid at a constant temperature will always occupy the same volume regardless of pressure (if you change the temperature, you will likely change the density)

• A mass of compressible gas at a constant temperature will occupy different volumes depending on the applied pressure, i.e., the average density depends on the average pressure and temperature

think : how to squeeze a can / bottle of soda

p1 > p2 > p3

F

The equation of state of a fluid relates its system properties:

density, pressure and temperature. The question is: how much density is changing under variation of temperature or pressure ? Variation of pressure in water: bulk modulus of elasticity: E E= - dp / (dV/V) = 2.2 GN/m2 (water) dV/V=-dp / 2.2 109 N/m2

if dP=10 atm, (1 atm ~ 105 Pa = 105N/m2) dV/V=106 /2.2 109 ~ -0.05% a huge pressure difference induce a negligible volume difference density is constant under pressure variation fro water air is also often considered uncompressible (though E air=105 N/m2<< E water), unless it is flowing really fast (Mach number: M=V/c >0.3 ) compressibility effects are manifested when changes in volume of fluid elements are due to change in pressure. Only in a shock wave pressure changes are so strong to compress fluid parcels.

if dp>0 (compression) the fluid will shrink dV<0

What is the simplest equation of state ? density = constant

Ideal Gas Law

• The ideal gas law relates the absolute pressure (p) to the density (ρ) and absolute temperature (T) in a gas:

p = ρR T R is the gas constant

(standard air Rair = 287J/kg K)

T Kelvin p Pascal = N/m2

Note that compressibility accounts for the effect of variation of pressure on the density at different locations (X,t) in the domain of the equation of motion. Think about shock wave locations. However if we are in standard atmospheric conditions and subsonic velocity the density remains weakly varying with elevation z. It is calculated using the ideal gas law, but only on average (z), not at every point (X,t)

Example

2A blackboard

s

displacement not a velocity

dy

dsG

shear modulus

Contrast to solid Strain-deformation in fluids the def. rate involves a ds/dt a velocity

For very thin layers, we assume a linear velocity profile: dV/dy ~ ΔV/Δy

shear stress is strain rate

The strain rate shear stress also defined as deformation rate,

shear stress

y

[N/m2]

y

BLACKBOARD 2A+

Deformation of a fluid parcel under shear

• 2B blackboard

Kinematic viscosity 𝜈 =𝜇

𝜌 𝜈 =

𝜇

𝜌 =

𝑚2

𝑠

Dynamic viscosity 𝜇 = 𝑁𝑠

𝑚2

Dynamic - Force - Newton

Kinematic - motion - meter, second

Note: combination of fluid properties , as dynamic viscosity and fluid density, result in a new fluid property , 𝜈

note: increasing T results in - decreasing dynamic viscosity for a liquid - increasing dynamic viscosity for a gas

Dynamic Viscosity

shear stress (strain rate)

strain rate

strain rate

Torque shear

stress

velocity variation dV/dy

y

strain rate

slope changes with the shear rate

building bullet proof vests! running on starch

surfactants

specific heat , c property that describe the capacity of a fluid to store thermal energy e.g. high c are usually chosen as working fluids in a thermal cycle c is defined as the amount of thermal energy able to increase by 1 degree a unit mass of a given substance. [c] = Joule / Kg Kelvin = J/(Kg K) Note: the more energy a liquid is able to store, the more energy is necessary to increase its temperature (large c). In the SAFL wind tunnel we had 30% solution of ethylene glycol circulating on the floor to maintain a low constant temperature. water has a high thermal capacity (large c) and ethylene glycol has a low freezing point

Surface tension

How does this insect stay on top of the water? Is it floating?

Why doesn’t the

liquid spread on the leaf?

water strider

https://vimeo.com/57163295

https://www.youtube.com/watch?v=GRkzfs4_PSk

What is surface tension?

When a small amount of fluid is released in a gas, it will form a stable drop.

Why does this happen?

Molecules at the liquid surface have a much stronger attraction to each other than the molecules within the liquid

This attraction creates a skin or membrane

stretching over the liquid.

think about a tiny membrane, like a plastic foil covering the fluid domain

Force due to Surface Tension • The value of the surface tension σ depends on the liquid and the

other fluid in contact (as air). For water-air surface 𝜎 = 0.073N/m

• The direction of this force is always parallel to the surface of the liquid-gas interface.

• The magnitude of the force due to surface tension σ is

where L is the length over which the surface tension acts

This is not the whole story.

• When a liquid drop contacts a solid surface, the drop will deform

• The deformation depends on the liquid AND gas AND solid. e.g. • the deformation of the drop modify the direction of the surface tension force

LFs

Liquid/gas interface

water mercury

Note : why L? because the intersection between two surfaces is a line/curve [m]

Surface tension – contact angles The mercury molecules do not “like” being in contact with the glass

surface and pull away from it.

Untreated glass surfaces and water molecules “like” to be in contact

The surface tension force acts in the direction shown, all around the contact line between the drop and the glass.

The contact angles define the direction of the surface tension force

Hg in air on glass: σ = 0.48 N/m and θc = 130° Water in air on glass: σ = 0.073 N/m and θc ~ 0°

c

bottom view

bottom view

LFs

LFs

Surface tension – “wetting” Water wets many surfaces well. To prevent this from

happening, a waterproofing agent can be added to the surface to increase the wetting angle

Adding a waterproofing agent results in water droplets that can easily be removed. Such a surface is said to be hydrophobic (literally “water-fearing”)

Note: the surface tension of the water is not changed. The contact angle

is the only thing changed.

Note: super hydrophobic surfaces are studies for drag reduction purpose

Example 1

• To what height above the reservoir will water at room temperature rise in a glass tube if the inside tube diameter d =1.6mm?

• How does this change if the water is replaced by a fluid with the same surface tension, but θc = 60°?

BLACKBOARD 2E

Example 2

• The surface tension at a liquid-gas free surface allows us to carefully place an object such as steel needle onto the liquid without it sinking

• A steel needle of length 0.025 m rests on a water surface. How heavy can the needle be and still float?

W

F𝜎 F𝜎

𝑊 = 2𝐹𝜎 = 2𝜎𝐿

Example

A board 1 m x 1m that weighs 20 N slides down an inclined ramp. The board is separated from the ramp by a thin film of oil whose dynamic viscosity is 0.05 N s / m2 The slope θ = 30o, and the velocity of the board = 2.0 cm/s.

What is the thickness of the film? What assumptions do you make to find the answer?

2C Blackboard