1. Introduction, fluid properties (1.1, 2.8, 4.1, and handouts) · PDF fileVVR 120 Fluid Mechanics 1. Introduction, fluid properties (1.1, 2.8, 4.1, and handouts) • Introduction,

VVR 120 Fluid VVR 120 Fluid MechanicsMechanics

1. Introduction, fluid properties (1.1, 2.8, 4.1, and handouts)

• Introduction, general information• Course overview• Fluids as a continuum• Density• Compressibility• ViscosityExercises: A1


Fluid mechanics

• Fluid properties (2)• Hydrostatics (3)• Basic equations (6)• Pipe flow (5)• Flow around submerged

bodies (1)• Channel flow (3)• Repetition (2)


Applications of fluid mechanics

• Transport of gas and fluids in pipes and channels• Societal supply of safe energy and water• Energy production (oil, hydropower, nuclear energy,

natural gas)• Environmental engineering and water treatment (channels,

basins, filtering)• Process technology (relationship temperature, pressure,

and energy)• Protection against climate extremes/catastrophes

(flooding, harbours, wind forces)


FLUID AS A CONTINUUM

• A fluid is considered to be a continuum in which there are no holes or voids ⇒ velocity, pressure, and temperature fields are continuous.

• Validity criteria: Smallest length scale in a flow >> average spacing between molecules composing the fluid.



DENSITY (ρ)

Mass/ unit volume (kg/m3)

Density decreases normally with increasing temperatureρwater = ρ(T,S,p)

i.e., dependent on- Temperature- Salt content (ρ ≈ 1000 + 0.741⋅S, S in per mille;

S = 3.5% in ocean ⇒ ρ = 1026 kg/m3)- Pressure (but only a small variability)


OTHER DEFINITIONS• Weight = mass × gravity acceleration

(W = mg, [N = kg⋅m/s2]) (Eqn. 1.4)

• Weight density (or specific weight)= density × gravity acceleration(w = ρg, [N/m3 = kg⋅/(m2s2)]) (Eqn. 1.6) (Note w = γ in exercises)

• Specific volume ν = reciprocal of density(ν = 1/ρ, [m3/kg])

• Relative density (or specific gravity), s, is the density normalized with the density of water at a specific temperature and pressure(normally 4°C and atmospheric pressure):s = R.d. = ρ/ρwater (often = ρ/1000) (Eqn. 1.7)

• Power P [W = J/s = kg⋅m2/s3 = Nm/s]; P = T ω (T = torque, ω = angular velocity [rad/s, 360o = 2πrad]


Example – density. The specific weight of water at ordinary temperature and pressure is 9.81 kN/m3. The specific gravity of mercury is 13.56. Compute the density of water and the specific weight and density of mercury.


COMPRESSIBILITY

• All fluids can be compressed by application of pressure ⇒ elastic energy being stored

• Modulus of elasticity (“elastitetsmodul”) describes the compressibility properties of the fluid and is defined on the basis of volume


• Modulus of elasticity:

E=-dp/(dV/V1) [Pa]

• For liquids, region of engineering interest is when V/V1 ∼ 1 ⇒

• Ewater ~ 2⋅109 Pa (function of temperature)

Ep

VV Δ

−≈Δ


A1 What pressure must be applied to water to reduce its volume 1 % ?


Example – compressibility. At a depth of 8 km in the ocean the pressure is 81.8 MPa. Assume that the specific weight of sea water at the surface is 10.05 kN/m3 and that the average volumemodulus of elasticity is 2.34*109 N/m2 for the pressure range.

A) What will be the change in specific volumebetween that at the surface and at that depth?B) What will be the specific volume at that depth?C) What will be the specific weight at that depth


IDEAL FLUIDA fluid in which there is no friction

REAL FLUIDA fluid in which shearing forces always exist whenever motion takes place due to the fluid’s inner friction – viscosity.


VISCOSITY

• Viscosity is a measure of a fluid’s “inner friction” or resistance to shear stress.

• It arises from the interaction and cohesion of fluid molecules.

• All fluids posses viscosity, but to a varying degree. For instance, syrup has a considerably higher viscosity than water.


DEFINITION OF DYNAMIC VISCOSITY - μ

Shearing of thin fluid film between two plates. The upper plate has an area A.

• Experiments have shown that for a large number of fluids:

F ~ AV/h (if V and h not too large)

• Linear velocity profile ⇒ V/h = dv/dy

yy


• Introduction of the proportionality constant μ, named dynamic viscosity, gives Newton’s viscosity law shear force:

μ [Pa⋅s or kg/ms]

• ν = μ/ρ [m2/s] - Kinematic viscosity

• No-slip condition – water particles adjacent to solid boundary has zero velocity (observational fact)

dydv

hV

AF μμτ === (Eqn. 4.1-4.2) N/m2


μ (Pa·s)


Implication of viscosity: a fluid cannot sustain a shear stress without deformation


Implications of Newton’s law:• τ, μ independent of pressure (in contrast to solids)• no velocity gradient ⇒ no shear stress

Restriction of Newton’s law:• law only valid if the fluid flow is laminar in which viscous

action is strong


• Laminar flow: smooth, orderly motion in which fluid elements appears to slide over each other in layers (little exchange between layers).

• Turbulent flow: random or chaotic motion of individual fluid particles, and rapid mixing and exchange of these particles through the flow

Turbulent flow is most common in nature.


Newtonian – non-Newtonian fluidsExamples non-Newtonian fluids:Plastics, blood, suspensions, paints, foods

ii dy

duττμττ >=− ,

Shear vs. rate of strain re-lations for non-Newtonian fluids:

Bingham plastic

n>1: Shear-thickening fluid, n<1: Shear-thinning fluid

n

dy

du)(μτ =


Example – viscosity. A space, 3 cm wide, between two plane horizontal surfaces is filled with SAE 30 Western lubricating oil at 20°°C. What force is required to drag a very thin plate of 1 m2 area through the oil at a velocity of 0.1 m/s if the plate is 1 cm from one surface?

Documents

1. Introduction, fluid properties (1.1, 2.8, 4.1, and handouts) · PDF fileVVR 120 Fluid Mechanics 1. Introduction, fluid properties (1.1, 2.8, 4.1, and handouts) • Introduction,