Fluid Basics

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Fluid mechanics is the study of fluids at rest or in motion and their interaction with solids or other fluids at the boundaries[1]. More specialised categories of fluid mechanics are:

  1. Hydraulics: the study of flow in pipes or channels
  2. Hydrodynamics: the study of incompressible fluids
  3. Aerodynamics: the study of fluids over loads
  4. Gas Dynamics: the study of fluids undergoing density variations

A fluid is any material in its liquid or gas form. The distinction between these states is that:

  • A solid can resist an applied stress by deforming
  • A fluid deforms continuously under the influence of an applied stress

The difference between a liquid and a gas is that

  • a liquid generally maintains uniform density and has a defined volume
  • a gas will continue to expand to fill a volume of space

Contents

Textbook Readings

Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010), pp. 2-13 and 38-42.

Fluid Properties and Conditions

No Slip Condition

[2]The "no slip" condition refers to the condition that fluids interacting with solids have zero velocity at the contact point. This is due to the viscosity of the fluid which, when high, is also responsible for creating the boundary layer- the flow region next to the contact point between the solid and the fluid.

Density

[3]Density is defined as mass per unit volume. That is:

ρ = m/V kg/m³
Where
m = mass (kg)
V = Volume (m³)

Density varies with temperature and pressure. Densities to remember are:

  • Air at 20°C, ρ = 1.204 kg/m³
  • Water at 20°C, ρ = 995 kg/m³

For ideal gasses, the equation PV = mRT can be rearranged to show that ρ = m/V = P/RT. The relationship between density, temperature and pressure, as well as Reynold's number is quite clear from this'.

Specific Gravity

[4]Sometimes the density of a fluid is reported relative to the density of other well known substances. This measure is known as the specific gravity or the relative density and is defined as the ratio of the density of a substance to the density of another well known substance at a specific temperature. The most common substance used is water so that specific gravity is calculated by:

SG = ρ/ρH2O
Where
ρH2O = 1000 kg/m³. It is the density of water at 4°C.

This value is quite handy to use without conversion back to actual density because it will yield pressure values in kilopascals (kPa) rather than pascals

Cavitation

[5]Cavitation is the tendency for a liquid to vaporise (as bubbles) when its pressure drops below its vapour pressure. The vapour pressure is the same as the saturation pressure for pure substances. These bubbles reduce flow efficiency and can be very dangerous to propellor blades and other equipment.

Viscosity

[6]Viscosity, ɥ kg/m.s, is a measure of a liquid's internal resistiveness to stress. For Newtonian fluids (the only fluids dealt with in this course), viscosity is linearly proportional to the stress applied and the rate of deformation. That is

Viscosity.png
Where
τ = shear stress (Pa)
du/dy = rate of deformation

Viscosity is calculated by measuring the velocity of a fluid passing between two solid plates. It can be shown that since

Viscosity1.png (N)
Where
F = force applied (N)
A = contact area between liquid and solid (m²)

That

Viscosity2.png
Where
v = liquid/plate velocity (m/s)
ι = distance between solid plates (m)

Viscosity can be thought of being the same as the coefficient of friction in solid mechanics, especially since they take the same symbol and are both linearly proportional to the applied stress.

Surface Tension

[7]Surface tension is the force that keeps a fluid in a shape, like a drop. The coefficient of surface tension has the symbol σs and has units N/m.

Capillary Effect

[8]The capillary effect is the rise or fall of a liquid in a tube that is inserted into that liquid. The capillary rise is inversely proportional to the radius of the tube and the density of the liquid. The rise of the liquid, h, is given by:

h = (2σs.cos ø)/ρgR (m)
Where
σs = coefficient of surface tension
ø = angle of meniscus or the angle that the tangent to the liquid surface makes with the tube
ρ = density (kg/m³)
g = gravity (m/s2)
R = radius of tube (m)

Fluid Flows

[9]Different types of fluid flows are (in pairs):

  1. Viscous - where viscosity effects are significant.
  2. Inviscid - where viscosity effects are negligible


  1. Internal - such as in a pipe or channel
  2. External - such as over a wing


  1. Laminar - smooth, ordered
  2. Turbulent - chaotic, random


  1. Steady - no change of properties (such as temperature, pressure, density etc.)
  2. Unsteady - change of properties


The last two flow types are:

  1. Compressible - where fluid density is not uniform, such as in a gas
  2. Incompressible - Where fluid density is uniform, such as in a liquid

Mach Number

Gasses can be approximated as incompressible when their mach number is roughly M < 0.3, where M is the (dimensionless) mach number defined as:

M = v/c
Where
v = fluid flow (or velocity) (m/s)
c = speed of sounds (~ 340 m/s)

References

"Textbook" refers to Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010).

  1. Textbook, p. 2
  2. Textbook p.10
  3. Textbook p. 38
  4. Textbook p. 39
  5. Textbook pp. 41-42
  6. Textbook pp. 51-52
  7. Textbook p. 55
  8. Textbook p. 58
  9. Textbook pp.10 - 12
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