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    Topics covered in this chapter

  • Equations of motion of a Newtonian fluid
  • The Reynolds number
  • Dissipation of Energy by Viscous Forces
  • The energy equation
  • The effect of compressibility
  • Resume of the development of the equations
  • Special cases of the equations
    • Restrictions on types of motion
      • Isochoric motion
      • Irrotational motion
      • Plane flow
      • Axisymmetric flow
      • Parallel flow perpendicular to velocity gradient
    • Specialization on the equations of motion
      • Hydrostatics
      • Steady flow
      • Creeping flow
      • Inertial flow
      • Boundary layer flow
      • Lubrication and film flow
    • Specialization of the constitutive equation
      • Incompressible flow
      • Perfect (inviscid, nonconducting) fluid
      • Ideal gas
      • Piezotropic fluid and barotropic flow
      • Newtonian fluids
  • Boundary conditions
    • Surfaces of symmetry
    • Periodic boundary
    • Solid surfaces
    • Fluid surfaces
    • Boundary conditions for the potentials and vorticity
  • Scaling, dimensional analysis, and similarity
    • Dimensionless groups based on geometry
    • Dimensionless groups based on equations of motion and energy
    • Friction factor and drag coefficients
  • Bernoulli theorems
    • Steady, barotropic flow of an inviscid, nonconducting fluid with conservative body forces
    • Coriolis force
    • Irrotational flow
    • Ideal gas

Reading assignment

Chapter 2&3 in BSL
Chapter 6 in Aris

Equations of motion of a newtonian fluid

We will now substitute the constitutive equation for a Newtonian fluid into Cauchy's equation of motion to derive the Navier-Stokes equation.

Cauchy's equation of motion is

ρ α i = ρ D v i D t = ρ f i + T i j , j or ρ a = ρ D v D t = ρ f + T

The constitutive equation for a Newtonian fluid is

T i j = ( - p + λ Θ ) δ i j + 2 μ e i j or T = ( - p + λ Θ ) I + 2 μ e

The divergence of the rate of deformation tensor needs to be restated with a more meaningful expression.

e i j , j = 1 2 x j v i x j + v j x i = 1 2 2 v i x j x j + 1 2 x i v j x j = 1 2 2 v i + 1 2 x i ( v ) . or e = 1 2 2 v + 1 2 ( v )


T i j , j = - p x i + ( λ + μ ) x i ( v ) + μ 2 v i or ρ D v D t = ρ f - p + ( λ + μ ) Θ + μ 2 v

Substituting this expression into Cauchy's equation gives the Navier-Stokes equation.

ρ D v i D t = ρ f i - p x i + ( λ + μ ) x i ( v ) + μ 2 v i or ρ D v D t = ρ f - p + ( λ + μ ) Θ + μ 2 v

The Navier-Stokes equation is sometimes expressed in terms of the acceleration by dividing the equation by the density.

a = D v D t = v t + ( v ) v = f - 1 ρ p + ( λ ' + v ) Θ + v 2 v

where v = μ / ρ and λ ' = λ / ρ . ν is known as the kinematic viscosity and if Stokes' relation is assumed λ ' + ν = ν / 3 . Using the identities

2 v ( v ) - × ( × v ) w × v

the last equation can be modified to give

a = D v D t = f - 1 ρ p + ( λ ' + 2 ν ) Θ - ν × w .

If the body force f can be expressed as the gradient of a potential (conservative body force) and density is a single valued function of pressure (piezotropic), the Navier-Stokes equation can be expressed as follows.

a = D v D t = - [ Ω + P ( p ) - ( λ ' + 2 ν ) Θ ] - ν × w where f = - Ω and P ( p ) = p d p ' ρ ( p ' )

Assignment 6.1

Do exercises 6.11.1, 6.11.2, 6.11.3, and 6.11.4 in Aris.

The reynolds number

Later we will discuss the dimensionless groups resulting from the differential equations and boundary conditions. However, it is instructive to derive the Reynolds number N R e from the Navier-Stokes equation at this point. The Reynolds number is the characteristic ratio of the inertial and viscous forces. When it is very large the inertial terms dominate the viscous terms and vice versa when it is very small. Its value gives the justification for assumptions of the limiting cases of inviscid flow and creeping flow.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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