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- P = ρ g cos β = ρ g sin α , α = π / 2 - β

The equations of motion are as follows.

0 = d P d x 0 = ρ g cos β - d τ x z d x 0 = ρ g cos β + μ d 2 v z d x 2 , Newtonian fluid

The boundary conditions are zero stress at the gas-liquid interface and no slip at the wall.

v z = 0 , x = 0 τ x z = 0 , x = h

The shear stress profile can be determined by integration and application of the zero stress boundary condition.

τ x z = ρ g h cos β x h - 1 , 0 x h τ w = - ρ g h cos β

The velocity profile for a Newtonian fluid can be determined by a second integration and application of the no slip boundary condition.

v z = ρ g h 2 cos β 2 μ 2 x h - x h 2 , 0 x h v z , max = ρ g h 2 cos β 2 μ

The average velocity and volumetric flow rate can be determined by integration of the velocity profile over the film thickness.

v z = ρ g h 2 cos β 3 μ Q = ρ g W h 3 cos β 3 μ

The film thickness, h, can be given in terms of the average velocity, the volume rate of flow, or the mass rate of flow per unit width of wall ( Γ = ρ h v z ) :

h = 3 μ v z ρ g cos β = 3 μ Q ρ g W cos β 3 = 3 μ Γ ρ 2 g cos β 3

Unsteady viscous flow

Suddenly accelerated plate. (BSL, 1960) A semi-infinite body of liquid with constant density and viscosity is bounded on one side by a flat surface ( the x z plane). Initially the fluid and solid surface is at rest; but at time t = 0 the solid surface is set in motion in the positive x -direction with a velocity U . It is desired to know the velocity as a function of y and t . The pressure is hydrostatic and the flow is assumed to be laminar.

The only nonzero component of velocity is v x = v x ( y , t ) . Thus the only non-zero equation of motion is as follows.

ρ v x t = μ 2 v x y 2 , y > 0 , t > 0

The initial condition and boundary conditions are as follows.

v x = 0 , t = 0 , y > 0 v x = U , y = 0 , t > 0 v x = 0 , y , t > 0

If we normalize the velocity with respect to the boundary condition, we see that this is the same parabolic PDE and boundary condition as we solved with a similarity transformation. Thus the solution is

v x = U e r f c y 4 μ t / ρ

The presence of the ratio of viscosity and density, the kinematic viscosity, in the expression for the velocity implies that both viscous and inertial forces are operative.

The velocity profiles for the wall at y = 0 suddenly set in motion is illustrated below.

Developing Couette flow. The transient development to the steady-state Couette flow discussed earlier can now be easily derived. We will let the plane y = 0 be the surface with zero velocity and let the velocity be specified at y = L . The initial and boundary conditions are as follows.

v x = 0 , t = 0 , 0 < y < L v x = 0 , y = 0 , t > 0 v x = U , y = L , t > 0 or v x = 0 , t = 0 , - L < y < L v x = - U , y = - L , t > 0 v x = U , y = L , t > 0

It should be apparent that the two formulations of the boundary conditions give the same solution. However, the latter gives a clue how one should obtain a solution. The solution is antisymmetric about y = 0 and the zero velocity condition is satisfied. A series of additional terms are needed to satisfy the boundary conditions at y = ± L . The solution is

v x = U n = 0 e r f c ( 2 n + 1 ) L - y 4 ν t - e r f c ( 2 n + 1 ) L + y 4 ν t ν = μ ρ

Asignment. 8.3

Flow of a fluid with a suddenly applied constant wall stress. This problem is similar to that of flow - < x < , y > 0 , t > 0 near a wall - < x < , y = 0 suddenly set into motion, except that the shear stress at the wall is constant rather than the velocity. Let the fluid be at rest before t = 0 . At time t = 0 a constant force is applied to the fluid at the wall, so that the shear stress τ y x takes on a new constant value τ o at y = 0 for t > 0 .
  1. Start with the continuity and Navier-Stokes equations and eliminate the terms that are identically zero. Differentiate the resulting equation with respect to distance from the wall and multiply by viscosity to derive an equation for the evolution of the shear stress. List all assumptions.
  2. Write the boundary and initial conditions for this equation.
  3. Solve for the time and distance dependence of the shear stress. Sketch the solution.
  4. Calculate the velocity profile from the solution of the shear stress. The following equation will be helpful.
  5. Suppose the fluid is not infinite but rather there is another wall at a distance 2h away from the original wall and it was also set into motion but in the opposite direction with the same wall shear stress. What are the stress and velocity profiles for the fluid between the two walls? Sketch and express as series solutions.
  6. What are the steady state stress and velocity profiles for the problem of part (e)? Sketch and express as analytical solutions.

Questions & Answers

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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
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.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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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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