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    ui-1/2 = 0.5*(u(i,j)+u(i-1,j)) if ui-1/2 >0 then (uw)i-1/2 = ui-1/2 wi-1 ; b'=b + $\delta$ Re ui-1/2else (uw)i-1/2 = ui-1/2 wi ; e'=e + $\delta$Re ui-1/2endif  ui+1/2 = 0.5*(u(i,j)+u(i+1,j)) if uI+1/2 >0 then (uw)i+1/2 = ui+1/2 wi ; e'=e - $\delta$  Re ui+1/2else (uw)i+1/2 = ui+1/2 wi+1 ; a'=a - $\delta$  Re ui+1/2endif

The y-direction will be similar except the aspect ratio, , must be included. These modifications to the coefficients must be made before the coefficients are updated for the boundary conditions.

Calculation of pressure

The vorticity-stream function method does not require calculation of pressure to determine the flow field. However, if the force or drag on a body or a conduit is of interest, the pressure must be computed to determine the stress field. Here we will derive the Poisson equation for pressure and determine the boundary conditions using the equations of motion. If the pressure is desired only at the boundary, it may be possible to integrate the pressure gradient determined from the equations of motion. The dimensionless equations of motion for incompressible flow of a Newtonian fluid are as follows.

d v * d t * = v * t * + v * * v * = v * t * + * ( v * v * ) = - * P * + 1 R e * 2 v * where v * = v U , x * = x L , t * = U L t , P * = P ρ U 2 , * = L P = p - ρ g z , R e = ρ U L μ

Here all coordinates were scaled with respect to the same length scale. The aspect ratio must be included in the final equation if the coordinates are scaled with respect to different length scales. We will drop the * and use x and y for the coordinates and u and v as the components of velocity.

The following derivation follows that of Hoffmann and Chiang (1993). The equations of motion in 2-D in conservative form is as follows.

u t + ( u 2 ) x + ( u v ) y = - P x + 1 R e 2 u x 2 + 2 u y 2 v t + ( u v ) x + ( v 2 ) y = - P y + 1 R e 2 v x 2 + 2 v y 2

The Laplacian of pressure is determined by taking the divergence of the equations of motion. We will carry out the derivation step wise by first taking the x-derivative of the x-component of the equations of motion.

t u x + 2 u x u x + 2 u 2 u x 2 + u x v y + u 2 v x y + v x u y + v 2 u x y = - 2 P x 2 + 1 R e x 2 u

Two pair of terms cancels because of the equation of continuity for incompressible flow.

u x u x + v y = 0 u 2 v x y + u 2 u x 2 = u x u x + v y = 0 Thus t u x + u x 2 + u 2 u x 2 + v x u y + v 2 u x y = - 2 P x 2 + 1 Re x 2 u

Similarly, the y-component of the equations of motion become

t v y + v y 2 + v 2 v y 2 + u y v x + u 2 v x y = - 2 P y 2 + 1 R e y 2 v

The x and y-components of the above equations are now added together and several pairs of terms cancels pair-wise.

t u x + v y = 0 2 u x 2 + 2 v x y = x u x + v y = 0 2 u x y + 2 v y 2 = y u x + v y = 0 x 2 u x 2 + 2 u y 2 + y 2 v x 2 + 2 v y 2 = 2 x 2 u x + v y + 2 y 2 u x + v y = 0

Therefore the equations reduce to

u x 2 + v y 2 + 2 u y v x = - 2 P x 2 + 2 P y 2

The left-hand side can be further reduced by consideration of the continuity equation as follows.

u x + v y 2 = u x 2 + v y 2 + 2 u x v y = 0 from which u x 2 + v y 2 = - 2 u x v y

Upon substitution, we have

- 2 P x 2 + 2 P y 2 = 2 u y v x - u x v y

This equation can be written in terms of the stream function.

2 P x 2 + 2 P y 2 = 2 2 ψ x 2 2 ψ y 2 - 2 ψ x y 2

Questions & Answers

can someone help me with some logarithmic and exponential equations.
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I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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is it 3×y ?
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J, combine like terms 7x-4y
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how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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types of nano material
abeetha Reply
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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