Fluid mechanics

[pic 1]

SCHOOL OF SCIENCE AND ENGINEERING

Spring 2018

Fluid Mechanics

EGR 3301

PROJECT

By

Youssef Tahri

Houssame Limami

        In the x direction, the left hand side of the Navier Stokes equation is zero, the pressure force acts in the x- direction and the velocity only depends upon z.

Assumptions:

1.  The wall is infinite in the s-y plane (y is out of the page for a right-handed coordinate system).
2.  The flow is steady, i.e.  [pic 3]

1.  The flow is parallel and fully developed (we assume the normal component of velocity,, is zero, and we assume that the streamwise component of velocity  is independent of streamwise coordinate s)[pic 4][pic 5]

1.  The fluid is incompressible and Newtonian, and the flow is laminar

1.  Pressure P = constant =  at the free surface. In other words, there is no applied pressure gradient pushing the flow; the flow establishes itself due to a balance between gravitational forces and viscous forces along the wall. Atmospheric pressure is constant everywhere since we are neglecting the change of air pressure with elevation.[pic 6]
2.  The velocity field is purely 2-dimensional, which implies that  and [pic 7][pic 8]

1.  Gravity acts in the negative z direction. We can express this mathematically as           . In the s-n plane,  and [pic 9][pic 10][pic 11]

The boundary conditions:

(1) No slip at the wall: at n = 0, [pic 12]

(2) At the free surface (n = h) there is no shear, which in this coordinate system at the vertical free surface means   . [pic 13]

(3) P =  at n = h.[pic 14]

Continuity equation:

[pic 15]

                              We took coordinate (s,n) in place of (x,z) (so: &  )[pic 16][pic 17]

[pic 18]

So, [pic 19]

n momentum equation:  According to the Incompressible Navier-Stokes equation:

[pic 20]

        [pic 21]

• un=0 because parallel flow
• We have a steady flow so   [pic 22]

[pic 23]

We integrate this equation to solve for the pressure:

Pressure:  [pic 24]

• P =  at n = h, so [pic 25][pic 26]

Thus, [pic 27]

s momentum equation:  

  [pic 28]

By integrating the previous equation twice, we get [pic 29]

At n = 0,  [pic 30]

And at (n = h),   [pic 31][pic 32][pic 33]

Thus, velocity field:    [pic 34]

Average Velocity:                         at     n = h/2[pic 35]

For the mass flow rate, L is width, which is on the y-axis, so the cross sectional area is L*dn

               [pic 36]

[pic 37]

Step 1 All the relevant parameters in the problem are listed in functional form:

List of relevant parameters:

[pic 38]

Step 2 The primary dimensions of each parameter are listed

Step 3 As a first guess, j is set equal to 3, the number of primary dimensions represented in the problem (m, L, and t).

Reduction: [pic 51]

If this value of j is correct, the expected number of s is[pic 52]

Number of expected:                        [pic 53][pic 54]

Step 4

We need to choose two repeating parameters since j = 3. Since this is often
the hardest (or at least the most mysterious) part of the method of repeating
variables, several guidelines about choosing repeating parameters are listed
in Table 7–3.

• We can’t choose a dependent variable, which  so we’ll choose 3 (j=3) from the remaining (6).[pic 55]
• We can’t choose 3 parameters that form by themselves a dimensionless group: in our case we can choose any combination from the 6 remaining parameters
• We must represent all the primary dimensions, therefore we can’t choose (L, h, ) or (g, h , L)[pic 56]
• We shouldn’t include dimensionless parameters: we can’t choose [pic 57]
• We can’t choose 2 parameters with same dimensions or that differ by one exponent (exp: volume & length –l3 & l): we can’t choose L along with h
• Choose common parameters: it is wise to not choose  => we’re left with either g, h,  or L, g, [pic 58][pic 59][pic 60]

Therefore, we choose:

Repeating parameters: g, h and [pic 61]

Step 5 The dependent  is generated:[pic 62]

Then, we write the  equation with the dependent parameter [pic 63][pic 64]

1)

 =                   {) }[pic 65][pic 66][pic 67][pic 68][pic 69][pic 70]

...