Problem 1 (3 Points) Consider the axisymmetric flow u which in cylindrical coordinates can be...

Question

Problem 1 (3 Points) Consider the axisymmetric flow u which in cylindrical coordinates can be written as u(r, z, t) = ur(r, z, t)er + uz(r, z, t)ez 1. What can you say about the streamlines of the flow? 2. What is the shape of the vortex tubes in this flow? 3. Show that for this flow, the vorticity equation reduces to D Dt ? r  = 0 where ? is the magnitude of the vorticity. In other words, the vorticity of any fluid element changes in time in proportion to r. Problem 2 (3 Points) A steady Beltrami flow is a velocity field u(r) for which the vorticity is always parallel to the velocity: ? × u = f(r)u, where f is a scalar function of three variables. 1. Show that u(x, y, z) = hB sin y + C cos z, C sin z + A cos x, A sin x + B cos yi, where A, B, and C are constants, is a Beltrami field and find its corresponding f. 2. Returning to general Beltrami fields, show that if a steady Beltrami field u is the steady velocity field of a fluid with constant density, then f is necessarily constant on streamlines. 3. If in addition the fluid is inviscid, give an expression for the pressure in terms of ? and u. Problem 3 (6 Points) In this problem, we consider the steady flow of a viscous fluid with uniform and constant density in cylindrical geometry. 1. Assume u = hur, u?, uzi, with uz = Kz, u? = u?(r), and K a constant. Find ur. 2. Show that the vorticity ? is entirely in the ez direction: ? = ?ez. What is the expression for ? in terms of u?? 3. Use the vorticity equation to derive a differential equation for ?. Solve this differential equation to find an expression for ?. 4. Use the previous question to find an expression for u?. Congratulations: you constructed an exact solution to the steady Navier-Stokes equation!
	
Document Preview: Introduction to Fluid Dynamics { Problem Set 7
March 6, 2017
Due April 13, 2017 after class
Problem 1 (3 Points)
Consider the axisymmetric ow u which in cylindrical coordinates can be written as
u(r;z;t) =u (r;z;t)e +u (r;z;t)e
r r z z
1. What can you say about the streamlines of the ow?
2. What is the shape of the vortex tubes in this ow?
3. Show that for this ow, the vorticity equation reduces to
 
D !
= 0
Dt r
where ! is the magnitude of the vorticity. In other words, the vorticity of any uid element changes
in time in proportion to r.
Problem 2 (3 Points)
A steady Beltrami ow is a velocity eld u(r) for which the vorticity is always parallel to the velocity:
ru =f(r)u, where f is a scalar function of three variables.
1. Show that u(x;y;z) =hB siny +C cosz;C sinz +A cosx;A sinx +B cosyi, where A, B, and C are
constants, is a Beltrami eld and nd its corresponding f.
2. Returning to general Beltrami elds, show that if a steady Beltrami eld u is the steady velocity eld
of a uid with constant density, then f is necessarily constant on streamlines.
3. If in addition the uid is inviscid, give an expression for the pressure in terms of  and u.
Problem 3 (6 Points)
In this problem, we consider the steady ow of a viscous uid with uniform and constant density in cylindrical
geometry.
1. Assume u =hu ;u ;ui, with u =Kz, u =u (r), and K a constant. Find u .
r  z z   r
2. Show that the vorticity ! is entirely in the e direction: ! = !e . What is the expression for ! in
z z
terms of u ?

3. Use the vorticity equation to derive a dierential equation for !. Solve this dierential equation to
nd an expression for !.
4. Use the previous question to nd an expression for u .

Congratulations: you constructed an exact solution to the steady Navier-Stokes equation!
1Problem 4 (4 Points)
Consider the steady ow of a viscous uid of constant density, given by the equation:
1
2
uru =  rp +r u

Show that for any closed streamline C, we have
I
r!dr =...

Robert · Accepted Answer

Solution: 1 
(a)  As we know that stream line is an imaginary line or series of imaginary lines in a flow 
field which are drawn in such a way that tangent to this line at any point at any instant 
represent the direction of the instantaneous velocity vector at that point.
Equation of stream line can be given as:
 ⃗      ⃗⃗⃗⃗    
Where V is velocity vector and ds is space vector.
Given velocity vector:
 ⃗   (     )    (     )     (     )   
And  
  ⃗⃗⃗⃗              
So we get: 
*  (     )     (     )  +   *           +     
So this is a vector cross product which can be solved and gives:
  
 
  
  
  (     ) 
(b) Vorticity is a measure of rotation of a fluid particle, it is twice of the rotation vector 
For the 2-d cylindrical flow it is given as:

Problem 1 (3 Points) Consider the axisymmetric flow u which in cylindrical coordinates can be written as u(r, z, t) = ur(r, z, t)er + uz(r, z, t)ez 1. What can you say about the streamlines of the...

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