Determine the paths of fluid particles that originate at x0 = (2,3). Problem 3 – 1.6.3 in text When...

Question

Determine the paths of fluid particles that originate at x0 = (2,3). Problem 3 – 1.6.3 in text When carrying out the flow decomposition, determine a, , and G. Also determine the directions in which the principle stresses act by determining the angle ß from the +x-axis made by each of the two eigenvectors of the rate of deformation matrix (E). Problem 4 – streamlines, particle paths, and streak lines Assume that a transient, three-dimensional flow has velocity components given by where a, b, and c are constants. The objective of this problem is to compare and contrast the streamlines in this flow with the paths of the fluid particles. (a) Find the equations governing the streamline that passes through the point (1,1,1) at time t. Note that the other points included on this streamline change with time. (b) Calculate the path of a particle that starts at x0=(x,y,z)=(1,1,1) at t=0. (c) Use the results of part (a) to determine the conditions under which the streamlines and particle paths coincide. (d) Determine the streakline of tracer particles released from x0 between t=0 and t=1. (e) For the 2-D flow with uz=0, construct a plot of the streamline passing through (1,1) at t=0 and t=1, the path of the particle initially at x0 for t [0,1], and the streak line of tracer particles released from x0 between t=0 and t=1 for the following parameters: i. a=1, b=1 ii. a=0.5, b=2 (There should be one plot for i. and 1 plot for ii, with 4 labeled curves on each plot.)

Robert · Accepted Answer

2
3.ux(x,y,t) = w(t).(2x+3y)       uy = w(t).(-x-2y)
B = w(t)[ a   b]   =  w(t)[ 2   3]  
               c    d                -1  -2 
Ω = w(t)(b-c/2) = 4/2 = 2w(t)
α = (a+d)w(t) = 0
E = 1/2 [ a-d  b+c]w(t)   = 1/2 [4  2]w(t)    = [ 2   1]w(t)
               c+b d-a                       2 -4                 1  -2 
the eigen value of E is denoted by λ and the corresponding eigen vector is given by
f = [ fx ]
        fy
 
| E – λI | = 0
| 2-λ  1   | = 0
1 -2-λ
λ = +_√5w(t)                                                          {   +_ means (plus or minus) }
G = λ = +_√5w(t)
[ 2 +​_√5   1     ] [ fx ]   = [ 0 ]             
      1    -2+_√5     fy           0
fy/fx = - (2 +_ √5)
β = tan-1(fy/fx) = tan-1(- (2 +_ √5))
[ In the textbook u provided the answer for β in the example was left in terms of tan-1, so even I left the answer in the same terms]
4.

Determine the paths of fluid particles that originate at x0 = (2,3). Problem 3 – 1.6.3 in text When carrying out the flow decomposition, determine a, , and G. Also determine the directions in which...

Solution

Answer To This Question Is Available To Download

Related Questions & Answers

Submit New Assignment