(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The velocity components in a plane (i.e., 2-D) flow are measured at four points as indicated in the sketch at the bottom of this page (see attached doc. I wrote in the coords). The velocity components at the respective points are, in units of cm/sec:

[tex]u_a=5, v_a=2, u_b=7, v_b=3, u_c=6.5, v_c=3.5, u_d=5.5, v_d=1.5[/tex]

a) Estimate the percentage change in volume of a fluid parcel at the central point o.

b), c) no questions thus far on my part

2. Relevant equations

divergence of V = 0 for incompressible fluid (assumption made in problem since we haven't done compressible flow yet)

3. The attempt at a solution

So, I figured that since the divergence of V = partial(u)/partial(x) + partial (v)/partial(y), the partial(v)/partial(x) and partial(u)/partial(y) components don't need to be calculated (i.e., v_a, v_b, u_c, u_d are not relevant). Is this a correct assumption? If so, I took

(u_b-u_a)/2(dx)

= (7-5)/2(3) = (1/3)

and (v_d-v_c)/2(dy)

= (1.5-3.5)/2(2) = (1/2)

so div(V) = (1/3)-(1/2)

= -(1/6)

Thus, the total change in volume at the central point is (-1/6).

Does this make sense, or am I approaching the problem incorrectly?

Part b) asks whether this flow satisfies conservation of mass for an incompressible fluid. I would think that since there is a net change in volume, that it wouldn't satisfy it. Is this correct?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Percentage change in volume of a fluid parcel

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