FLUID MECHANICS FUND. (LL)-W/ACCESS
FLUID MECHANICS FUND. (LL)-W/ACCESS
4th Edition
ISBN: 9781266016042
Author: CENGEL
Publisher: MCG CUSTOM
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Chapter 4, Problem 73P
To determine

The magnitude of the shear strain rate about point P in the xy plane is given by : εxy=12(uy+vx).

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Explanation of Solution

The shear strain rate is half of the rate of decrease of the angle between two perpendicular lines which intersect at the point. Initially at time t1, line a and b are perpendicular and intersect at point P. The angle between line a and b is αab at time t2.

The following figure shows the fluid element at time t2.

  FLUID MECHANICS FUND. (LL)-W/ACCESS, Chapter 4, Problem 73P

Figure-(1)

At time t2, the point A moves a distance of (u+uxdx)dt to right and (v+vxdx)dt to up and the point B moves a distance of (u+uydy)dt to the right and (v+vydy)dt up. The point P moves a distance of udt to right and vdt up.

Refer to Figure-(I), to obtain the values of the horizontal distance between point P and A as dx+uxdxdt, the vertical distance between point P and A as vxdxdt, the horizontal distance between point P and B as uydydt and the vertical distance between point P and B as dy+vydydt.

Write the expression for the angle α at point A

  tanαa=vxdxdtdx+uxdxdtαa=tan1( v xdxdtdx+ u xdxdt)

The distance dx+uxdxdt is approximate to dx.

  αa=tan1( v xdxdtdx)=tan1(vxdt)vxdt

Write the expression for the angle α at point B

  tanαb=uydydtdy+vydydtαb=tan1( u ydydtdy+ v ydydt)

The distance dy+vydydt is approximate to dy.

  αb=tan1( u ududtdu)=tan1(uydt)uydt

Write the expression for the angle αab at time t1.

  (αab)t1=π2

Refer to figure (I), to obtain the value of (αab)t2 as αb+π2αa at time t2.

  (αab)t2=αb+π2αa   ...... (I)

Substitute vxdt for αa and uydt for αb in Equation (I).

  (αab)t2=uydt+π2vxdt

Write the expression for the shear strain rate.

  εxy=12ddtαabεxy=12ddt(( α ab ) t 2( α ab ) t 1)   ...... (II)

Substitute uydt+π2vxdt for (αab)t2 and π2 for (αab)t1 in Equation (II).

  εxy=12ddt(uydt+π2vxdtπ2)εxy=12ddt(uydtvxdt)εxy=12ddt(uydt+vxdt)

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Chapter 4 Solutions

FLUID MECHANICS FUND. (LL)-W/ACCESS

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