Fluid Mechanics: Fundamentals and Applications
Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 4, Problem 101P

Combine your results from Prob. 4—100 to form the two-dimensional strain rate tensor in the xy-plane, ε i j = ( ε y x ε y y ε x x ε x y ) Are the x- and y-axes principal axes?

Expert Solution & Answer
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To determine

The two-dimensional strain rate tensor εijin the xy-plane.

Answer to Problem 101P

The two-dimensional strain rate tensor εijin the xy-plane is:

  εij=(014μ×dPdx(2yh)14μ×dPdx(2yh)0)

Explanation of Solution

The given figure represents Poiseuille flow between two plates.

Fluid Mechanics: Fundamentals and Applications, Chapter 4, Problem 101P

Figure-(1)

Write the expression for two-dimensional strain rate tensor along xyplane.

εij=(εxxεxyεyxεyy)...... (I)

Here, the strain rate tensor is εij, strain rate in xdirection is εxx, strain rate along the ydirection is εyyshear strain rate in xyplane is εxyand shear strain rate in yxdirection is εyx.

Write the expression for the velocity component along xdirection.

u=12μ×dPdx(y2hy)...... (II)

Here, the velocity along xdirection is u, viscosity of the fluid is μ, the pressure gradient is dPdx, distance between the two plates is h.

Write the expression for the velocity component is ydirection.

v=0...... (III)

Write the expression for linear strain rate along the xdirection.

εxx=dudx...... (IV)

Substitute u=12μ×dPdx(y2hy)for u in Equation (IV).

εxx=d(12μ×dPdx(y2hy))dx=d(12μ×dPdx(y2))dxd(12μ×dPdx(hy))dx=00=0

Write the expression for linear strain rate along the ydirection.

εyy=dvdy...... (V)

Substitute 0 for v in the Equation (V).

εyy=dvdy=d(0)dy=0

Write the expression for shear strain rate in xyplane.

εxy=12(dudy+dvdx)...... (VI)

Write the expression for shear strain rate in yxplane.

εyx=12(dvdx+dudy)

...... (VII)

Differentiate Equation (II) with respect to y.

dudy=d(12μ×dPdx(y2hy))dy=12μ×dPdx(d(y2hy)dy)=12μ×dPdx(2yh)

Differentiate Equation (III) with respect to x.

dvdx=d(0)dxdvdx=0

Calculation:

Substitute 12μ×dPdx(2yh)for dudyand 0for dvdxin Equation (VI).

εxy=12(12μ×dPdx(2yh)+0)=12(12μ×dPdx(2yh))=(14μ×dPdx(2yh))

Substitute 12μ×dPdx(2yh)for dudyand 0for dvdxin Equation (VII).

εyx=12(0+12μ×dPdx(2yh))=12(12μ×dPdx(2yh))=(14μ×dPdx(2yh))

Substitute (14μ×dPdx(2yh))for εxyand (14μ×dPdx(2yh))for εyxin Equation (I).

εij=(0(14μ×dPdx(2yh))(14μ×dPdx(2yh))0)

Therefore, the two-dimensional strain rate tensor εijin the xy-plane is:

  εij=(014μ×dPdx(2yh)14μ×dPdx(2yh)0)

Conclusion:

The two-dimensional strain rate tensor εijin the xy-plane is

  εij=(014μ×dPdx(2yh)14μ×dPdx(2yh)0)

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