Page 374 - Introduction to Continuum Mechanics

358 Interpretation of A and u

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Thus, (A+-//) is the proportionality constant relating the viscous mean normal stress to the
rate of change of volume. It is known as the coefficient of bulk viscosity. The total mean normal
stress is given by

and it is clear that the so-called pressure is in general not the mean normal stress, except when
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either A = 0 or (A+-^) is assumed to be zero.

Given the following velocity field:

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for a Newtonian liquid with viscosity p. = 0.982 mPa-s (2.05 xlO lb-s/ft ). For a plane
whose normal is in the e rdirection, (a) find the excess of the total normal compressive stress
over the pressure/?, and (b) find the magnitude of the shearing stress.
Solution. From

we have

Now, from Eq. (i),

Therefore

(b)

Thus, the magnitude of shearing stress equals 1.96 mPa.