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Chapter 4: Problem 54

A gas flows along the \(x\) axis with a speed of \(V=5 x \mathrm{m} / \mathrm{s}\) and eressure of \(p=10 x^{2} \mathrm{N} / \mathrm{m}^{2},\) where \(x\) is in meters. (a) Determine the time rate of change of pressure at the fixed location \(x=1\) (b) Determine the time rate of change of pressure for a fluid particle flowing past \(x=1 .\) (c) Explain without using any equations why the answers to parts (a) and (b) are different.

Short Answer

Expert verified
The time rate of change of pressure at the fixed location \(x=1\) is \(0 \mathrm{N/m^2\,s}\). The time rate of change of pressure for a fluid particle flowing past \(x=1\) is \(100 \mathrm{N/m^2\,s}\). The difference is due to the movement of the fluid particle in the pressure gradient.

Step by step solution

01

Determine the rate of change of pressure at x=1

It is given that the pressure vary with position \(x\) as \(p = 10x^2\). The change in pressure with time at a fixed location is given by the partial derivative of pressure with respect to time. Since the pressure does not directly depend on time, the rate of change of pressure at the fixed location \(x=1\) will be zero.
02

Determine the rate of change of pressure for a fluid particle flowing past x=1

The change in pressure along the fluid particle is given by the convective derivative of the pressure. It can be computed using the formula:\[\frac{dp}{dt} = \frac{∂p}{∂t} + V \frac{dp}{dx}\]As mentioned before, \(\frac{∂p}{∂t}\) is zero, while \(\frac{dp}{dx}\) is the rate of change of pressure with respect to position. Given \(p = 10x^2\), \(\frac{dp}{dx}\) is \(20x\). At \(x =1\) and given \(V = 5\), by subbing these into the formula:\[\frac{dp}{dt} = 0 + 5 \times 20 \times 1 = 100 \mathrm{N/m^2\,s}\]
03

Explain the difference between answers to parts (a) and (b)

The rate of change of pressure at a fixed point is zero because there's no movement to cause a variation in pressure. On the other hand, for a fluid particle flowing along the x axis, it experiences changes in pressure due to its movement in the pressure gradient. Hence, the rate of change of pressure along a flowing fluid particle is non-zero.

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