Question

A gas flows along the x axis with a speed of V = 5x m/s and a pressure of p = 10x2 N/m2, 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

Answer 1

Answer:

a) $(\frac{dP}{dt} )_{x=1}=20\times 1=20\ Pa.s^{-1}$

b) $\frac{dP}{dt}>20x$

c) The answers to parts a) and b) are different because the rate of change of pressure is a function of position and in the two cases the position changes. In the later case the position is greater than 1 and the former case it is equal to 1.

Given:

expression of the speed of gas flow, $v=5x\ m.s^{-1}$

expression for the pressure of gas, $P=10x^2\ Pa$

a)

Time rate of change of pressure at x=1:

$\frac{d}{dt} P=\frac{d}{dt} (10x^2)$

$\frac{dP}{dt} =20x$ ............................(1)

now at point x=1

$(\frac{dP}{dt} )_{x=1}=20\times 1=20\ Pa.s^{-1}$

b)

From equation (1):

$\frac{dP}{dt} =20x$

When x>1 then the rate of change in pressure:

$\frac{dP}{dt}>20x$

c)

The answers to parts a) and b) are different because the rate of change of pressure is a function of position and in the two cases the position changes. In the later case the position is greater than 1 and the former case it is equal to 1.