Bernoulli Equation Manipulation

Classical Mechanics Level 3

Suppose we have some water flowing in a pipe and obeying the Bernoulli Principle .

$P + \frac12 \rho v^2 + \rho gh = \text{constant}$

In the above equation, $P$ is the water pressure, $\rho$ is the water mass density, $v$ is the flow speed, $g$ is the local gravitational acceleration, and $h$ is the height of the fluid (measured along an axis parallel to gravity).

Suppose that the height is constant along the flow line, and that the water is accelerating at a rate of $0.2 \, \text{m/s}^2$ . Determine the magnitude of the spatial pressure gradient (in $\text{N/m}^3$ ) (where $x$ is the distance along the pipe).
$\Big | \frac{d P}{d x} \Big | = ?$

Details and Assumptions:
- Water density is $1000 \, \text{kg/m}^3$
- Give your answer as a positive number

The answer is 200.0.

1 solution

Aryaman Maithani
Jun 5, 2018

$\because\text{h is constant:}$

$P+\frac{1}{2}\rho v^2 = K$

$\text{Differentiating w.r.t. x:}$

$\frac{dP}{dx} + \rho v\frac{dv}{dx} = 0$

$\implies \frac{dP}{dx} = - \rho v\frac{dv}{dx}$

$\implies \Big| \frac{dP}{dx} \Big| = \rho v\frac{dv}{dx} = \rho \frac{dx}{dt} \frac{dv}{dx} = \rho \frac{dv}{dt}$

$\implies \Big| \frac{dP}{dx} \Big| = \rho a = 1000\cdot 0.2$

$\implies \Big| \frac{dP}{dx} \Big| = \boxed{200N/m^3}$

Bernoulli Equation Manipulation

The answer is 200.0.

1 solution

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