## Related questions with answers

The Bernoulli equation is valid for steady, inviscid, incompressible flows with constant acceleration of gravity. Consider flow on a planet where the acceleration of gravity varies with height so that $g=g_{0}-c z$, where $g_{0}$ and c are constants. Integrate “F = ma” along a streamline to obtain the equivalent of the Bernoulli equation for this flow.

Solution

VerifiedAccording to second law of Newton

$\sum F_{s} =\sum ma_{s}$

Applying Bernoulli's equation

$\begin{align*} dP+\dfrac{1}{2}\rho dV^{2}+\gamma dZ &= 0\\ & \text{Replace}\\ \gamma&= \rho g\\ g &=g_{0}-cz\\ dP+ d(\dfrac{1}{2}\rho V^{2})+\rho(g_{0}-cz)dz &=0\\ & \text{integrating equation}\\ \int _{1}^{2}dp+\int _{1}^{2}d(\dfrac{1}{2}\rho V^{2})+\rho \int _{1}^{2}(g_{0}-cz)dz &=0\\ [p]_{1}^{2}+\dfrac{1}{2}\rho[v]_{1}^{2}+\rho g_{0}[z]_{1}^{2}- \rho c[\dfrac{z^{2}}{2}]_{1}^{2}& =0\\ p_{2}-p_{1} +\dfrac{1}{2}\rho[v_{2}^{2}-v_{1}^{2}]+\rho g_{0}[z_{2}-z_{1}] -\dfrac{1}{2}\rho c[z_{2}^{2}-z_{1}^{2}] &=0\\ p+\dfrac{1}{2}\rho V^{2} +\rho g_{0}z-\dfrac{1}{2}\rho c z^{2} &=k\\ \end{align*}$

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