The Bernoulli equation is valid for steady, inviscid,...

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 $=$ $m \mathbf{a}^{\prime \prime}$ along a streamline to obtain the equivalent of the Bernoulli equation for this flow.

Key Concepts

Bernoulli Equation

The Bernoulli equation represents the principle of conservation of mechanical energy in fluid flow along a streamline. It relates pressure, kinetic energy, and potential energy under assumptions like steady, inviscid, and incompressible flow, and is used to analyze how these quantities change along a flow path.

Steady, Inviscid, Incompressible Flow

This concept involves the assumptions that fluid properties do not change with time (steady), there is no viscosity (inviscid), and density remains constant (incompressible). These simplifications allow the derivation of energy conservation relations such as the Bernoulli equation by eliminating terms related to time dependence and viscous dissipation.

Variable Gravitational Acceleration

When gravitational acceleration varies with position, the usual potential energy term in the Bernoulli equation must be modified. This concept involves accounting for the spatial variation of gravity—in this case, linearly decreasing with height—thereby affecting the balance between pressure and kinetic energy in the fluid.

Streamline Integration

Integrating the forces acting on a fluid element along a streamline leads to the derivation of the Bernoulli equation. This method involves the integration of the momentum or energy equations to obtain a relation that combines pressure, kinetic, and potential energy contributions, especially when external forces like variable gravity are present.

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