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Chapter 3: Problem 7

The Bemoulli 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 \(" \mathbf{F}=m \mathbf{a}^{\prime \prime}\) along a streamline to obtain the equivalent of the Bernoulli equation for this flow.

Short Answer

Expert verified
The equivalent of the Bernoulli equation for a flow where the gravitational acceleration \(g=g_{0}-c z\) is \(- ∫dP + ∫ (g_0 - c z) dz = \frac{1}{2}v^2 + constant\)

Step by step solution

01

Express Forces Acting on the Fluid Element

First, identify forces that impact the fluid element. In general, these include gravity and pressure forces. The pressure force in the direction of the streamline flow is \(F_{pressure} = - A \Delta P\), where A is the cross-sectional area of the flow path and ∆P is the pressure difference. For gravity, based on the description of the problem, the gravitational force is \(F_{gravity} = ρ A dz (g_{0} - c z)\), where ρ is the density of the fluid and dz is the vertical distance element along the streamline.
02

Apply Newton's Second Law

Combine the forces from Step 1 in Newton’s second law of motion, which states that the sum of the forces is equal to the mass times the acceleration. With the assumption that the flow is steady, the components of acceleration in the direction of the flow can be attributed to changes in velocity magnitude and direction. Write the equation as \(- A \Delta P + ρ A dz (g_{0} - c z) = ρ A dl dv/dt\), where dl is the incremental flow path length and dv/dt is the time rate of change of the fluid's velocity.
03

Simplify the Equation

Next, simplify the equation from Step 2. Divide the whole equation by \(A \, dl\) and rearrange the terms. This results in \(- ∆P/ρ + (g_0 - c z) dz = dv dv/dt\).
04

Integrate the Equation

Lastly, integrate both sides of the equation left after simplification. First, integrating \(- ∆P/ρ\) provides a general energy equation from static pressure. Integrating \((g_0 - c z) dz\) provides the potential energy contribution from height z with a varying gravitational pull. The right-hand side will yield a kinetic energy term. This gives: \(- ∫dP + ∫ (g_0 - c z) dz = \frac{1}{2}v^2 + constant\). This is the Bernoulli equation equivalent for the specific case where the acceleration due to gravity varies with height.

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