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The Bernoulli Equation (continued)

Σ F_s = dm a_s P dn db - (P + ∂P/∂s ds) dn db – dW sin θ = dm v dv/ds

Now dm = ρ dndbds and dW = γ dndbds = ρg dndbds

- (∂P/∂s) dn db ds - ρg dndbds sin θ = ρ dndbds v dv/ds

By geometry from the figure below sin θ = dz/ds

So - (∂P/∂s) dn db ds - ρg dndbds (dz/ds) = ρ dndbds v dv/ds

- ∂P/∂s - ρg dz/ds = ρ v dv/ds or ∂P + ρ v dv + ρg dz = 0

Integration of this equation gives the Bernoulli equation:

P + ½ ρ V² + ρg z = Constant

where P = static pressure at point on streamline, ρ = mass density of fluid, g = acceleration of gravity,

V = speed of fluid at point on streamline, z = elevation at point on streamline