The Bernoulli Equation
In a Nutshell: The Bernoulli
equation holds along each streamline. It
governs the accelerated motion
of the fluid particle under the assumptions that viscosity is negligible, the
density is constant,
and the flow is steady (independent of time). Bernoulli Equation: The
strategy in developing this equation is to take the Lagrangian viewpoint and
determine the accelerated
motion of a general fluid particle using Newton’s 2nd , F
= m a Integration of the this
differential equation of fluid particle motion yields Bernoulli’s equation. The
figure below shows a free body diagram of a typical fluid element using
intrinsic coordinates, s
and n where s denotes the coordinate along the streamline and n the coordinate normal to the streamline. Forces on the fluid particle come from
pressure acting on the fluid particle and on gravity
giving rise to the elemental weight, dW, of the fluid particle. Now
the acceleration of the fluid particle using intrinsic coordinates is Note:
v = v(s) a = dv/dt = as
es + an en = dv/dt
es + v2
/R en as = dv/dt
= ∂v/∂s ds/dt (using the chain rule) and
ds/dt = v So
the components of acceleration of the fluid particle are as = v
∂v/∂s and an = v2 /R The element of mass, dm =
ρ dsdbdn. The equations
describing the motion of the fluid particle are Σ
Fs = dm as and
Σ Fn = dm an .
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Copyright © 2019 Richard C. Coddington
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