Introduction to this concept
Considered as aderivational statement from the principle of conservation of energy, Bernoulli’s equationdepicts any fluid’s dynamic behavior and studies about it and the reason if its motion. In order to find the derivational equation of an ideal fluid’s dynamic functioning, we will be taking the help of Newton’s 2nd law of motion.
Before we proceed to find out this equation, we have to keep (µ = 0) and (p = constant) of that liquid.
For an effective action on any fluid element (liquid or air), there are many forces those act on them. Few of those forces are:
Fµ= pressure force,
Fµ =viscous force,
Ft = turbulence force,
Fe= compressibility force, and
F g = gravity force.
To derive Bernoulli’s equation, we will be using pressure force and gravity.
In a certain aspect, you can see a fluid element’s dynamic equilibrium. This equilibrium is because of all the action of the forces that are stated above. You are already aware that ‘p’ is kept as aconstant which represents the intensity of fluid pressure.
The equation that we get after using Newton’s 2nd law of motion is, Mass x acceleration in a certain direction = ‘Σ’ forces in ‘s’ direction
This in equational form will be,
pdA – (p+ δp/δs ds)dA – ρgdA dscosθ
After resolving this equation, you will get,
pdA ds x as
Force or acceleration in ‘s’ direction will give velocity as this equation where v = v(s, t).
as = dv/dt
= dv/dsx δs/δt + δv/δt
= v x δv/δs + δv/δt
= v x δv/δs
Acceleration of fluid in steady flow will be δv/δt=0
After putting the value of steady flow in the previous equation, next step of the equation will stand to be,
ρdAds x v δv/δs=- δp/δsdsdA – ρgdAdz
pvδv/δs = – δp/δs – ρg dz/ds
so, finally, we get the result.
Now for calculating the fluids which have an incompressible and steady flow, the equation will be,
δp/ρ + gdz + vdv = 0
p is kept constant for that incompressible flow
After keeping ‘p’ as constant, a new equation is formed (stated above) and that equation is known as Euler’s equation of motion.
p/p+ g z + v2/2 (this is constant)
p/pg + v2/2g + z (this too is constant)
In entirety, these equations depict Bernoulli’s equation.