Different forms of Bernoulli's equation

[Jak243]

I was reading some textbooks on doppler echo for medicine and came across this version of the bernoulli equation for blood flow in the aorta. $$P_{1} - P_{2}= 1/2 \rho (v{_{2}}^{2}- v{_{1}}^{2}) + \rho \int_{1}^{2} \frac{\overrightarrow{dv}}{dt}\cdot \overrightarrow{ds} + R(\overrightarrow{v})$$ They call the second term on the rhs the flow acceleration and the last term on the rhs is the viscous friction. Can someone help explain the origin of the flow acceleration term and also offer a derivation of this form of the bernoulli equation from conservation of energy. Most simple derivations of the bernoulli equation only yield the familiar C = 1/2 ρ v^2 + ρgh + p

 

[Chestermiller]

I was reading some textbooks on doppler echo for medicine and came across this version of the bernoulli equation for blood flow in the aorta. $$P_{1} - P_{2}= 1/2 \rho (v{_{2}}^{2}- v{_{1}}^{2}) + \rho \int_{1}^{2} \frac{\overrightarrow{dv}}{dt}\cdot \overrightarrow{ds} + R(\overrightarrow{v})$$ They call the second term on the rhs the flow acceleration and the last term on the rhs is the viscous friction. Can someone help explain the origin of the flow acceleration term and also offer a derivation of this form of the bernoulli equation from conservation of energy. Most simple derivations of the bernoulli equation only yield the familiar C = 1/2 ρ v^2 + ρgh + p

The simple version you wrote is for a steady state flow in which the fluid velocity at any location in the fluid is not changing as a function of time. In the first equation, the 2nd term on the rhs is a correction to allow for the velocity to be changing with time. Google "unsteady state Bernoulli equation." The third term corrects for viscous drag at the walls of the conduit. In blood flow applications, particularly capillaries, the term can dominate. Google "viscous flow in a pipe."

 

[boneh3ad]

Have you looked into trying to derive this yourself at all? I mean, just from looking at the terms you ought to be able to get a sense of what it is trying to do. The Bernoulli equation is essentially an energy balance that usually deals with energy due to static pressure, kinetic energy (dynamic pressure), and sometimes gravity. Clearly this neglects gravity, but the second RHS term ought to look pretty familiar to you in terms of an energy balance, and the last term is basically a catchall term for viscous terms.