Bernoulli's Equation

Rewrite (2.16) as

$${\partial {\bf v}\over \partial t} + \nabla \left ({1\over ... ...bla \times {\bf v}) = -{1\over \rho} \nabla p - \nabla (gz),$$

where $({\bf v}\cdot \nabla ){\bf v} = \nabla ({1\over 2}v^{2}) - {\bf v}\times (\nabla \times {\bf v})$ and ${\bf g} = -\nabla (gz) = - g{\bf e}_{z}$. Thus,

$${\partial {\bf v}\over \partial t} - {\bf v}\times (\nabla ... ...\nabla \left ({p\over \rho} + {1\over 2} v^{2} + gz\right ).$$ (2.19)

When the vorticity $\omega = \nabla \times {\bf v}$ the flow is called irrotational. If, in addition, the flow is steady, (2.19) reduces to

$$0 = - \nabla \left ({p\over \rho} + {1\over 2} v^{2} + gz\right ).$$

or, on integrating,

$${p\over \rho} + gz + {1\over 2} v^{2} = \hbox{ constant.}$$ (2.20)

This is called Bernoulli's equation. It says that the total energy (pressure plus gravitational plus kinetic energy) is constant. The implication is that the pressure falls where the fluid flows faster and visa-versa. For example, if neglect gravity, ${\bf g} = 0$, then for the air flow over an aeroplane wing we find that, as shown in Figure 2.6,

Figure 2.6: Airflow over an aeroplane wing. Flow is faster of the upper surface of the wing and so the pressure is lower there.

the air travels further over the top surface than the lower surface. Hence, it must travels faster over the top surface. From Bernoulli's equation (2.20) we see that if the velocity is higher then the pressure must be lower. With a lower pressure on the upper surface and a higher pressure on the lower surface, there is a vertical pressure force that generates the lift.