# Can Bernoulli's theorem be applied to laminar flow?

Since in streamline flow the velocity of a particle at any point is same, Bernoulli's theorem can be applied to streamline flow, but could it be applied to laminar flow?

• What terms change with laminar compared to turbulent flow?
– user207455
May 21 '19 at 15:21

Bernoulli's theorem is applicable only if fluid is ideal, streamline flow, irrotational flow. This theorem is just energy conservation. In turbulent flow viscosity comes into play. Viscous force will dissipate energy of the system into other form such as thermal energy. But laminar flow is same as streamline flow. Hence we can apply to laminar flow.

In fact, there are many possible expressions of the Bernoulli's equation counting in various effects. I have found Benny Lautrup's Physics of Continuous Matter to be very comprehensive on the subject. Especially the enthalpy based formulation persists falls of many idealizations.

Nevertheless, the classic form

$$\frac{1}{2}\rho_0u^2 + p + mgh =\text{const.}$$

does not require any particular kind of flow organization (that would be strange), but there conditions on the physical magnitudes and their treatment such as

• zero viscosity
• incompressible fluid
• validity on just one streamline or vanishing vorticity within the fluid (this one is extra hard to fullfill)

Nevertheless, as I have pointed out above, there are ways to work around. The trade off are more complicated expressions providing a bit lower degree of intuition on the system behavior.

The simple Bernoulli equation as written above DOES NOT apply to laminar flow in a pipe. The reason is the velocity profile across the pipe in laminar flow is not uniform, it's parabolic. Therefore, the Bornoulli term 1/2 v^2 averaged over the cross section of the pipe is not equal to 1/2 (V)avg^2. ( Vavg is typically what one is referring to as "flow velocity" in terms of mass flow rate/rho*A). The precisely correct Bernoulli equation over the area of a pipe has a kinetic energy correction factor "alpha" in front of the kinetic energy term to correct for this integration error. The reason alpha may be ignored for turbulent flow is it's close to = 1.0 because turbulent flow is mostly flat across a pipe's cross section, but for laminar flow this is not true: (V)avg = 1/2 Vmax at the center. It turns out, Alpha for laminar flow = 2. See Victor Ugaz's of Texas A&M lecture 6.1 on U-tube.

• Please use LaTeX equations for math formatting. Dec 28 '20 at 8:58