# 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 at 15:21
• For flow to be laminar, the fluid needs to have some viscosity. But Bernoulli's Principle applies to inviscid fluids only. These flow turbulently at any flow speed. – Gert May 21 at 16:10
• "Since in streamline flow the velocity of a particle at any point is same, Bernoulli's theorem can be applied to streamline flow..." This is bad logic. You can have streamlines in viscous flow, for example viscous flow around a sphere, but Bernoulli's equation cannot be applied in these scenarios. – Drew May 21 at 18:17
• @Gert Inviscid flows do not "flow turbulently". Turbulence is a viscous effect. – D. Halsey May 21 at 18:39

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