Bernoulli Equation and Flow in a pipe - paradox

Question

I am studying Bernoulli equation and am facing a problem. The Bernoulli equation is applicable along a streamline and in steady flow condition (I guess this condition is to make sure it can be applied at all times).

Now say I want to calculate the flow velocity at a particular cross section of a pipe of varying cross sections. Now assume, I have put a Gauge pressure meter between these two particular cross sections, which gives me the change in pressure between these two cross sections along the pipe.

So, $\Delta P = \frac{\rho (v_1^2 - v_2^2)}{2} + \rho g \Delta z$

Here, $v_2$ is the velocity at the cross section at which we want to calculate the velocity. Now we know $\Delta P$, we know $\Delta z$ (assume we are calculating along a horizontal streamline).

Now interestingly, in all the literature the velocity they calculate, everyone assumes it to be uniform over the cross section. Why? The Bernoulli equation is applicable along a streamline, and every starting point will have a different ending point, and hence a different streamline. Why on earth, everyone assumes the velocity is uniform over cross section?

Answer 1

The explanation for this assumption is the same for most assumptions: because it makes the problem easier. This equation is (generally) applied to a single streamline due to the assumptions that were made when the equation was derived. Many mistakenly ascribe Bernoulli's Law to the principle of conservation of energy, when in reality it is a direct consequence of Newton's linear momentum equation. From the relatively straightforward force analysis of a differential fluid mass, it can be shown that

$$-\frac{\partial p}{\partial s}=\rho a_s=\rho v\frac{\partial v}{\partial s}, \tag{i}$$

$$+\frac{\partial p}{\partial n}=\rho a_n=\rho\frac{v^2}{R}, \tag{ii}$$

where $p$ is the static pressure, $\rho$ is the fluid density, $a$ is the local acceleration, $v$ is velocity, $R$ is the local radius of curvature, and $s$ and $n$ are curvilinear coodinates along and normal to the streamline, respectively. A partial differential is used because the pressure and velocity (in general) change in both the $n$ and $s$ directions. Now, if we limit our analysis to changes only along the streamline, we can replace the original partial differentials in (1) with exact differentials. Rearranging, this gives us

$$\frac{dp}{ds}+\rho V\frac{dV}{ds}=0, \tag{iii}$$

which can be simplified further into the classic differential Bernoulli equation:

$$\frac{dp}{\rho}+VdV=0. \tag{iv}$$

It is this version of the equation (with it's inherent assumptions) that is then integrated to give the classic textbook version of Bernoulli's Eqn. mentioned earlier.

$$p+\frac{1}{2}\rho V^2=p_0$$

Why would we do this? Well, there are several flow situations in which it is approximately valid (e.g. irrotational flows), where the stagnation pressure is uniform everywhere and needs to be calculated only once. For viscous flows, the equation can still be used to determine the stagnation pressure at a given location in the flow, but there should be no expectation that the stagnation pressures will be equal between streamlines.

Answer 2

The literature does not all assume constant velocity over a cross section. See for example the Hagen Poiseuille equation and its derivation.

Infinitesimal cylindrical shells are considered, each having a different velocity.