Bathtub Vortex Pressure difference

[Timtam]

The problem statement

Example 1 A single particle
I have a particle being forced by a radial centripetal force onto a smaller radius

$$l_1=m_c.v_1.r_1$$
$$L_1=L_2$$
$$L_2=m_c.v_2.r_2$$
$$m_c.v_1.r_1=m_c.v_2.r_2$$
$$v_1.r_1=v_2.r_2$$
$$v_2=v_1\frac{r_2}{r_1}$$

Its increase from $v_1$ to $v_2$ is explained by Conservation of Angular Momentum
Example 2 The bathtub vortex-A particle in a fluid

I have the same particle entering a control volume- with the same angular momentum $L_1=m_c.v_1.r_1$ as its drawn towards the drain its angular velocity increases as radius decreases $v=\frac{1}{r}$

$$v_1=\frac{1}{r_1}$$
$$v_1.r_1=1$$
$$v_2=\frac{1}{r_2}$$
$$v_2.r_2=1$$
$$v_2.r_2=v_1.r_1$$
$$v_2=v_1\frac{r_2}{r_1}$$
So in both examples the increase in velocity is explained just by Conservation of Angular momentum

Yet applying Bernoulli's and Energy conservation the increase in velocity is explained by a proportional decrease in pressure ...and we do see a pressure decrease in a vortex . (This pressure gradient ,once created is also explained to be the radial force)

My question
If the increase in Angular velocity is explained just by Conservation of Momentum - Why does the pressure decrease?

Shouldn't such a velocity change, due to a pressure change, be in addition of the radius change and the pressure change ?

Where does that decrease in potential energy go if not into an additional increase in velocity (kinetic energy) over the one expected by Angular Momentum Conservation ?

 

[boneh3ad]

These ideas are two sides of the same coin. At any rate, the pressure gradient pointing toward the center of the vortex must be negative because the resultant force must serve as the centripetal force that holds a particle (or fluid parcel, if you will) into its circular "orbit" around the center. If a particle takes a trajectory that causes it to move with or against that gradient, then it will speed up or slow down accordingly.

 

[Timtam]

Ah thanks @boneh3ad ! Ok I think I understand it , the pressure difference is purely radial so contributes only a radial acceleration , in the stream wise direction there is no pressure gradient so no acceleration (increase in velocity) in the stream wise direction. Is this correct

I am still a little unsure of what mechanism causes it to develop in the first place ??

 

[boneh3ad]

That's the general idea for a generic vortex. For a bathtub vortex, you've also got the effect of having mass being lost to the "sink' in the center. If you think about it in terms of potential flow, for example, you would have a superposed point vortex and point sink in order to model that flow. Then the flow field is a little bit more complicated but still fairly simple in the grand scheme of things.