### #7 The optimal concentration of nectar-II

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Please finish reading episode 6 before reading this episode. Today we are going to talk about the capillary suction.

Suggested Video
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In previous episode, we have written down the equation governing the nectar with density ρ, viscosity μ and mean flow velocity u in a mouthpiece with length L, and inner radius a according to Newton’s second law:
$(\pi&space;a^2L\rho)\textup{d}u/\textup{d}t&space;=&space;+\Delta&space;P(\pi&space;a^2)&space;-&space;(\pi&space;a^2L\rho)g&space;-&space;8\pi\mu&space;uL$

For capillary sucker such as hummingbirds, they lack the ability to produce negative pressure (recently there are some researches finding against this hypothesis and we will talk about it later), so the pressure force term = 0. However, there is an extremely narrow tunnel on their tongue which allows fluid to arise by capillary flow. Assume the contact angle between nectar and tongue θ, and the surface tension of nectar σ, the original equation now reads:
$(\pi&space;a^2L\rho)\textup{d}u/\textup{d}t&space;=&space;+2\pi&space;a\sigma&space;\cos\theta&space;-&space;(\pi&space;a^2L\rho)g&space;-&space;8\pi\mu&space;uL$

And we use the same reasoning to discard the terms considering inertial force and gravitational force as we did in previous episode. The equation now reads
$0&space;=&space;+2\pi&space;a\sigma&space;\cos\theta&space;-&space;8\pi\mu&space;uL$

Now here is the problem. The capillary sucker must retract their tongue after the whole narrow tunnel is filled with nectar in order to permit the next filling. (Please refer to the suggested video to observe this process.) Namely the height of liquid column is a function of time, or h=h(t). And since u=h’(t), the above equation could now be written as
$0&space;=&space;+2\pi&space;a\sigma&space;\cos\theta&space;-&space;8\pi\mu&space;hh'$
With the initial condition h(0)=0, it is easy to solve (using hh’ = (0.5h^2)’) h(t)
$h(t)=&space;(a\sigma&space;t\cos\theta/2\mu)^{1/2}。$

Now let’s consider the average volumetric flow rate Q. Assume the time required to fill up the narrow tunnel on the tongue T, and the time required for tongue retraction and re-protrusion T0. The average volumetric flow rate Q could be calculated as
$Q&space;=&space;\frac{\pi&space;a^2h(T)}{T+T0}&space;\propto&space;\frac{(T/\mu)^{1/2}}{T+T0}$
According to previous experiments, (T)^0.5 / (T+T0) does not depend on viscosity. Therefore we now know the power law between average volumetric flow rate and viscosity is
$Q&space;\propto&space;\mu^{-1/2}$
Which is the same as the active sucker and is supported by experiments.

Someone may have noticed that we automatically neglect the effect of capillary flow in the governing equation of active sucker. This could be readily justified by calculation the ratio between 2 forces:
$\frac{2\pi&space;a\sigma&space;\cos\theta}{\Delta&space;P(\pi&space;a^2)}&space;=&space;\frac{2\sigma&space;\cos\theta}{\Delta&space;Pa}&space;\sim&space;\frac{\sigma}{\Delta&space;Pa}$

That is to say, the greater the inner radius, the smaller the contribution from capillary forces. The surface tension of water is about 70mN/m, the negative pressure generated by butterflies is about 10kPa, and the inner radius of the mouthpiece of butterflies is 100μm. The above ratio is about 0.1 and is therefore negligible.

Nonetheless, recent literatures (see reference) suggested that hummingbirds could generate negative pressure by actively changing the shape of the narrow tunnels in their tongues. However, both hypothesis yield the power law Q μ^(-1/2) and is therefore not differentiable with this approach.

Besides, what truly matters for creatures is the energy flow rate rather than the volumetric flow rate. Assume the energy density of sugar c, density of nectar ρ, and the sugar concentration s. It is readily to show that
$E'=c\rho&space;sQ$
Previous experiments show that the density of nectar varies negligibly with the concentration of sugar. We could therefore write down
$E'\propto&space;s\mu(s)^{1/2}$
With the experimental results of μ(s), the researchers found that the optimal nectar concentration is about 30% ~ 40%.

In the next episode, we will discuss the physics of viscous dipper. Stay tuned!
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