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#9 The optimal concentration of nectar-IV

Please finish reading the episode 8: http://biophys3min.blogspot.tw/2016/08/8-optimal-concentration-of-nectar-iii.html
Today we will talk about how to estimate the thickness of soy sauce on sashimi and the LLD theory. We will use a lot dimensional analysis.
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We have previously stated that, according to the fixed power hypothesis, the average volumetric flow of nectar obtained by viscous dipping follows:
Today we will talk about how to estimate "e." Obviously, the speed of retraction, the viscosity of nectar, how great the effect of gravity is, and the dimension of the mouthpiece would change the essence of the physical process. In 1922, Goucher and Ward first noticed the 2 main forces governing the mechanism of dipping sashimi into soy sauce. Because of viscous forces and the no-slip boundary condition, the liquid in direct contact with the sashimi must have their relative speed equal 0, and that is also why the sashimi would take some soy sauce away. However, the sashimi deforms the interface between air and soy sauce when we try to take it out. This deformation changes the area of interface and would therefore be resisted by surface tension. Assume the viscosity of soy sauce μ, the retraction speed of sashimi u,  the surface tension of soy sauce σ. Goucher & Ward defined a dimensionless quantity called "capillary number," or abbreviated as Ca (not calcium.)
The larger the capillary number, the more soy sauce we could get. Because the capillary number is dimensionless while the thickness of the soy sauce on sashimi e has unit of length.  If we could find a characteristic length L, we could reasonably expect the thickness of soy sauce to be expressed in the form:
in which f(Ca) is a function that we don't know yet. It should be an increasing function.

Fig. 1. Schematic drawing of the mouthpiece retracting from the liquid surface.

Now let's look back to our nectar problem. As shown in figure 1, for a column-shaped mouthpiece with radius a and retraction speed u, the transitional zone should look like this. According to the Laplace law, the liquid attached to the surface of mouthpiece will have a pressure difference from the reservoir with magnitude
Assume the length of transitional zone λ. If the liquid layer is very thin relative to the radius of mouthpiece, the pressure gradient in transitional zone could be estimated as
The viscous stress could be estimated as μu/e while the gradient of viscous stress could be estimated as μu/e^2. So the balance between surface tension gradient and the viscous stress gradient could be written in
Now there are 2 variable  λ & e but we have only 1 equation. The physicists before Landau, Levich and Derjaguin falsely assume λ to be a constant = radius a. However, this would yield a wrong scaling properties of volumetric flow.

Then what should λ be? We should note that both the liquid in transitional zone and those attached to mouthpiece have curvature, so there must be an associated pressure generated from this curvature. Laudau, Levich, and Derjaguin assumed that the magnitude of  λ depends on the balance between the pressures generated from these 2 curvatures. The pressure generated from the curvature of liquid attached to the mouthpiece is stated above to be σ/a. What about the pressure generated in transitional zone? Recall:
So the problem is equivalent to the estimation of curvature. We have stated in episode 3 (http://biophys3min.blogspot.tw/2016/08/3-scaling-law-of-locomotion-ii.html) that the curvature could be calculated by
since the thickness e is different along our coordinate x. By dimensional analysis, curvature could be estimated as
So the balance between the 2 pressures could be written as:
and we will get
After some simple manipulation, you should be able to derive the thickness of liquid layer to be
and we substitute back to our equation Q ^(-1/2) and we will have

Recall in episode 7 (http://biophys3min.blogspot.tw/2016/08/7-optimal-concentration-of-nectar-ii.html) we have mentioned that the energy flow rate E’ sQ, in which s means the concentration of sugar. So we could derive the optimized sugar concentration of nectar by maximizing
And we will found the optimal concentration to be 50%, which is different from the optimal concentration for the active sucker and capillary sucker (30 ~ 40%). However, in the real world, the concentration of nectar they are taking is not as high as we calculated (the observed nectar concentration taken by butterflies is about 20 ~ 25% while the one taken by honeybees is about 35%). Yet the concentration of nectar honeybees are taking is still much larger than the one took by butterflies. There are many reasons that might account for such difference. Among them, the most important one is probably that it is not beneficial for a flower to produce such concentrated nectar because the animal would be full enough and won't fly to the next flower. Flowers produce nectar to ask animals for helping their pollination rather than produce it for fun and try to make themselves generous.

In summary, the predicted optimal concentration of nectar is a bit different from the observed concentration. However, it successfully predicted that viscous dipper should take more concentrated nectar than active sucker or capillary sucker. They also explained the scaling law of average volumetric flow rate of nectar observed previously.

If we falsely assume λ = radius a, you will get Q μ^(-1/4). This is different from the experimentally observed Q μ^(-1/6).

*Suggested Reading:
W. Kim, T. Gilet, and J. W. M. Bush. Optimal concentrations in nectar feeding. PNAS 108(40): 16618-16621. (2011)
*Reference about the Landau, Levich, and Derjaguin Theory
D. Quer´e´. Fluid coating on a fiber. Annu. Rev. Fluid Mech. 31:347–84. (1999)