# “kawaii” in DE (^_^)

JK! Warotan.

Recently, the Japanese word “kawaii” is getting popular around the world.
It means cute, little, lovely and so on.

On the other hand, it is well-known that an advection term in the incompressive Navier-Stokes equation is kawaii.
Yes! DE of the title is short for a Differential Equation!!
This page pursues kawaisa which is the noun of kawaii in a DE, trying to define kawaii equation or term with some mathematical quantities.

I pursue kawaisa in a DE, so the term should have some kind of differentiation.
I would like to give some physical image to the term if I can. (maybe it’s difficult.)

Why am I writing this kind of a very humorous page?
Well, some exchange on twitter with my follower made me do so.

It’s common to tweet that the Navier-Stokes equation is kawaii and I did so,
then one of my followers replied to it and I said
“wait until I make a very kawaii equation!!”
That brought me to write this foolish page (o_o)ha.

## Kawaisa in DE

First, I would like to define a quantity to calculate kawaisa from an objective and quantitative view.
However, the concept of kawaisa is very subjective and difficult to quantitate.

Therefore,
starting from the incompressive Navier-Stokes equation which is thought of kawaii equation by the public and considering why it is,
we will try to quantitate kawaisa in a DE.

## An incompressive Navier-Stokes equation

The incompressive Navier-Stokes equation LOOKS as follows;

$$\begin{eqnarray} \frac{\partial {\bf u}}{\partial {\rm t}}+\left({\bf u}\cdot\nabla\right){\bf u} = -\frac{1}{\rho}\nabla p+\nu\Delta {\bf u}+{\bf K}, \end{eqnarray}$$

and this time we seek for kawaisa in a DE,
so we do not go into the physical meanings of each term.

In public, people think the second term of the left hand side called an advection term is kawaii;

$$\begin{equation} T_{{\rm cute}} = \left({\bf u}\cdot\nabla\right){\bf u} \end{equation}$$

I did trial and error to find which of $${\bf u}$$ or $${\bf v}$$ is kawaii when I wrote this equation.
I think roundness of a character is important for kawaisa.
At this stage, here is the key to defining kawaisa of the NS equation.

I sent out questionnaires on twitter as a test

The number of samples is too few to count on,
and we should interpret it as either is ok for the moment.

By the way, do we as a human watch some kawaii things in details when we think so?
The total balance is important, not the details and what’s more,
the roundness is important. 1(e.g. marumoji in Japanese)

Here we consider kawaisa in DE as a function of total roundness and the pattern of the characters should remind us emoticons.

## Defining kawaisa in DE quantitatively

Above all, kawaisa in DE can be defined quantitatively.
However, what kind of mathematical quantity should we use?

First of all, the curvature of the characters came up to our mind.
But it is difficult to define because we can’t calculate the curvature of an unsmooth line.
In addition to that, it is not so easy to get some relationships between the curvature and kawaisa and seems to be unsolvable.

Therefore, let us consider two tangential lines from any two points on a character and define the maximum angle $$\theta$$ at the cross point as the desired value.
For example, the above $${\bf u}$$ has $$\theta_{{\rm max}}=\pi /2$$; 2 tangential lines giving the maximum angle $$\theta_{{\rm max}}$$

We also should put parentheses around the characters as it looks like emoticons.

## Candidates of kawaii alphabets

This definition of kawaii equation makes it difficult to determine the most kawaii term.
If we list them of $$\theta_{{\rm max}} \neq \pi /2$$, we get only 7 characters
i,k,l,v,w,x,z.
The rest 19 characters can be kawaii.

OK. Let’s write some.

$$\begin{eqnarray} T_{{\rm adv}} &=& \left({\bf a}\cdot\nabla\right){\bf a} \\ T_{{\rm adv}} &=& \left({\bf f}\cdot\nabla\right){\bf f} \\ T_{{\rm adv}} &=& \left({\bf a}\cdot\nabla\right){\bf f} \end{eqnarray}$$

Uh-oh…!?
All looks kawaii!?

Especially

$$\begin{equation} T_{{\rm adv}} = \left({\bf a}\cdot\nabla\right){\bf f} \end{equation}$$
is so kawaii!!

My goodness!!

## Summary

Now, that’s all what I wanted to write.

This time, I tried to pursue kawaisa in DE but finally I was trapped in kawaisa of the advection term unnoticeably.

But that led me to write some funny blogs, so it’s ok.