3.Helmholtz Decomposition theorem is wrong(Chapter 3)

3. Classification of flows

＜3−1＞ a irrotational flow(vector)

The rotation of the flow F(=ui+vJ) is given as follow

∇×F=∂ｖ／∂ｘ−∂u／∂ｙ

So, the characteristic of the irrotational vector(flow)F can be shown by the following expression.

∇×F=∂ｖ／∂ｘ−∂u／∂ｙ＝０　　　・・・・・・・・・・・3.1）

The expression 1) is a necessary and sufficient condition for that there exist some scalar function χ which is given as below

ｄχ＝ｕ・ｄｘ＋ｖ・ｄｙ　　　・・・・・・・・・・・・・3.2）

That is, not having rotation is a necessary and sufficient condition for that the flow include χ、that is ∇・W in (C.4) in the appendix C in the book written by Holton.

In other wards, you can say that if there exist some rotation in the flow, the flow does not include χ, or ∇・W.

You can get below expression from the definition of the total differential.

ｄχ＝∂χ／∂ｘ・ｄｘ＋∂χ／∂ｙ・ｄｙ

Then, we can get the following expressions for an irrotational flow.

ｕ＝∂χ／∂ｘ , ｖ＝∂χ／∂ｙ ・・・・・・・・・・・・・3.3）

Here, you can get the x-component and y-component of the flow by differentiating χ. So, the function χ is called Potential velocity.

＜3−2＞　a nondivergent flow(vector)

　The divergence of flow F(=ui+vJ) is given as follow,

div F＝∂ｕ／∂ｘ＋∂ｖ／∂ｙ

So, the characteristic of the nondivergent flow (vector) F can be shown by the following expression.

div F＝∂ｕ／∂ｘ＋∂ｖ／∂ｙ＝０

And you can rewrite as below

∂ｕ／∂ｘ−∂（−ｖ）／∂ｙ＝０　　 　　・・・・・・・・・3.4）

The expression 3.4) is a necessary and sufficient condition for that there exist some scalar function A and the total differential of it is given as below

ｄA＝ｕ・ｄｘ＋(-ｖ)・ｄｙ　　　・・・・・・・・・・・・・3.5）

So, not having divergence is a necessary and sufficient condition for that the flow include scalar function A given with 3.5).

In other wards, if the flow has divergence, there does not exist A.

Here we consider vector A(0,0. A).

Then ∇×A = (∂A／∂y)I -(∂A／∂x)j.

Where you can put A(c,c,A) instead A(0,0,A). c is constant for space.

Then if scalar A exist, we can consider the vector function A(0,0,A).

Then, you can say that if there exist some divergences in the flow, the flow does not include A. Then W given in the (c.4) does not exist too.

　And we can get the following equation from the definition of the total differential.

ｄA＝∂A／∂ｘ・ｄｘ＋∂A／∂ｙ・ｄｙ

Then, we can get the following expressions for a nondivergent flow.

ｕ＝∂A／∂y , -ｖ＝∂A／∂x ・・・・・・・・・・・・・3.6）

If you draw A on x-y plane, the isolines of A shows the direction of the flow. So A is called stream function.

Have you known the relationship between A and stream function which is given as a scalar function?

＜3−3＞　Classification of flows

From the two examinations above-mentioned, all flows can be classified into the following four.

The first one is nondivergent and irrotational. That flow has χ and A. That is, the flow has both velocity potentials and the stream functions in itself.

The second one is divergent and irrotational. That flow has only χ or velocity potentials.

The third one is nondivergent and rotational. That flow has only A or stream function.

And the forth flow is divergent and rotational. That flow does not have neither χ nor A.

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