＜3−1＞ a irrotational flow(vector)
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 belowｄχ＝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.ｄχ＝∂χ／∂ｘ・ｄｘ＋∂χ／∂ｙ・ｄｙ
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.
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,
So, the characteristic of the nondivergent flow (vector) F can be shown by the following expression.
And you can rewrite as below
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
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).
And we can get the following equation from the definition of the total differential.
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?continue
＜3−3＞ Classification of flowsFrom 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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