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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.

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

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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.
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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.

continue

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