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April 10, 2019 at 11:02 am |

so many situations are given here for thus observed prof dr mircea orasanu and prof drd horia orasanu and followed that these can be developments

May 1, 2019 at 12:37 am |

here we mentionthat can be appear other as observed many and that submitted to prof dr mircea irasanu and prof drd horia orasanu as followed with important

May 1, 2019 at 12:38 am |

and

May 7, 2019 at 2:20 pm |

now must mentions other as is natural indeed and stated prof dr mircea orasanu and prof drd horia orasanu and followed withidea is that if something isn’t an element of both and , then it must be in the complement of one or the other set. This leads to the following result for sets :

Again, I won’t prove this, but the idea is as follows: if something is an element of the left hand side, then it must be in the complement of at least one of the which means it can’t be in ALL of them and therefore can’t be in the intersection.

July 10, 2019 at 11:01 am |

in these cases are mentioned the situations as observed prof dr mircea orasanu and prof drd horoa orasanu and followed with As we have seen, a two-dimensional velocity field in which the flow is everywhere parallel to the – plane, and there is no variation along the -direction, takes the form

(5.16)

Moreover, if the flow is irrotational then is automatically satisfied by writing , where is termed the velocity potential. (See Section 4.15.) Hence,

(5.17)

(5.18)

On the other hand, if the flow is incompressible then is automatically satisfied by writing , where is termed the stream function. (See Section 5.2.) Hence,

(5.19)

(5.20)

Finally, if the flow is both irrotational and incompressible then Equations (5.17)-(5.18) and (5.19)-(5.20) hold simultaneously, which implies that

(5.21)

(5.22)

It immediately follows, from the previous two expressions, that

(5.23)

or

(5.24)

Likewise, it can also be shown that

(5.25)

We conclude that, for two-dimensional, irrotational, incompressible flow, the velocity potential and the stream function both satisfy Laplace’s equation. Equations (5.21) and (5.22) also imply that

(5.26)

In other words, the contours of the velocity potential and the stream function cross at right-angles.

The function is known as the stream function. Moreover, the existence of a stream function is a direct consequence of the assumed incompressible nature of the flow.

Consider two points, and , in addition to the fixed point . (See Figure 5.2.) Let and be the fluxes from right to left across curves and . Using similar arguments to those employed previously, the flux across is equal to the flux across plus the flux across . Thus, the flux across , from right to left, is . If and both lie on the same streamline then the flux across is zero, because the local fluid velocity is directed everywhere parallel to . It follows that . Hence, we conclude that the stream function is constant along a streamline. The equati