The potential function of a flow is log (x2 + y2 ), then construct the flux function and complex potential function.
Answers
Answer:
The definition of Cauchy–Riemann equations can lead to the definition of the complex potential F(z) as following
F(z)=ϕ(x,y)+iψ(x,y)(10.4.1.1)
where z=x+iy . This definition based on the hope that F is differentiable and continuous in other words analytical. In that case a derivative with respect to z when z is real number is
dFdz=dFdx=dϕdx+idψdx(10.4.1.2)
On the other hand, the derivative with respect to the z that occurs when z is pure imaginary number then
dFdz=1idFdy=−idFdy=−dϕdy+dψdy(10.4.1.3)
Equations (2) and (3) show that the derivative with respect to z depends on the orientation of z . It is desired that the derivative with respect z will be independent of the orientation. Hence, the requirement is that the result in both equations must be identical. Hence,
∂ϕ∂x=∂ψ∂y∂ϕ∂y=−∂ψ∂x(10.4.1.4)
In fact, the reverse also can be proved that if the Cauchy–Riemann equations condition exists it implies that the complex derivative also must be exist. Hence, using the complex number guarantees that the Laplacian of the stream function and the potential function must be satisfied (why?). While this method cannot be generalized three dimensions it provides good education purposes and benefits for specific cases. One major advantage of this method is the complex number technique can be used without the need to solve differential equation. The derivative of the F is independent of the orientation of the z
W(z)=dFdz(10.4.1.5)
This also can be defined regardless as the direction as
W(z)=dFdx=∂ϕ∂x+i∂ψ∂x(10.4.1.6)
Using the definition that were used for the potential and the stream functions, one can obtain that
dFdz=Ux−iUy(10.4.1.7)
The characteristic complex number when multiplied by the conjugate, the results in a real number (hence can be view as scalar) such as
WW¯¯¯¯¯=(Ux−iUy)(Ux+iUy)=Ux2+Uy2(10.4.1.8)
In Bernoulli's equation the summation of the squares appear and so in equation (??). Hence, this multiplication of the complex velocity by its conjugate needs velocity for relationship of pressure–velocity. The complex numbers sometimes are easier to handle using polar coordinates in such case like finding roots etc. From the Figure the following geometrical transformation can be written
Ux=Urcosθ−Uθsinθ(10.4.1.9)
and
Ux=Ursinθ+Uθsinθ(10.4.1.10)
Using the above expression in the complex velocity yields
W=(Urcosθ−Uθsinθ)−i(Ursinθ+Uθcosθ)(10.4.1.11)
Combining the r and θ component separately
W=Ur(cosθ−isinθ)−Uθ(cosθ−isinθ)(10.4.1.12)
It can be noticed the Euler identity can be used in this case to express the terms that, are multiplying the velocity and since they are similar to obtain
Answer:
the potential function is log(
2 + 2) , find the complex potential function