Fluid Flows - Complex potential function and general equation of a streamline

In summary, the conversation revolves around a complex potential function and a stream function for a given problem. The speaker is seeking help in determining the equation of a general streamline for the problem. Another person offers their method of solving the problem, using differential equations and integrating to find the stream lines and equi-potential lines. The speaker is then advised to determine a constant in order to find the equation of a specific stream line.
  • #1
fr33d0m
4
0
Hi All

I'm having some problem with this question and was hoping that someone could help me with it. I think I have the first two bits but the rest I'm totally stuck
q2a.JPG


For part (ii) I have the complex potential function as (3x^2)/2 - (3y^2)/2 +6y and the stream function as x(3y-6). Is this correct and if so, how do I use these to calculate part (iii). In other words how can i determine the equation of a general streamline for q? Can someone please help me?


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  • #2
It's been a long time since I have done problems like that but here's how I would do it (perhaps awkwardly):
In the complex plane, writing z= x+ iy, the velocity function, [itex]q(z)= dx/dt+ i dy/dt= \overline{z}+ 6i= x- iy+ 6i= x+ (6-y)i[/itex] which gives the two differential equations dx/dt= 3x and dy/dt= 6- 3y. You can integrate those directly or divide one by the other to eliminate t and get dy/dx= (6-3y)/3x. I would be inclined to go ahead and cancel "3"s. dy/dx= (2-y)/x. That is a separable equation: dy/(2-y)= dx/x and integrating -ln(2-y)= ln(x)+ C or 1/(2-y)= C'x so x(2-y)= C', a family of hyperbolas. That is, except for a factor of 3 which is now incorporated in the constant, what you have for the stream lines. The "equi-potential" lines are always orthogonal to the stream lines so they are given by
dy/dx= -x/(2- y). (2-y)dy= -xdx so 2y- y2/2= -x2/2+ C That is the same as x2/2- y2/2+ 2y= C or x2- y2+ 4y= C' (C'= 2C), the family of hyperbolas orthogonal to the stream lines. That is also, except for the factor of 3, what you have.

To answer (c), determine C' in x(2-y)= C'. For example, the stream line through (0,0) has both x and y= 0 so 0(2-0)= C'. C'= 0 so x(2-y)= 0 which is only satisfied by (0,0) (that's your "degenerate" stream line). The stream line through 1+i= (1,1) has x=y= 1 so 1(2-1)= C'. C'= 1 and the stream line is the hyperbola x(2-y)= 1. From dx/dt= x, we see that, at x= 1> 0, x is increasing. The direction of flow is to the right.
 

1. What is a complex potential function in fluid flow?

A complex potential function is a mathematical function used to describe the flow of a fluid, specifically in two-dimensional, inviscid, and irrotational flow. It is a combination of the velocity potential function, which describes the speed and direction of flow at a given point, and the stream function, which describes the flow around a body.

2. How is a complex potential function related to the general equation of a streamline?

The general equation of a streamline is derived from the complex potential function. It is a contour plot that represents the path of a fluid particle in a flow field. The real part of the complex potential function gives the equation of the streamline, while the imaginary part represents the stream function.

3. What is the significance of the Cauchy-Riemann equations in fluid flow?

The Cauchy-Riemann equations are a set of conditions that must be satisfied for a complex function to be analytic and have a complex potential. In fluid flow, these equations ensure that the flow remains irrotational and that the streamlines are continuous and without any breaks or jumps.

4. How does the complex potential function help in solving fluid flow problems?

The complex potential function simplifies the equations for fluid flow, making it easier to solve complex problems. It also allows for the use of analytical techniques, such as the method of conformal mapping, to study and analyze fluid flow around different objects.

5. Can the complex potential function be used for all types of fluid flow?

No, the complex potential function is only applicable to ideal, two-dimensional, inviscid, and irrotational flow. In real-world scenarios, fluid flow is often more complex and may require the use of other mathematical models and equations to accurately describe it.

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