Curl and Cauchy-Riemann Conditions problem

In summary, the conversation discusses the expression for the velocity of a two-dimensional flow of liquid and how to verify that the incompressibility and irrotationality of the flow lead to the equations \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} and \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}. The solution involves using the continuity equation and the fact that the liquid is incompressible.
  • #1
Saketh
261
2
Problem

The velocity of a two-dimensional flow of liquid is given by

[tex]
\textbf{V} = \textbf{i}u(x, y) - \textbf{j}v(x, y).
[/tex]​

If the liquid is incompressible and the flow is irrotational show that

[tex]
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}
[/tex]​

and

[tex]
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
[/tex]​

My Work

I evaluated [itex]\nabla \times \textbf{V} = 0[/itex] through a determinant, and ended up with this expression:

[tex]\textbf{i}\frac{\partial v}{\partial z} + \textbf{i}\frac{\partial u}{\partial z} - \textbf{k}\left ( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial z} \right ) = 0
[/tex]​

Through this, I was able to verify:

[tex]
\frac{\partial u}{\partial y} = -\frac{\partial{v}}{\partial x}
[/tex]​

I could not verify the other expression. How can I verify the other expression - I've tried everything I can think of. It seems simple, but I am missing something.

Thanks in advance.
 
Last edited:
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  • #2
Hello Saketh,

to get

[tex]
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}
[/tex]

you need to use the fact that div V = 0 for incrompressible liquids.

This follows from the continuity equation:

[tex]
\frac{\partial \rho}{\partial t} + \nabla(\rho\,\textbf{V})= 0
[/tex]

and using the fact that the liquid is incompressible, hence the mass density [itex]\rho=const.[/itex]

Regards,

nazzard

P.S.: There's a small typo in your current solution: one \frac command is missing
 
Last edited:
  • #3
Oh, I thought that incompressible just meant that the fluid flow is behaving ideally, and had no mathematical significance.

Now that you tell me [tex]\nabla \cdot \textbf{V} = 0[/tex], the answer is obvious. Thanks!
 

Related to Curl and Cauchy-Riemann Conditions problem

1. What are the Curl and Cauchy-Riemann Conditions?

The Curl and Cauchy-Riemann Conditions are two mathematical conditions used to determine if a given function is analytic. These conditions are used in complex analysis, a branch of mathematics that deals with functions of complex variables.

2. How are the Curl and Cauchy-Riemann Conditions related?

The Curl Condition and the Cauchy-Riemann Condition are closely related and are often used together. The Curl Condition is a necessary condition for a function to be analytic, while the Cauchy-Riemann Condition is both necessary and sufficient for a function to be analytic.

3. What is the significance of the Curl and Cauchy-Riemann Conditions in complex analysis?

The Curl and Cauchy-Riemann Conditions are important because they allow us to determine if a function is analytic, which means it can be represented by a power series. Analytic functions have many useful properties and are used in a variety of applications, such as in physics, engineering, and economics.

4. How are the Curl and Cauchy-Riemann Conditions used to solve problems?

The Curl and Cauchy-Riemann Conditions are used to check if a given function satisfies the necessary conditions for being analytic. If the conditions are satisfied, the function can then be represented by a power series, which can be used to calculate its values at any point. This allows us to solve complex analysis problems involving analytic functions.

5. Can the Curl and Cauchy-Riemann Conditions be applied to other fields of mathematics?

Yes, the Curl and Cauchy-Riemann Conditions have applications in other fields of mathematics, such as differential geometry and differential equations. In these fields, they are used to study the properties of vector fields and to solve certain types of differential equations.

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