# Differential equation and fluids

• Niles
In summary, the conversation is about a question involving a differential equation and a link to a book. The person asking the question is struggling with how to show that a function satisfies the differential equation, and another person is offering to help but needs more information and clarification. They mention applying an operator twice and putting the proposed solution into the equation to arrive at the required ordinary differential equation. The person asking for help is asked to show their work and provide more context for the question.
Niles

## Homework Statement

Hi guys

Please take a look at the second question in 10.20 (the question with the diff. equation):

I have found the expression for the operator E, but how do I show that f(r) satiesfies the diff. equation?

## The Attempt at a Solution

First thing would be to insert "psi"(r,theta) in E - but where to go from here? I am very lost.

sincerely Niles.

Niles, It would be nice if you showed the work done for obtaining the differential operator E. It is not explicitly given in your link and I have no clue whether my result is correct or not. At a certain point in time I was pressed with my nose against the screen trying to read the link because the thing does not zoom in properly. It's a fairly bad reading link. It would help to post the question in it's entire form and also to show some of your work. Principally it is just applying the operator twice to the stream function and the equation you get then has the proposed solution $$\Psi(r,\theta)=f(r)\cdot sin^2(\theta)$$. The only thing to do is to put it in the equation and you will arrive at the required ordinary differential equation. As stated, I can't check anything over here because I don't have the book and it seems it is a rather lengthy algebraic journey, which I will undertake to help you, but I need to make sure that my operator is correct.

## 1. What are differential equations?

Differential equations are mathematical equations that describe how a quantity changes in relation to other variables. In the context of fluid mechanics, they are used to model the behavior of fluids and how they flow and interact with other objects.

## 2. What is the significance of differential equations in fluid dynamics?

Differential equations are essential in fluid dynamics because they allow us to mathematically describe and predict the behavior of fluids. They help us understand how forces, pressure, and other factors influence the flow of fluids and how this affects their movement and properties.

## 3. How are differential equations used in analyzing fluid flow?

Differential equations are used to develop mathematical models of fluid flow, which can then be solved to predict the behavior of fluids. These equations take into account various factors such as viscosity, pressure, and velocity to describe how fluids move and interact with their surroundings.

## 4. Are there different types of differential equations used in fluid dynamics?

Yes, there are different types of differential equations used in fluid dynamics, such as Euler's equation, Navier-Stokes equation, and Bernoulli's equation. Each equation has its specific use and assumptions, and they all play a crucial role in understanding and analyzing fluid flow.

## 5. Can differential equations be solved analytically for all fluid flow problems?

No, not all fluid flow problems can be solved analytically using differential equations. In some cases, the equations may be too complex to solve, or there may not be a known closed-form solution. In such cases, numerical methods and computer simulations are used to approximate the solution.

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