# Fluid Flow: Principal Rates of Deformation/Principal Axes

• jhuleea
In summary, the conversation discusses a problem regarding finding the principal rates of deformation and principal axes for a specified flow, along with satisfying a continuity equation. The conversation also includes a solution attempt and a question about the correct method to proceed. One participant suggests that the solution seems correct, with the exception that the continuity equation may need to be adjusted. Another participant calculates the principal values and directions differently, resulting in different values.

#### jhuleea

Hi all,

I've been stumped on this problem for over a month. Any guidance would alleviate my overwhelming frustration. Here is the original problem statement:

Find the principal rates of deformation and principal axes for the flow given by: u = (x,y) and v = 0, satisfying the continuity equation (density = rho = constant)
$$\frac{\partial u_i}{\partial x_i} = 0$$​

Attached to this post is my attempt to work out the solution. I'm not sure how to proceed on, so your help would be greatly appreciated. Thanks!

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jhuleea said:
Hi all,
Find the principal rates of deformation and principal axes for the flow given by: u = (x,y) and v = 0, satisfying the continuity equation (density = rho = constant)
I guess that continuity should be du_i/dx_i=0 instead of div(u_3)=0.

Anyhew, your solution seems correct up till and including the principal directions (with the note that u=u(y) only, i.e. 2-D shear flow). When I calculate the principal values and directions in the old fashion way (as eigenvalues/eigenvectors of the strain rate tensor), I get the same directions, but principal values are +/- du/dy.

Cheers //Rope

Hello,

Thank you for sharing your attempt at solving this problem. It seems like you have a good understanding of the problem and have made some progress. Let me try to provide some guidance to help you move forward.

To find the principal rates of deformation, we need to first find the velocity gradient tensor, which is given by:

\frac{\partial u_i}{\partial x_j} = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{bmatrix}

In this case, since v = 0, the velocity gradient tensor simplifies to:

\frac{\partial u_i}{\partial x_j} = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ 0 & 0 \end{bmatrix}

Next, we can calculate the eigenvalues and eigenvectors of this velocity gradient tensor. The eigenvalues represent the principal rates of deformation and the corresponding eigenvectors represent the principal axes.

I noticed that in your attempt, you have used the velocity components (u and v) instead of the partial derivatives in the velocity gradient tensor. Make sure to use the partial derivatives to get the correct values.

I hope this helps you to move forward and solve the problem. If you have any further questions, please don't hesitate to ask. Best of luck!

## 1. What is fluid flow and why is it important?

Fluid flow is the movement of a substance, such as a liquid or gas, from one location to another. It is important because it is a fundamental process in many natural and industrial systems, including weather patterns, ocean currents, and the operation of pumps and turbines.

## 2. What are principal rates of deformation?

Principal rates of deformation refer to the rates at which a fluid is being stretched or compressed along different directions. These rates are represented by three principal axes, which are perpendicular to each other and describe the direction and magnitude of deformation.

## 3. How are principal rates of deformation related to fluid flow?

Principal rates of deformation are key parameters in the study of fluid flow as they describe the changes in a fluid's shape and volume as it moves. These rates are used to understand the behavior of fluids in different scenarios, such as in pipes, channels, or around obstacles.

## 4. What factors affect principal rates of deformation in fluid flow?

Several factors can influence the principal rates of deformation in fluid flow, including the fluid's viscosity, density, and velocity. The geometry of the flow system, such as the shape of the channel or the presence of obstacles, can also impact these rates.

## 5. How are principal rates of deformation measured and calculated?

Principal rates of deformation can be measured experimentally using tools like strain gauges or by tracking the movement of particles in the fluid. They can also be calculated mathematically using equations that take into account the properties of the fluid and the flow system.