We will study different situations, like:

One plate moving.

Both plates moving.

Both plates moving in different directions.

Different types of fluid.

For each situation we will establish specific conclusions

**Couette flow**

**Àlex March Vidal**

Oriol Esteban Timoneda

Oriol Esteban Timoneda

Introduction

Objectives

This is a picture of an horizontal velocity profile between two plates, also known as Couette flow.

Our study will only have one difference respect to Couette flow: the conditions WON'T be STACIONARY.

What are we talking about

**Methods & Results**

Analytical method

Conclusions & Error

ERROR between the analytical solution and the numerical one

**Fluid Mechanics – Francesca Ribas**

UPC Aerospace

UPC Aerospace

What equations we use

The equations used for solving our problem will be the Continuity and Navier-Stokes equations.

We will work in 2D, so we will only consider the velocity on the X direction, and will depend on Y (u(y)).

Regarding the moment equation, we will only consider this equation in the X direction, because the other moment equations regarding the other directions will be NEGLIGIBLE or VANISHED.

What is Couette flow

Who was Couette

We can solve the problem analitically, calculating the velocity profile. For that reason, we will have to consider stationary conditions, to simplify the equation.

This analytical resolution will allow us check if our numerical solution is correct with a minimum of precision.

In fluid dynamics, Couette flow is the laminar flow of a viscous fluid in the space between two parallel plates, one of which is moving relative to the other.

The flow is driven by virtue of viscous drag force acting on the fluid and the applied pressure gradient parallel to the plates.

This type of flow is named in honor of Maurice Marie Alfred Couette, a Professor of Physics at the French university of Angers in the late 19th century.

For solving the velocity profile in a simple and fast way, we will use discretization with finite differences.

By the left side of the equation we will use the forward derivative, and for the right side the central derivative.

Using the numerical method, we can consider NON stationary conditions, because that fact don’t make our way of resolution harder to solve.

Numerical method

Results

Cases

Upper plate moving

In this case we can see the velocity profile corresponding to the upper plate moving.

Lower plate moving

In this case we can see the velocity profile corresponding to the lower plate moving.

Both plates moving

In this case, both were moving at the same direction (and same speed), so the velocity profile tends to be uniform, like the steady case (red line) shows.

Both plates moving in different directions

In this case, both were moving at the same direction (and same speed), so the velocity profile tends to be uniform, like the steady case (red line) shows.

SAE 30 W OIL

SAE 30 W OIL

SAE 30 W OIL

SAE 30 W OIL

HONEY

Ethylene Glycol

SAE 30W OIL

SAE 30W OIL

In our case we used these fluids:

SAE 30W OIL

Ethylene Glycol

0

HONEY

x 9

SAE 30W OIL

HONEY

Ethylene Glycol

x 9

x 16

Upper plate moving

(Honey)

Upper plate moving

(Ethylene Glycol)

In this case, we can see that the Honey velocity evolution varies a lot with SAE 30W OIL. The high viscosity forces all the fluid to move at a time.

On the other side, the Ethylene Glycol velocity evolution also varies because its viscosity is lower. So the fluid velocity only changes on the closer part of the wall (moving one).

Courant–Friedrichs–Lewy condition

What is this condition?

Use?

Is a necessary condition for convergence while solving certain partial differential equations

9'4 · E^-3

Ethylene Glycol Case

if

= 2% ~ 0'0006

For calculating the error we have taken the velocity in the middle point, at the final time. That means our numerical solution is almost stationary, like our analytical.

3,46% error

One plate moving: Analytical 0,5 Numerical 0,4827

3,45% error

Both same direction: Analytical 1 Numerical 0,9655

3,46% error

Both diff. direction: Analytical 0,5 Numerical 0,4827

~3'46% APROX.

As a conclusion to our project and after doing all the tests, we can say that:

In order to properly display the full velocity profile, we would have to tend to infinite, which is "equal" to a lot of iterations.

With only 100 iterations, our results were not very different from the steady ones at the last iteration (error of 3,46%, like we have seen).

The numerical methods really help us with the solution of a very difficult analytic problem, simplifying it to just a few equations and operations.

Conclusions

Thanks for your attention