- #1

tomwilliam2

- 117

- 2

## Homework Statement

I'm trying to find the Force per unit length due to a pressure function across a cylinder immersed in a fluid.

## Homework Equations

Bernouilli's equation,

$$\mathbf{F}_s = -\mathbf{n} \int_s p dA$$

## The Attempt at a Solution

I've got an expression for p which takes the form $$p = p(\theta)$$.

My expression for the unit vector is $$\mathbf{n}=\cos \theta \mathbf{i} + \sin \theta \mathbf{j}$$

I then get an equation:

$$\mathbf{F}_s/length = -(\cos \theta \mathbf{i} + \sin \theta \mathbf{j}) \int_s p(\theta) r d\theta$$

Now it's clear from the solution that I need to have the trig functions inside the integral...but is this mathematically justified? I know that the values of the unit vector change as we move around the surface, but does that mean I can just include them into the integral? Where the i-component for example would become:

$$\mathbf{F}_i / length = \int_s -p(\theta)r \cos \theta d\theta$$

Is this right?

Thanks in advance