Deriving 2D Navier Stokes EQs

[Saladsamurai]

Homework Statement

I need to derive the 2D N-S equations for steady, incompressible, constant viscosity flow in the xy-plane. I need to use a control volume approach (as opposed to system approach) on a differential control volume (CV) using the conseervation of linear momentum.

Homework Equations

Conservation of Linear Momentum for a CV is given by:

$$\mathbf{F} = \frac{\partial{}}{\partial{t}}\int_v\mathbf{V}\rho\,dv + \int_{A_{in}}\mathbf{V}\rho(\mathbf{V}\cdot\hat{n})\,dA_{in} + \int_{A_{out}}\mathbf{V}\rho(\mathbf{V}\cdot\hat{n})\,dA_{out} \qquad(1)$$

The Attempt at a Solution

My problem is with the application of (1). I have a control volume with dimensions dx,dy,1 (unit depth). Now (1) is a vector equation so it can (and probably should) be divided up into its components. We know it is steady and hence the first term in (1) goes to zero.

Note that $\mathbf{V} = u\hat{i} + v\hat{j}$

x-momentum:

$$F_x = \int u\rho \left [u\hat{i}\cdot(-\hat{i})\right ]\,dy(1) + \int (u+\frac{\partial{u}}{\partial{x}}\,dx)\rho(u+\frac{\partial{u}}{\partial{x}}\,dx)\hat{i}\cdot\hat{i}\,dy(1)$$

Now I know that I can simplify some things here, but

i) Does this look correct?
ii) What are the bounds of my integrals?

Any advice is appreciated.

 

[mark726]

Your approach looks good so far. The only thing that needs to be adjusted is the bounds of your integrals. Since you are dealing with a 2D flow in the xy-plane, your control volume should have bounds of x and y. Therefore, your integrals should be:

F_x = \int_{x_1}^{x_2}\int_{y_1}^{y_2} u\rho \left [u\hat{i}\cdot(-\hat{i})\right ]\,dy\,dx
+
\int_{x_1}^{x_2}\int_{y_1}^{y_2} (u+\frac{\partial{u}}{\partial{x}}\,dx)\rho(u+\frac{\partial{u}}{\partial{x}}\,dx)\hat{i}\cdot\hat{i}\,dy\,dx

Where x_1 and x_2 are the bounds of your control volume in the x-direction, and y_1 and y_2 are the bounds in the y-direction. These bounds will depend on the specific problem you are trying to solve.

Keep up the good work and don't hesitate to ask for further clarification if needed. Best of luck with your derivation!