Simple vector-decomposition (fluid mechanics)

[Niles]

Homework Statement

I am reading an introductory book on fluid mechanics (freely available on the WWW) and I came across this on page 10:

Consider an infinitesimal plane surface element, dS = ndS, located at point r. Here, dS is the area of the element, and n its unit normal. The fluid which lies on that side of the element toward which n points is said to lie on its positive side, and vice versa. The net flux of x-momentum across the element (in the direction of n) is π_x · dS, which implies (from Newton’s second law of motion) that the fluid on the positive side of the surface element experiences a force π_x ·dS in the x-direction due to short-range interaction with the fluid on the negative side.

I don't understand the underlined part. We have a force-vector π_x, which we decompose to find the overlap with some surface element dS. Shouldn't the force entering dS be directed along dS, and not π_x as the author claims?

Thanks in advance.

 

[Chestermiller]

Homework Statement

I am reading an introductory book on fluid mechanics (freely available on the WWW) and I came across this on page 10:



I don't understand the underlined part. We have a force-vector π_x, which we decompose to find the overlap with some surface element dS. Shouldn't the force entering dS be directed along dS, and not π_x as the author claims?

Thanks in advance.

It's hard to get past all this unusual notation, with the symbol n being used for two different things (momentum flux and normal), but I think I know what they are trying to say. Let $\vec{N_x}$ represent the net momentum flux (vector) through a surface of constant x. This momentum flux vector is not necessarily normal to the surface, since the fluid velocity is not necessarily normal to the surface. We can represent this momentum flux vector by:
$$\vec{N_x}=ρv_xv_x\vec{i_x}+ρv_xv_y\vec{i_y}+ρv_xv_z\vec{i_z}$$
In this example, since the surface of interest is one of constant x, $\vec{dS}=\vec{i_x}dS$, the component of the momentum flux normal to the surface is $ρv_xv_x\vec{i_x}$, and the component of the momentum flux tangential to the surface is $ρv_xv_y\vec{i_y}+ρv_xv_z\vec{i_z}$.