# A fluid element taken in the velocity boundary layer region

Consider a fluid flowing over a flat plate with a free stream velocity of $$U_\infty$$. We know that a thin boundary layer region will be formed. The shear stress acting at any distance y (measured perpendicular to the plate) is given by Newton's law of viscosity as,

$$\tau_y = \mu \frac{\partial u }{\partial y}$$

where $$u$$ represents velocity which is a function of $$x$$ and $$y$$

I was interested in knowing how can I show the stresses acting on a fluid element taken at y. For instance, we take a stress element in mechanics of materials, and show shear stresses with complementary ones, acting on it. However in case of a fluid, an infinitesimal fluid particle is not at rest, it's moving so I'm not sure if complimentary stresses will appear.

In a nutshell, I wanted to know how the stresses on a fluid element will look like, if I took one in the boundary layer region. If somebody could help.

• Check out the Navier-Stokes equation and FEA. You can also read the help files in just about any of the CFD programs available as they explain as well. Mar 7, 2022 at 5:52

## 1 Answer

The complementary stresses are still needed: without them, for a differentially small, rectangular-parallelepipedal fluid element, the net torque on the fluid element would contain three differentially-small factors of side lengths of the element, whereas the moment of inertia of the fluid element would contain five differentially-small factors of side lengths of the element. Hence, in the absence of complementary stresses, the reciprocal of the angular acceleration would be differentially small, i.e. the angular acceleration would be infinite.