We were discussing the basic concept of streamline
and equipotential line, dimensional
homogeneity, Buckingham
pi theorem, difference
between model and prototype, basic
principle of similitude i.e. types of similarity and various forces acting on moving fluid in
the subject of fluid mechanics, in our recent posts.

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Now we will go ahead to start a new topic i.e. Boundary
layer theory, in the subject of fluid mechanics with the help of this post.

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**Boundary
layer theory **

When a real fluid will flow over a solid body or a
solid wall, the particles of fluid will adhere to the boundary and there will
be condition of no-slip.

We can also conclude that the velocity of the fluid
particles, close to the boundary, will have equal velocity as of the velocity
of boundary.

If we assume that boundary is stationary or velocity
of boundary is zero, then the velocity of fluid particles adhere or very close to
the boundary will also have zero velocity.

If we move away from the boundary, the velocity of
fluid particles will also be increasing. Velocity of fluid particles will be changing
from zero at the surface of stationary boundary to the free stream velocity (U)
of the fluid in a direction normal to the boundary.

Therefore, there will be presence of velocity gradient
(du/dy) due to variation of velocity of fluid particles.

The variation in the velocity of the fluid
particles, from zero at the surface of stationary boundary to the free stream
velocity (U) of the fluid, will take place in a narrow region in the vicinity
of solid boundary and this narrow region of the fluid will be termed as
boundary layer.

Science and theory dealing with the problems of
boundary layer flows will be termed as boundary layer theory.

According to the boundary layer theory, fluid flow
around the solid boundary might be divided in two regions as mentioned and
displayed here in following figure.

**First region**

A very thin layer of fluid, called the boundary
layer, in the immediate region of the solid boundary, where the variation in
the velocity of the fluid particles, from zero at the surface of stationary
boundary to the free stream velocity (U) of the fluid, will take place.

There will be presence of velocity gradient (du/dy) due
to variation of velocity of fluid particles in this region and therefore fluid
will provide one shear stress over the wall in the direction of motion.

Shear stress applied by the fluid over the wall will
be determined with the help of following equation.

**𝜏**

**= µ x (du/dy)**

**Second region**

Second region will be the region outside of the
boundary layer. Velocity of the fluid particles will be constant outside the
boundary layer and will be similar with the free stream velocity of the fluid.

In
this region, there will be no velocity gradient as velocity of the fluid
particles will be constant outside the boundary layer and therefore there will
be no shear stress exerted by the fluid over the wall beyond the boundary
layer.

Further we will go ahead to find out the some basic concepts and definitions in the respect of boundary layer theory in the subject
of fluid mechanics, with the help of our next post.

Do you have any suggestions? Please write in comment
box.

### Reference:

Fluid mechanics, By R. K. Bansal

Image courtesy: Google