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Sunday, 28 October 2018

BOUNDARY LAYER THEORY IN FLUID MECHANICS

We were discussing the basic concept of streamline and equipotential linedimensional homogeneityBuckingham pi theoremdifference between model and prototypebasic 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. 

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. 

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 

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