CHEMICAL ENGINEERING

SECTION 11

The Boundary Layer

PROBLEM 11.1

Calculate the thickness of the boundary layer at a distance of 75 mm from the leading
edge of a plane surface over which water is flowing at a rate of 3 m/s. Assume that the
flow in the boundary layer is streamline and that the velocityuof the fluid at a distance
yfrom the surface can be represented by the relationuDaCbyCcy^2 Cdy^3 ,wherethe
coefficientsa,b,c,anddare independent ofy. The viscosity of water is 1 mN s/m^2.

Solution

At a distanceyfrom the surface:uDaCbyCcy^2 Cdy^3.
WhenyD0,uD0, and henceaD0.
The shear stress within the fluid:R 0 D

∂u/∂y yD 0 and since
∂u/∂y is constant for
small values ofy,
∂^2 u/∂y^2 yD 0 D0.
At the edge of the boundary layer,yDυanduDus, the main stream velocity.

∂u/∂yD0anduDbyCcy^2 Cdy^3

∴ ∂u/∂yDbC 2 cyC 3 dy^2 and∂^2 u/∂y^2 D 2 cC 6 dy

WhenyD0,∂^2 u/∂y^2 D0, and hencecD0.

WhenyDυ,uDbυCdυ^3 Dus

and: ∂u/∂yDbC 3 dυ^2 D 0

∴ bD 3 dυ^2

∴ dDus/ 2 υ^3 andbD 3 us/ 2 υ

The velocity profile is given by,uD 

3 usy/ 2 υ

us/ 2 

y/υ^3

or: u/usD 1. 5
y/υ  0. 5
y/υ ^3 (equation 11.12)

The integral in the momentum equation 11.9 is now evaluated, and substituting from
equations 11.14 and 11.15 into equation 11.9:

υ/xD 4. 64 Rex^0.^5

RexD 

0. 075 ð 3 ð 1000 / 1 ð 10 ^3 D 225 , 000

υ/xD 

4. 64 ð 225 , 000 ^5 D 0. 00978

and: υD
0. 00978 ð 0. 075 D 0 .000734 m or 0.734 mm

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