u-a+by-cy”, where y is the perpendicolar distance measured from the surface of the flat plate and a, b and c are constants. Determine the expression of velocity distribution in dimensionless form as $\frac{\square}{\square}=(\frac{y}{δ})$ where, U is main stream velocity at boundary layer thickness δ. Further also find boundary layer thickness in terms of Reynolds number.

Velocity distribution: u=$a+by-cy^2$

The following condition must be satisfied:

i)At y=0 , u=0

∴u=$a+by-cy^2$

0=a+0-0 ∴a=0

ii)At y=δ, u=U

∴U=$bδ-cδ^2$………………………(i)

iii)At y=$δ, \frac{du}{dy}$=0

∴$(\frac{du}{dy})_(y=δ)=\frac{d}{dy} (a+by-cy^2 )=b-2cy=b-2cδ=0$…………(ii)

Substituting values of b=2cδ from (ii) in (i), we get

U=$2cδ^2-cδ^2$

$U=cδ^2$

$c=\frac{U}{δ^2}$

Hence from the velocity distrubution is:

u=$\frac{2Uδ}{δ^2} y-\frac{U}{δ^2} y^2$

$u=\frac{2U}{δ y}-\frac{U}{δ^2} …