The boundary-layer thickness, δ, on a smooth flat plate in an incompressible flow without pressure gradients depends on the freestream speed, U, the fluid density, rho, the fluid viscosity, μ, and the distance from the leading edge of the plate, x. (a) Express these variables in dimensionless form and (b) calculate dimensionless parameter (proportional to x) with x = 0.142 m, rho = 225 kg/m3, U = 0.133 m/s, μ = 0.2 × 10-4 N-s/m2.

a) (σ/x)=f((μ)/(ρUx)) dimensionless form

b) Re=212467.5

Explanation:

a) There are 5 parameters: σ, U, ρ, μ, x, thus n=5. M, L, T are primary variables (m=3). The numbers of variables are 3 (r=3). According to Buckingham n-m(5-3=2) given by π1 and π2. The dimensions of parameters are:

σ=L

U=LT^1

ρ=ML^-3

μ=ML^-1T^-1

x=L

π1=ρ^aU^bx^cσ=(ML^-3)^a(LT^-1)^b(L^c)L=M^0L^0T^0

if we equalize the coefficients on both sides of the equation:

M:a=0

T:-b=0

L:-3a+b+c+1=0

c=-1

π1=ρ^0U^0x^-1σ

π1=σ/x=L/L=1

π2=ρ^dU^ex^fμ=(ML^-3)^d(LT^-1)^e(L^f)(ML^-1T^-1)=M^0L^0T^0

if we equalize the coefficients on both sides of the equation:

M:d+1=0, d=-1

T:-e-1=0, e=-1

L:-3d+e+f-1=0, f=-1

π2=ρ^-1U^-1x^-1μ

π2=(μ)/(ρUx)=1

The realation is:

π1=f(π2)

dimensionless form is (σ/x)=f((μ)/(ρUx))

b) The variables are expressed in a diamensionless form that is named Reynold`s number. Replacing values:

Re=(ρUx)/μ=(225*0.133*0.142)/(0.2x10^-4)=212467.5

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