Question

Water flows near a flat surface and some measurements of the water velocity, u, parallel to the surface, at different heights y above the surface are obtained. After an analysis of the data, the lab technician reports that the velocity distribution in the range 0 < y < 0.1 ft is given by the equation u = 0.81 + 9.2y + (4.1 times 103 y3) With u in ft/s and y in ft.

(a) Do you think this equation would be valid in any system of units? Explain,
(b) Do you think this equation is correct? Explain.

Answer 1

a) since u from equation1 and u from equation2 are not the same ( 1.7825 ft/s ≠ 3.165 ft/s ) then this equation is not valid for any system of units.

b) The velocity according to the equation at y=0 is equal to 0.81 ft/sec but since the fluid is flowing on flat surface which is stationary, this value is wrong hence the equation is NOT CORRECT

Step-by-step explanation:

Given that;

range ⇒ 0 < y < 0

1ft is given by the equation u = 0.81 + 9.2y + (4.1 × 10³y³)

so u=velocity of water at different layers

y= height of the layer

a)

consider BG system of units

u(ft/s) = 0.81 + 9.2y + (4.1 × 10³y³)

and consider y=0.05 ft

u = 0.81 + 9.2(0.5) + (4.1 × 10³(0.5³)

u = 0.81 + 0.46 + 0.5125

u = 1.7825 ft/s lets say this is equation 1

now consider the SI system units

u(m/s) = 0.81 + 9.2y + (4.1 × 10³y³)

also consider y=0.05ft

1ft = 3.048×10⁻¹ (from conversion table)

so 0.05ft = 0.01524m

we substitute

u(m/s) = 0.81 + 9.2(0.01524m) + (4.1 × 10³(0.01524m)³)

u = 0.81 + 0.1402 + 1.4512×10⁻²

u = 0.9647 m/s

1m/s = 3.281 ft per seconds ( conversion table)

so

0.9647 m/s = 0.9647(3.281)

u = 3.165 ft/s lets say this is equation 2

now since u from equation1 and u from equation2 are not the same ( 1.7825 ft/s ≠ 3.165 ft/s ) then this equation is not valid for any system of units.

b)

we know that the velocity of water at the surface contact is zero

u=0

so from the equation

u = 0.81 + 9.2y + (4.1 × 10³y³)

at y = 0

u = 0.81 + 9.2(0) + (4.1 × 10³(0)³)

u = 0.81 ft/s

The velocity according to the above equation at y=0 is 0.81 ft/sec but since the fluid is flowing on flat surface which is stationary this value is wrong hence the equation is NOT CORRECT