Question 1

Water of density p flows past a smooth flat plate of length L and large span and a laminar

boundary layer of thickness d develops on each side. The upstream velocity may be assumed

to have the uniform value Uo, whilst the downstream velocity profile has the form:

where y is the distance perpendicular to the plate. There is negligible pressure change along the

length of the plate. A suitable control volume to analyse the problem is shown below. Depth

H is any depth greater than the downstream boundary-layer thickness and cancels during your

working.

(a) Use continuity to calculate the apparent displacement of streamlines d* (this is called the

displacement thickness) as a function of boundary-layer depth d

(b) Calculate the difference between momentum fluxes (per unit span) at inlet and outlet of

the control-volume shown

(c) Hence deduce the viscous drag force (per unit span) on one side of the plate.

(d) Noting that there is a boundary layer on both sides of the plate, define a suitable overall

drag coefficient and calculate its value.

(e) Explain, with reference to the appropriate concepts and definitions of fluid mechanics, how

the solution to the above questions would change should the plate be inlined downwards

30° to the horizontal.