Mechanics of Fluids

360 Boundary layers, wakes and other shear layers
be 2U sin θ , where U denotes the velocity far upstream and
θ the angle between the radius and the initial flow direction.
Neglecting changes of elevation and assuming that the pressure
in the wake is sensibly uniform and that there are no discontinuities
of pressure anywhere, estimate the position at which
the boundary layer separates if the measured drag coefficient is
1.24. Why will this estimate be somewhat in error?
8.10 If, for the cylinder and air flow mentioned in Problem 8.9,
the boundary layer is entirely laminar, use Thwaites’s method
(with λ =−0.09 at separation) to show that the position of
separation is at θ ≏ 103.1 ◦ .
8.11 Air of kinematic viscosity 15 mm 2 · s −1 and density
1.21 kg · m −3 flows past a smooth 150 mm diameter sphere at
60 m · s −1 . Determine the drag force. What would be the drag
force on a 150 mm diameter circular disc held perpendicular
to this air stream?
8.12 Calculate the diameter of a parachute (in the form of a
hemispherical shell) to be used for dropping a small object
of mass 90 kg so that it touches the earth at a velocity no
greater than 6 m · s −1 . The drag coefficient for a hemispherical
shell with its concave side upstream is approximately 1.32 for
Re > 10 3 . (Air density = 1.22 kg · m −3 .)
8.13 Determine the diameter of a sphere of density 2800 kg · m −3
which would just be lifted by an air-stream flowing vertically
upward at 10 m · s −1 . What would be the terminal velocity
of this sphere falling through an infinite expanse of water?
(Densities: air 1.21 kg · m −3 ; water 1000 kg · m −3 . Dynamic
viscosities: air 18.0 µPa · s; water 1.0 mPa · s.)
8.14 When water (kinematic viscosity 1.2 mm 2 · s −1 ) flows steadily
at the rate of 18.5 L · s −1 through 25 m of a 100 mm diameter
pipe, the head loss is 1.89 m. Estimate the relative roughness,
the velocity on the axis of the pipe and the shear stress at
the wall.
8.15 Gas is pumped through a smooth pipe, 250 mm in diameter,
with a pressure drop of 50 mm H2O per kilometre of pipe.
The gas density is 0.7 kg · m −3 and the kinematic viscosity is
18 mm 2 · s −1 . Use eqn 8.52 to determine the rate of flow and
the shear stress at the wall of the pipe.