Tank draining with boundary layer $-M.$ For a laminar boundary layer on a flat plate, the solution in Section $8\mathrm{.}2$ gave the following expression for the wall shear stress at a point of distance $x$ from the leading edge:

$\frac{{}_w}{\frac{1}{2}{v}_x^2}=\frac{0\mathrm{.}656}{\sqrt[\phantom{2}]{R{e}_x}}$

Derive an expression for the total drag force $F$ exerted by a plate of length $L$ on the fluid, per unit distance normal to the plane of Fig. $8\mathrm{.}3.$ Your answer should give $F$ in terms of $v_x,L,,$ and $$ or

$u.$

$($b$)$

Fig. P$8\mathrm{.}3($a$)$ Tank draining; $($b$)$ detail of boundary layer.

As shown in Fig. P$8\mathrm{.}3($a$),$ a viscous liquid is draining steadily from a largediameter tank. As it enters the pipe of diameter $D$ and length $L,$ it has a flat velocity profile, and a thin laminar boundary layer subsequently forms, as in Fig. P$8\mathrm{.}3($b$).$ Derive an expression for the velocity of the liquid $v_x$ in the mainstream, in terms of any or all of $D,L,g,,$ and $.$ Assume that the boundary layer is sufficiently thin so that $v_x$ is essentially constant; also, since the velocities are small, neglect any inertial terms in a momentum balance. Evaluate $v_x$ if $L=1m,D=0\mathrm{.}1m,=1,000k\frac{g}{m^3},g=9\mathrm{.}81\frac{m}{s^2},$ and $=1k\frac{g}{m}s.$