When a wall next to an infinite fluid is put into sudden motion at a speed of $U$ at time $t=$

$0,$ we found that the velocity profile of the fluid can be given by

$\frac{v_x}{U}=1-$erf$(\frac{y}{\sqrt[\phantom{2}]{4vt}})$

While this equation satisfies the Navier$-$Stokes equation and boundary and initial

conditions, it can be argued that it is flawed in describing the velocity far from the wall,

as the equation implies that the instant the wall starts to move, all of the fluid will start to

move. In a real fluid, only the fluid within a boundary layer will move, with the boundary

layer growing over time.

Borrowing in part from creeping flow around a sphere, we can propose an approximate

velocity profile,

$\frac{v_x}{U}=1-\frac{3}{2}\frac{y}{(t)}+\frac{1}{2}(\frac{y}{(t)}{)}^3$

when where $(t)$ is the boundary$-$layer thickness. When $y(t)($i$.$e$.$ outside

the boundary layer$),$ the fluid velocity is $0.$ While this is arguably more realistic, it is an

approximation and does not actually satisfy the Navier$-$Stokes equation. Instead, the

equation can be satisfied if we can integrate both sides of the simplified Navier$-$Stokes

equation:

$\frac{d}{dt}{\displaystyle \underset{0}{\overset{}{}}}{v}_{x}dy=v{\displaystyle \underset{0}{\overset{}{}}}\frac{de{l}^2{v}_x}{del{y}^2}dy$

$\frac{d}{dt}{\displaystyle \underset{0}{\overset{(t)}{}}}{v}_{x}dy=-v\frac{del{v}_x}{\text{d}\text{e}\text{l}\text{y}}{|}_y=0$

Substitute the approximate velocity profile into the above equation and solve for the

boundary layer width, $,$ as a function of time.