#$7.$ Consider water flowing over a flat plate under laminar

conditions as depicted in Figure $5\mathrm{.}19.$ Assume that the

edge of the boundary layer is at a position where $\frac{u}{U}$

$=0\mathrm{.}99.$ For conditions a$-$c below, plot the depth of the

boundary $(y)$ as a function of the distance $(x)$ from the edge

of the plate, for the range of $x=[1,100]m.$ Use the values

in Table $5\mathrm{.}3$ on page $227.$

$($a$)$ water at $20C$ and an uniform flow velocity of $0\mathrm{.}1\frac{m}{s}$

$($b$)$ water at $20C$ and an uniform flow velocity of $1\frac{m}{s}$

$($c$)$ water at $5C$ and an uniform flow velocity of $0\mathrm{.}1\frac{m}{s}$

$($d$)$ Comment on the effect of physical$/$flow properties on

thickness of the boundary layer.

Figure $5\mathrm{.}19.$ Laminar boundary layer along a flat plate

immersed in a uniform flow of velocity $U.$ Note that there

might also be a boundary layer below the plate, but we are

not interested in it here.

Therefore, our analysis begins with the $x$ component of

the Navier$-$Stokes equations, which for steady two$-$

dimensional flow with no free surface is

$(5\mathrm{.}107).$

$u\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}+v\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{y}}=-\frac{1}{}\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{x}}+v(\frac{de{l}^{2}u}{del{x}^2}+\frac{de{l}^{2}u}{del{y}^2})$