B$7.($a$)$ In the usual notation, the momentum equation $($Euler$\text{'}$s equation$)$ for an inviscid flow is

$\frac{Du}{Dt}=-\frac{1}{}$gradp$+F.$

Under the assumptions that $p=p()$ and that $F$ is conservative, derive Bernoulli's equation for a

steady, irrotational flow. $[$Explain carefully the steps used in deriving this result. You may use

the vector identity $u(gradu)=$grad$(\frac{1}{2}u*u)-(u.$grad$)u$ without proof.$]$

$[8$ marks$]$

$($b$)$ A flat plate aerofoil is placed horizontally in an infinite fluid medium which has fixed, uniform

density $.$ Far upstream and downstream the pressure is $p_0$ and the flow is uniform and horizontal

with speed $U.$ You may assume that the flow past the aerofoil is two$-$dimensional, and that you can

treat all the streamlines as being horizontal.

The aerofoil is placed so that it occupies $xx-U\{1+2f(x)\}U\{1-f(x)\}f(x)f(x)=\frac{x(a-x)}{2a^2}.x{p}_{\text{d}\text{i}\text{f}\text{f}}=p_{\text{l}\text{o}\text{w}\text{e}\text{r}}-p_{\text{u}\text{p}\text{p}\text{e}\text{r}}xL{\displaystyle \underset{0}{\overset{a}{}}}{p}_{\text{d}\text{i}\text{f}\text{f}}dxL=21a\frac{U^2}{80}0,$ where $xis$ measured from its leading $(i.e.$

forward$)$ edge, and with $x-$ far upstream.

You are given that the flow speed past the aerofoil $isU\{1+2f(x)\}$ just above the plate and

$U\{1-f(x)\}$ just below, where the function $f(x)is$ given $by$

$f(x)=\frac{x(a-x)}{2a^2}.$

State the slightly modification $to$ Bernoulli's equation for fluid $of$ constant density, and explain

why the potential $of$ the body forces, which $is$ usually present, $is$ not needed $in$ this case.

Now use Bernoulli's equation $to$ find the pressure $at$ position $xon$ the upper and lower surfaces

$of$ the aerofoil, and hence determine the pressure difference, $p_{\text{d}\text{i}\text{f}\text{f}}=p_{\text{l}\text{o}\text{w}\text{e}\text{r}}-p_{\text{u}\text{p}\text{p}\text{e}\text{r}},$ across the plate $at$

position $x.$

Given that the lift per unit span, $L,is$

${\displaystyle \underset{0}{\overset{a}{}}}{p}_{\text{d}\text{i}\text{f}\text{f}}dx$

show that $L=21a\frac{U^2}{80.}$

$[10$ marks$]$