$7\mathrm{.}7\mathrm{.}8$ A student is interested in the aerodynamic drag on spheres. She conducts a series of wind tunnel tests on a $10-cm-$diameter sphere. The air in the wind tunnel is at $50C$ and $101\mathrm{.}3$ kPa $.$ She presents the results of her tests in the form of a correlation,

$D=0\mathrm{.}0015V^{1\mathrm{.}92}$

where $D$ is the drag force in Newtons and $V$ is the velocity in $\frac{m}{s}.$ Another student realizes that the results would have been more effectively presented in terms of dimensionless parameters, say,

${}_1=f({}_2)=C{}_2^a$

where ${}_1$ and ${}_2$ are dimensionless parameters and $C$ and a are constants. What are the most appropriate dimensionless parameters $({}_1$ and ${}_2)$ and the corresponding values of $C$ and $a?$ What would be the drag on a $2-$ cm $-$diameter sphere placed in a $1\frac{m}{s},20C$ water stream?

${}_1=C_D$

${}_2=Re$

$a=-0\mathrm{.}08$

$C_D=0\mathrm{.}549R{e}^{{?}^{{?}^{?}}-0\mathrm{.}08}$

$D=0\mathrm{.}0497N$

${}_1=C_D$

${}_2=Re$

$a=-3\mathrm{.}92$

$C_D=0\mathrm{.}549R{e}^{-3\mathrm{.}92}$

$D=3\mathrm{.}99N$

${}_{\text{I}}=C_D$

${}_2=Re$

$a=-0\mathrm{.}08$

$C_D=0\mathrm{.}249$

$D=4\mathrm{.}97N$

${}_1=C_D$

${}_2=Re$

$a=-0\mathrm{.}08$

$C_D=0\mathrm{.}549Re{n}^{-0\mathrm{.}05}$

$D=0\mathrm{.}0994N$