A fluid is flowing in a pipe as shown in the figure below. The pressure drop per unitlength $(\frac{P}{L}[P\frac{a}{m}])$ is assumed to be determined by the inner diameter of pipe $(D[m]),$ the averagevelocity $(u[\frac{m}{s}]),$ the density $([k\frac{g}{m^3}]),$ and the viscosity of the fluid $([Pa*s])).$ Dimensional

analysis can clarify influences of the above factors on the pipe flow.

$1)$ By comparing the dimension of the product of $D^a,u^b,{}^c,$ and ${}^d(=D^a{u}^b{}^c{}^d)$ withthat of $\frac{P}{L},$ derive three different equations based on the exponents of $a,b,c,$ and $d.$

$2)$ Describe a function of the pressure drop $(\frac{P}{L})$ using a proportionality coefficient $(k),$

the exponent of $d,$ and the four parameters of $D,u,,$ and $.$

$3)$ According to the result in $2),$ describe an equation showing that the pressure drop$(\frac{P}{L})$ is a function of Reynolds number $(Re).$