$1.$ PHYSICAL MODELING: DIMENSIONAL ANALYSIS and SIMILITUDE

In the laminar flow of a viscous fluid through a horizontal pipe, the pressure drop $\backslash (\backslash$Delta $\backslash$mathrm$\{$p$\}\backslash )$ is a function of the velocity V and absolute viscosity $\backslash (\backslash$mu $\backslash )$ of the fluid, pipe diameter D $,$ and length L $($distance between the upstream and downstream point in the pipe$).$

a$.$ Using dimensional analysis, show that the equation for $\backslash (\backslash$Delta $\backslash$mathrm$\{$p$\}\backslash )$ for laminar flow in a horizontal pipe is equal to $\backslash (32\backslash$mu $\backslash$mathrm$\{$VL$\}/\backslash$mathrm$\{$D$\}^\{2\}\backslash ).$

b$.$ The pressure drop $\backslash (\backslash$Delta $\backslash$mathrm$\{$P$\}\backslash )$ in a $\backslash (5000-\backslash$mathrm$\{$m$\}\backslash )$ long horizontal oil pipel ine is $850$ Pa $,$ when the average velocity is $\backslash (0\mathrm{.}06\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$ The diameter of the pipeline is $30$ cm $.$ and the oil has a specific gravity S of $0\mathrm{.}86.$ If the flow is laminar and using the equation for the pressure drop, find the absolute viscosity of the oil.

c$.$ If the oil pipeline is to be tested in a $1$:$3$ model using water as the model fluid $(\backslash (\backslash$mu$=0\mathrm{.}001\backslash$mathrm$\{$~Pa$\}\backslash )-$s$),$ compute the model discharge for the given velocity using similitude.