1. Dimensional Analysis of Fluid Flow in a Pipe Consider fluid flow in a long cylindrical pipe of diameter D. A fluid with viscosity i and density p flows through the pipe with an average velocity of V. The internal surface of the pipe is rough, with the average height of the roughness being h. We wish to determine how the pressure gradient dp/dz depends on the properties of the system through dimensionless groups. (a) Identify the relevant parameters and their dimensions. (b) Determine the number of relevant non-dimensional groups. (c) Select the dimensionally-independent scaling variables. (d) Determine the dimensionless group that relates dp/dz to the other relevant dimen- sionless groups.

1. Dimensional Analysis of Fluid Flow in a Pipe Consider fluid flow in a long cylindrical pipe of diameter D. A fluid with viscosity i and density p flows through the pipe with an average velocity of V. The internal surface of the pipe is rough, with the average height of the roughness being h. We wish to determine how the pressure gradient dp/dz depends on the properties of the system through dimensionless groups. (a) Identify the relevant parameters and their dimensions. (b) Determine the number of relevant non-dimensional groups. (c) Select the dimensionally-independent scaling variables. (d) Determine the dimensionless group that relates dp/dz to the other relevant dimen- sionless groups.