maximum velocity at r = 0. The average velocity of the fluid in the pipe is

Q4. Consider simple laminar pressure driven flow in a circular pipe of radius R. We can write the equation for the velocity distribution in the pipe to be: V₂ = 2U|1 2U [1 - (R)²] where U= average velocity. Using the energy equation with the viscous dissipation term, calculate the temperature profile in the fluid due to viscous dissipation alone. Assume that

An incompressible fluid is flowing at a steady rate in a horizontal pipe. From a section, the pipe divides into two horizontal parallel pipes of diameters di and d2 (where d₁ = 4d₂) that run for a distance of L each and then again join back to a pipe of the original size. For both the parallel pipes, assume the head loss due to friction only and the Darcy-Weisbach friction factor to be the same. The velocity ratio between the bigger and the smaller branched pipes is

A U-tube of uniform cross-section has both arms open to the atmosphere. The U-tube contains three immiscible liquids, with density PA, PB, and pc, as depicted in the diagram below, where po is the atmospheric pressure. We denote h to be the depth of the open surface of liquid A on the left arm, relative to the open surface of liquid Con the right arm. I is the depth of the interface between liquid B and C on the right arm and 2l is the depth of the interface between liquid A and liquid B on the right arm. Po PA (c) Po PC PB е е (a) Explain why the liquid densities need to satisfy the condition PA PB PC. (b) Write down an expression for the pressure at depth z below the surface of liquid A in the left-hand part of the tube. Show that l < h < 2l if PB + PC < PA•