An Essential Need of Modern Civilization… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Viscous Fluid Flows in Ducts

Poiseuille Flow through Ducts Whereas Couette flow is driven by moving walls, Poiseuille flows are generated by pressure gradients, with application primarily to ducts. They are named after J. L. M. Poiseuille (1840), a French physician who experimented with low-speed flow in tubes. Regardless of duct shape, the entrance length can be correlated for laminar flow in the form

Fully Developed Duct Flow For x > L e, the velocity becomes purely axial and varies only with the lateral coordinates. v = w = 0 and u = u(y). The flow is then called fully developed flow. For fully developed flow, the continuity and momentum equations for incompressible flow are simplified as: With

These indicate that the pressure p is a function only of x for this fully developed flow. Further, since u does not vary with x, it follows from the x- momentum equation that the gradient dp/dx must only be a (negative) constant. Then the basic equation of fully developed duct flow is subject only to the slip/no-slip condition everywhere on the duct surface This is the classic Poisson equation and is exactly equivalent to the torsional stress problem in elasticity

Characteristics of Poiseuille Flow Like the Couette flow problems, the acceleration terms vanish here, taking the density with them. These flows are true creeping flows in the sense that they are independent of density. The Reynolds number is not even a required parameter There is no characteristic velocity U and no axial length scale L either, since we are supposedly far from the entrance or exit. The proper scaling of Poiseuille Equation should include , dp/dx, and some characteristic duct width h.

Dimensionless variables for Poiseuille Flow Dimensionless Poiseuille Equation

The Circular Pipe: Hagen-Poiseuille Flow The circular pipe is perhaps our most celebrated viscous flow, first studied by Hagen (1839) and Poiseuille (1840). The single variable is r* = r/R, where R, is the pipe radius. The equation reduces to an ODE: The solution of above Equation is: Engineering Conditions: The velocity cannot be infinite at the centerline. On engineering grounds, the logarithm term must be rejected and set C 1 = 0.

Engineering Solution for Hagen-Poiseuille Flow Conventional engineering flows: Kn < Micro Fluidic Devices : Kn < 0.1 Ultra Micro Fluidic Devices : Kn <1.0 The Wall Boundary Conditions

Macro Engineering No-Slip Hagen-Poiseuille Flow The no-slip condition: The macro engineering pipe-flow solution is thus For a flow through an immobile pipe:

Dimensional Solution to Macro Engineering No-Slip Hagen-Poiseuille Flow

The Capacity of A Pipe Thus the velocity distribution in fully developed laminar pipe flow is a paraboloid of revolution about the centerline. This is called as the Poiseuille paraboloid. The total volume rate of flow Q is of interest, as defined for any duct by

Mean & Maximum Flow Velocities The mean velocity is defined by The maximum velocity occurs at the center, r=0.

The Wall Shear Stress The wall shear stress is given by

Friction Factor  w is proportional to mean velocity. It is customary, to nondimensionalize wall shear with the pipe dynamic pressure. Two different friction factor definitions are in common use in the literature: Darcy Friction Factor This is called as standard Fanning friction factor, or skin-friction coefficient.

Hagen’s Pipe Flow Experiments Hagen was born in Köni gsberg, East Prussia, and studied there, having among his teachers the famous mathematician Bessel. He became an engineer, teacher, and writer and published a handbook on hydraulic engineering in He is best known for his study in 1839 of pipe-flow resistance, for water flow. At heads of 0.7 to 40 cm, diameters of 2.5 to 6 mm, and lengths of 47 to 110 cm. The measurements indicated that the pressure drop was proportional to Q at low heads.

Hagen’s Paradox