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

3. Engineering Solution for Hagen-Poiseuille Flow Conventional engineering flows: Kn < 0.001 Micro Fluidic Devices : Kn < 0.1 Ultra Micro Fluidic Devices : Kn <1.0 The Wall Boundary Conditions. or  u s uw uw Wall

4. Micro Engineering Mild Slip Hagen-Poiseuille Flow The first order slip condition: For a flow through an immobile pipe:

5. The micro engineering pipe-flow solution is thus Mean & Maximum Flow Velocities The Wall Shear Stress Friction Factor

6. Popular Creeping Flows Fully developed duct Flow. Flow about immersed bodies Flow in narrow but variable passages. First formulated by Reynolds (1886) and known as lubrication theory, Flow through porous media. This topic began with a famous treatise by Darcy (1856 Civil engineers have long applied porous-media theory to groundwater movement. http://www.ae.metu.edu.tr/~ae244/docs/FluidMechanics- by-JamesFay/2003/Textbook/Nodes/chap06/node17.html

7. Further Use of Mean Velocity for High Speed Flows

8. Compressible (Average) Frictional Flow in A Constant Area Duct ww Self similar compressible fully developed flow through ducts

9. Frictional Flow in A Constant Area Duct ww The shear stress is defined as an average viscous stress which is always opposite to the direction of flow for the entire length dx. Divide by  V 2

10. 1D steady real flow through constant area duct : momentum equation

12. One dimensional Frictional Flow of A Perfect Gas

13. Sonic Equation Differential form of above equation:

14. Energy equation can be modified as:

15. Differential Equations for Frictional Flow Through Constant Area Duct