MTH- 486: Fluid Mechanics Instructor: Dr. Fahad Munir Abbasi Assistant Professor Department of Mathematics Comsats Institute of Information Technology Islamabad, Pakistan

Highlights of the previous lecture Perfect fluid Equations of streamlines Equations of streamtubes Equations of streamlines or streamtubes in cylindrical coordinates Equations of surface perpendicular to streamlines Examples Summary Layout of lecture # 11

Buoyancy; Buoyant force; Center of buoyancy; Archimedes principle Stability of un-constrained submerged bodies in a fluid Stable, unstable and neutral equilibrium Brief review of what we have done so far Previously we studied about the…

Differential analysis of fluid motion Perfect fluid: A perfect fluid is the one for which the density, ρ = constant and the viscosity, μ = 0. The constant density of a perfect fluid implies that it is insensitive to the changes of temperature and pressure. Similarly, zero viscosity implies the fact that a perfect fluid is devoid of shear stresses and as a result does not entertain any energy loss. MTH-486, Fluid Mechanics Lec. # 11

Also the effects of surface tension and vapor pressure do not exist for a perfect fluid. In general, the flow of a perfect fluid is governed by pressure and gravitational forces only. MTH-486, Fluid Mechanics Lec. # 11

Equations of streamlines: Streamlines are always tangent to their respective velocity vectors at each point in any given flow field. MTH-486, Fluid Mechanics Lec. # 11

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Equations of streamtubes: In three-dimensional flow fields, streamlines are instead called streamtubes. The corresponding equations of streamtubes are similarly derived. MTH-486, Fluid Mechanics Lec. # 11

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Note: For steady-state flow fields, streamlines and streamtubes are fixed in a given flow domain; whereas in unsteady flow fields their positions are variable, and are functions of time. MTH-486, Fluid Mechanics Lec. # 11

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MTH-486, Fluid Mechanics Lec. # 11 (2)

Example: A two-dimensional steady state flow field is represented by the velocity field u = a, v = bx, where a=1 and b=2. Assume various values of the constant of integration and draw at least three streamlines for given velocity field. Solution: MTH-486, Fluid Mechanics Lec. # 11

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MTH-486, Fluid Mechanics Lec. # 11

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Perfect fluid Equations of streamlines Equations of streamtubes Equations of streamlines or streamtubes in cylindrical coordinates Equations of surface perpendicular to streamlines Examples Summary

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