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MEDE3005Notes

MEDE3005Notes - MEDE3005 TRANSPORT BIOLOGICAL SYSTEMS...

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MEDE3005 – TRANSPORT PHENOMENA IN BIOLOGICAL SYSTEMS 2008 – 2009 Academic Year Part I (11 hours, Dr. K. W. Chow) – Governing Equations of Fluid Dynamics (A) Elementary concepts of fluid flows Dependence on space and time (1) Conditions in a body of fluid can vary from point to point and, at any given point, can vary from one moment of time to the next. A flow is uniform if the velocity at a given instant is the same in magnitude and direction at every point in the fluid. If at any instant in time, the velocity changes from point to point, the flow is non–uniform . (2) A flow is steady if the fluid properties, e.g. velocity and pressure, at any given point do not change with time. A flow which is not steady, i.e. with time dependent properties, is termed unsteady . (3) Note that all these combinations are possible: (a) Steady uniform flows – flow properties independent on position and time; (b) Steady non–uniform flows – flow properties independent of time but depending on position; (c) Unsteady uniform flows – flow properties independent of position but depending on time; (d) Unsteady non–uniform flows – flow properties depending on both position and time. Real and ideal fluids (4) When a real fluid flows past a boundary, the fluid immediately in contact with the boundary will have the same velocity as the boundary (the no slip boundary condition , or friction will not permit a relative motion along the boundary). In most problems encountered in practice, this boundary is taken to be at rest. (5) A frictionless fluid which permits sliding along the boundary will be termed an ideal fluid . 1

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(6) In practice, the no slip condition must be enforced. A thin portion of fluid around the boundary, termed the boundary layer, will be the region where viscous effects are the largest. This is also the region where drag force, momentum transfer and energy loss are important considerations. Tremendous efforts have been spent over the years to understand this dynamics, and we shall attempt a brief introduction later this course. Compressible and incompressible flows (7) If the density of a fluid does not change in the flow, the fluid is incompressible (otherwise, compressible ). In practice, a liquid can be taken as incompressible. The dynamics of a gas will usually require compressible effects to be taken into account. However, as a working rule, a gas can still be treated as almost incompressible if the Mach number is small (about 0.1 or 0.2). The Mach number is the ratio of flow speed to the local sound speed. Given sound speed in air as roughly 340 m s –1 , it is remarkable that a gas is still roughly incompressible for a speed as fast as say 40 m s –1 . Compressibility must be considered when the Mach number reaches say 0.4 or 0.5. One, two and three (1D, 2D, 3D) dimensional flows (8) A flow is termed 1D, 2D or 3D if the flow depends on one, two or three spatial coordinates respectively.
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