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Note for Fluid Mechanics - FM by Suhas Mondal

  • Fluid Mechanics - FM
  • Note
  • West Bengal University of technology - WBUT
  • Civil Engineering
  • B.Tech
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Introduction to Fluid Mechanics Definition of a fluid A fluid is defined as a substance that deforms continuously under the action of a shear stress, however small magnitude present. It means that a fluid deforms under very small shear stress, but a solid may not deform under that magnitude of the shear stress. Fig.L-1.1a: Deformation of solid under a constant shear force Fig.L-1.1b: Deformation of fluid under a constant shear force By contrast a solid deforms when a constant shear stress is applied, but its deformation does not continue with increasing time. In Fig.L1.1, deformation pattern of a solid and a fluid under the action of constant shear force is illustrated. We explain in detail here deformation behaviour of a solid and a fluid under the action of a shear force. In Fig.L1.1, a shear force F is applied to the upper plate to which the solid has been bonded, a shear stress resulted by the force equals to, Where A is the contact area of the upper plate. We know that in the case of the solid block the deformation is proportional to the shear stress t provided the elastic limit of the solid material is not exceeded. When a fluid is placed between the plates, the deformation of the fluid element is illustrated in Fig.L1.3. We can observe the fact that the deformation of the fluid element continues to increase as long as the force is applied. The fluid particles in direct contact with the plates move with the same speed of the plates. This can be interpreted that there is no slip at the boundary. This fluid behavior has been verified in numerous experiments with various kinds of fluid and boundary material. In short, a fluid continues in motion under the application of a shear stress and can not sustain any shear stress when at rest. 2

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Fluid as a continuum In the definition of the fluid the molecular structure of the fluid was not mentioned. As we now the fluids are composed of molecules in constant motions. For a liquid, molecules are closely spaced compared with that of a gas. In most engineering applications the average or macroscopic effects of a large number of molecules is considered. We thus do not concern about the behavior of individual molecules. The fluid is treated as an infinitely divisible substance, a continuum at which the properties of the fluid are considered as a continuous (smooth) function of the space variables and time. To illustrate the concept of fluid as a continuum consider fluid density as a fluid property at a small region.(Fig.L1.2 (a)). Density is defined as mass of the fluid molecules per unit volume. Thus the mean density within the small region C could be equal to mass of fluid molecules per unit volume. When the small region C occupies space which is larger than the cube of molecular spacing, the number of the molecules will remain constant. This is the limiting volume above which th fect of molecular variations on fluid properties is negligible. A plot of the mean density versus the size of unit volume is illustrated in Fig.L1.2 (b). for all liquids and for gases at atmospheric Note that the limiting volume is about temperature. Within the given limiting value, air at the standard condition has approximately molecules. It justifies in defining a nearly constant density in a region which is larger than the limiting volume. In conclusion, since most of the engineering problems deal with fluids at a dimension which is larger than the limiting volume, the assumption of fluid as a continuum is valid. For example the fluid density is defined as a function of space (for Cartesian coordinate system, x, y, and z) and . This simplification helps to use the differential calculus for solving fluid time (t ) by problems 3

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Properties of fluid Some of the basic properties of fluids are discussed belowDensity : As we stated earlier the density of a substance is its mass per unit volume. In fluid mechanic it is expressed in three different ways1. Mass density ρ is the mass of the fluid per unit volume (given by Eq.L1.1) 2. Specific weight, w: - As we express a mass M has a weight W=Mg . The specific weight of the fluid can be defined similarly as its weight per unit volume. 3. Relative density (Specific gravity), S :Specific gravity is the ratio of fluid density (specific weight) to the fluid density (specific weight) of a standard reference fluid. For liquids water a is considered as standard fluid. Similarly for gases air at specific temperature and pressure is considered as a standard reference fluid. Units: pure number having no units Dimensio Typical vales : - Mercury- 13.6 Water-1 4

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