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Mechanics: Fluid

Fluid


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Fluid Properties
 Compressibility
 Speed of Sound
 Mach Number

Fluid Properties

Compressibility

Compressibility is used to described the easy of volume change for a fixed massed of fluid due to a pressure change. Bulk modulus is usually used to characterize the compressibility in an analogous manner to Young's modulus of elasticity for solids. For a given mass, image the volume change of a fluid is inversely proportional to its density. And the bulk modulus is expressed as

image

and can be approximated by a differential change as

image or image

For ideal gas compression or expansion under constant temperature,i.e. isothermal process,

image

imply image

And for the compression or expansion is reversible and adiabatic, i.e. frictionless and no heat exchange with surroundings under isentropic process,

image

where image; image

imply image

If the pressure changes are small, gases can be treated as incompressible fluid.

Speed of Sound

Speed of sound which describe the speed of pressure wave propagation in the fluid, is an important parameter in compressible flow. When a piston moves forward, the generated pressure wave will travel through the fluid at the speed of sound image. When the piston moves a constant incremental velocity image, there will have a small increase in local pressure at the wave front and a small difference in the fluid states across the wave front. Considering a moving control volume that encloses the wave front,  fluid will move toward the wave front into the control volume at a relative speed of image and fluid will also move away from the wave front and leave the control volume at a relative speed of image

image

From the conservation of mass,

image

From the steady flow energy balance equation, by neglecting potential energy change and there is no heat or work crosses the boundaries,

image

Imply image

The small disturbance can also be assumed as an isentropic process, imply

image

Therefore image or image

and it can be expressed as image
or image

and the seed of sound is a function of temperature only.

Mach Number

Mach number, Ma, is defined as the ratio of the fluid flow speed to the speed of sound in the same fluid at the same state, i.e.image and is another important parameter in compressible flow.


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