week3 - 2 Flow dynamics for an incompressible inviscid flow...

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Unformatted text preview: 2 Flow dynamics for an incompressible inviscid flow 2.1 Forces on a fluid Fluids move in response to the forces that act on each fluid particle. These forces are of two types: • Forces on a each parcel throughout the bulk of the fluid. These are known as body forces . In a gravitational potential, the force is proportional to the volume δV of the fluid element. In the typical case of a uniform gravitational field (as near the surface of the earth) the force is ρδV g . • The force is transmitted across the surface element δS of a fluid parcel, so the force is exerted by the fluid on the exterior of δV or vice versa. These are surface forces . When the fluid is a rest, the surface force must be in the direction of the normal, n . (The definition of a fluid is one that remains at rest unless a tangential force is applied). When a fluid is in motion, tangential components of the force on δS can occur. These are associated with viscosity which describes the effect that one layer of fluid in motion has on an adjacent layer. Defn An fluid is said to be inviscid when viscous effects are small enough to be ide- alised as zero. For an inviscid fluid, the ‘surface stress’ (i.e. Force per unit area) is in the direction n normal to the surface δS even when the fluid is in motion . In other words, the force exerted by the exterior fluid on the fluid inside δV is F s = − p n δS . The magnitude of this force is − pδS , where p ( x ,t ) is the pressure . It is directed inwards because fluids are usually in a state of compression. 2.2 Equation of motion We apply Newton’s law to a fixed (but arbitrary) volume V with surface S . The momen- tum within V may change as a result of (i) body forces (ii) surface forces, and (iii) fluid momentum flow out of V . x y z S dS dS V n n ( u · n ) δt ρ u Total momentum in V = integraldisplay V ρ u dV In time δt , the fluid momentum leaving a small section of surface δS is ρ u ( u · n ) δtδS , as illustrated in the Figure. So rate of momentum transport (i.e. the momentum flux) is ρ u ( u · n ) δS . Notice that the argument is very similar to the case of mass flux, discussed earlier....
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week3 - 2 Flow dynamics for an incompressible inviscid flow...

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