Fluid_Dynamics.pdf - Summary of the Equations of Fluid...

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Fluid Dynamics 1/22 Summary of the Equations of Fluid Dynamics Reference: Fluid Mechanics, L.D. Landau & E.M. Lifshitz 1 Introduction Emission processes give us diagnostics with which to estimate important parameters, such as the density, and magnetic field, of an astrophysical plasma. Fluid dynamics provides us with the capability of understanding the transport of mass, momentum and energy. Normally one spends more than a lecture on Astrophysical Fluid Dynamics since this relates to many areas of astrophysics. In following lectures we are going to consider one principal application of astrophysical fluid dynamics – accretion discs. Note also that magnetic fields are not included in the following. Again a full treatment of magnetic fields warrants a full course. 2 The fundamental fluid dynamics equations The equations of fluid dynamics are best expressed via conservation laws for the conservation of mass, mo- mentum and energy.
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Fluid Dynamics 2/22 2.1 Conservation of mass Consider the rate of change of mass within a fixed volume. This changes as a result of the mass flow through the bounding surface. Using the divergence theorem, The continuity equation Since the volume is arbitrary, V S v i n i Control volume for as- sessing conservation of mass. t ρ V d V ρ v i n i S d S = t ρ V d V x i ρ v i ( 29 V d V + 0 = ρ t ----- x i ρ v i ( 29 + V d V 0 = ρ t ----- x i ρ v i ( 29 + 0 =
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Fluid Dynamics 3/22 2.2 Conservation of momentum Consider now the rate of change of momentum within a vol- ume. This decreases as a result of the flux of momentum through the bounding surface and increases as the result of body forces (in our case gravity) acting on the volume. Let and then There is an equivalent way of thinking of , which is often useful, and that is, is the component of the force exerted on the fluid exterior to by the fluid interior to . n i Π ij n j V S Π ij Flux of i component of momentum in the j direction = f i Body force per unit mass = t ρ v i V d V Π ij n j S ρ f i V d V + d S = Π ij Π ij n j dS i th S S
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Fluid Dynamics 4/22 Again using the divergence theorem, Gravity For gravity we use the gravitational potential For a single gravitating object of mass t ρ v i ( 29 x j ∂Π ij + V d V ρ f i V d V = t ρ v i ( 29 x j ∂Π ij + ρ f i = f i x i ∂φ G = M φ G GM r --------- =
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Fluid Dynamics 5/22 and for a self-gravitating distribution where is Newton’s constant of gravitation. Expressions for The momentum flux is composed of a bulk part plus a part resulting from the motion of particles moving with respect to the centre of mass velocity of the fluid . For a perfect fluid (an approximation often used in as- trophysics), we take to be the isotropic pressure, then The equations of motion are then: 2 φ G 4 π G ρ = φ G G ρ x i ( 29 x i x i ------------------- d 3 x V = G Π ij v i ( 29 p Π ij ρ v i v j p δ ij + = t ρ v i ( 29 x j ρ v i v j
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