Equations of Fluid Dynamics: Conservation of Mass, Momentum, and Energy, Study notes of Thermodynamics

Download Equations of Fluid Dynamics: Conservation of Mass, Momentum, and Energy and more Study notes Thermodynamics in PDF only on Docsity! 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. 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 vi ni Control volume for as- sessing conservation of mass. t∂ ∂ ρ Vd V ∫ ρvini Sd S ∫–= t∂ ∂ ρ Vd V ∫ xi∂ ∂ ρvi( ) Vd V ∫+ 0= ρ∂ t∂ ----- xi∂ ∂ ρvi( )+    Vd V ∫⇒ 0= ρ∂ t∂ ----- xi∂ ∂ ρvi( )+ 0= 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: ∇2φG 4πGρ= φG⇒ G ρ xi′( ) xi xi′– -------------------d3x′ V′ ∫–= G Πij vi( ) p Πij ρviv j pδij+= t∂ ∂ ρvi( ) x j∂ ∂ ρviv j pδij+( )+ ρ xi∂ ∂φG–= t∂ ∂ ρvi( ) x j∂ ∂ ρviv j( )+⇒ xi∂ ∂p– ρ xi∂ ∂φG–= Fluid Dynamics 6/22 There is also another useful form for the momentum equation derived using the continuity equation. Hence, another form of the momentum equation is: On dividing by the density t∂ ∂ ρvi( ) x j∂ ∂ ρviv j( )+ vi t∂ ∂ρ ρ t∂ ∂vi vi x j∂ ∂ ρv j( ) ρv j x j∂ ∂vi+ + += vi t∂ ∂ρ x j∂ ∂ ρv j( )+ ρ t∂ ∂vi ρv j x j∂ ∂vi++= ρ t∂ ∂vi ρv j x j∂ ∂vi+= ρ t∂ ∂vi ρv j x j∂ ∂vi+ p∂ xi∂ ------– ρ φG∂ xi∂ ---------–= t∂ ∂vi v j x j∂ ∂vi+ 1 ρ -- p∂ xi∂ ------– φG∂ xi∂ ---------–= Fluid Dynamics 7/22 Differentiation following the motion This is a good place to introduce differentiation following the motion. For a function , the variation of following the motion of a fluid element which has coordinates is given by: Hence, the momentum equation can be written compactly as f xi t,( ) f xi xi t( )= df dt ----- t∂ ∂ f xi∂ ∂ f dxi dt -------+ t∂ ∂ f vi xi∂ ∂ f += = ρ dvi dt ------- xi∂ ∂p– ρ xi∂ ∂φG–= Fluid Dynamics 10/22 Specific enthalpy A commonly used thermodynamic variable is the specific enthalpy: In terms of the specific enthalpy, the equation becomes For a parcel of fluid following the motion, we obtain, after dividing by the time increment of a volume element, h ε p+ ρ -----------= kTds d ε ρ --    pd 1 ρ --   += kTds d ε p+ ρ -----------    d p ρ --   – pd 1 ρ --   + dh dp ρ -----–= = ρkT ds dt ----- dε dt ----- ε p+( ) ρ ---------------- dρ dt -----–= kT ds dt ----- dh dt ----- 1 ρ -- dp dt -----–= Fluid Dynamics 11/22 The fluid is adiabatic when there is no transfer of heat in or out of the volume element: The quantities , , etc. are perfect differentials, and these relationships are valid relations from point to point within the fluid. Two particular relationships we shall use in the following are: 2.3.1 Equation of state The above equations can be used to derive the equation of state of a gas in which the ratio of specific heats ( ) is a constant. Consider the following form of the entropy, internal energy, pressure relation: kT ds dt ----- 0= dε dt ----- ε p+( ) ρ ---------------- dρ dt -----– 0= dh dt ----- 1 ρ -- dp dt -----– 0= ⇒ ds dε dp ρkT t∂ ∂s t∂ ∂ε h t∂ ∂ρ –= ρkT xi∂ ∂s ρ xi∂ ∂h xi∂ ∂p–= γ cp cv⁄= ρkTds dε ε p+( ) ρ ---------------- dρ–= Fluid Dynamics 12/22 In a perfect gas, where is the mean molecular weight and Hence, (We can discard the since the origin of entropy is arbitrary.) p ρkT µmp ----------= µ p γ 1–( )ε= ε p+⇒ γε= µmp γ 1–( )εds dε γε ρ ---- dρ–= µmp γ 1–( )ds⇒ dε ε ----- γ ρ -- dρ–= µmp γ 1–( ) s s0–( )⇒ ε γ ρln–ln= ε ργ -----⇒ µmp γ 1–( ) s s0–( )[ ]exp µmp γ 1–( )s[ ]exp= = s0 Fluid Dynamics 15/22 We now eliminate the term using continuity, viz and we obtain t∂ ∂ρ t∂ ∂ρ xi∂ ∂ ρvi( )–= vi xi∂ ∂p – ρkT ds dt ----- t∂ ∂ε– h xi∂ ∂ ρvi( )– ρvi xi∂ ∂h –= ρkT ds dt ----- t∂ ∂ε– xi∂ ∂ ρhvi( )–= Fluid Dynamics 16/22 The term When the gravitational potential is constant in time, Hence, the energy equation ρvi xi∂ ∂φG– xi∂ ∂ ρφGvi( )– φG xi∂ ∂ ρvi( )+= xi∂ ∂ ρφGvi( ) φG t∂ ∂ρ ––= xi∂ ∂ ρφGvi( ) t∂ ∂ ρφG( ) ρ t∂ ∂φG+––= t∂ ∂φG 0= ρvi xi∂ ∂φG–⇒ xi∂ ∂ ρφGvi( ) t∂ ∂ ρφG( )––= t∂ ∂ 1 2 --ρv2     x j∂ ∂ 1 2 --ρv2v j   + vi xi∂ ∂P – ρvi φG∂ xi∂ ---------–= Fluid Dynamics 17/22 becomes Bringing terms over to the left hand side: When the fluid is adiabatic and we have the energy equation for a perfect fluid: t∂ ∂ 1 2 --ρv2     x j∂ ∂ 1 2 --ρv2v j   + ρkT ds dt ----- t∂ ∂ε– xi∂ ∂ ρhvi( )–= xi∂ ∂ ρφGvi( ) t∂ ∂ ρφG( )–– t∂ ∂ 1 2 --ρv2 ε ρφG+ +    x j∂ ∂ 1 2 --ρv2v j ρhv j ρφGv j+ +   + ρkT ds dt -----= ρkT ds dt ----- 0= t∂ ∂ 1 2 --ρv2 ε ρφG+ +    x j∂ ∂ 1 2 --ρv2v j ρhv j ρφGv j+ +   + 0= Fluid Dynamics 20/22 3.2 Energy conservation If we now take the scalar product of the momentum equation with we obtain that is, the same as before, but with the additional term Hence the energy equation becomes vi ρ t∂ ∂ 1 2 --v2     ρv j x j∂ ∂ 1 2 --v2    + vi xi∂ ∂P – ρvi φG∂ xi∂ ---------– vi x j∂ ∂σij+= vi x j∂ ∂σij x j∂ ∂ viσij( ) vi j, σij–= t∂ ∂ 1 2 --ρv2 ε ρφG+ +    x j∂ ∂ 1 2 --ρv2v j ρhv j ρφGv j viσij–+ +   + ρkT ds dt ----- vi j, σij–= Fluid Dynamics 21/22 The quantity is interpreted as the work done on the fluid by the viscous force; hence its appearance with terms that we associate with the energy flux. This is not the full story, however. When there is momentum transport associated with viscosity, there is also a heat flux, which is often represented as being proportional to the temperature gradient with a heat conduction coefficient , i.e. We then write the full energy equation as Conservation of energy is expressed by: and the entropy changes according to viσijn j q κ qi κ xi∂ ∂T –= t∂ ∂ 1 2 --ρv2 ε ρφG+ +    x j∂ ∂ 1 2 --ρv2v j ρhv j ρφGv j viσij– q j+ + +   + ρkT ds dt ----- vi j, σij– q j j,+= t∂ ∂ 1 2 --ρv2 ε ρφG+ +    x j∂ ∂ 1 2 --ρv2v j ρhv j ρφGv j viσij– q j+ + +   + 0= ρkT ds dt ----- vi j, σij q j j,–= Fluid Dynamics 22/22 The term represents viscous heating and the term represents escape of heat from the volume re- sulting from the heat flux. The viscous heating term can be written: remembering the definition of the shear tensor: vi j, σij q j j, σijvi j, 2ηsijsij ζvk k, 2+= sij 1 2 -- vi j, v j i, 2 3 --δijvk k,–+   =