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Conservation of the Linear Momentum CH.5. BALANCE PRINCIPLES Multimedia Course on Continuum Mechanics Overview Balance Principles Lecture 1 Convective Flux or Flux by Mass Transport Lecture 2 Lecture 3 Local and Material Derivative of a Volume Integral Lecture 4 Conservation of Mass Spatial Form Lecture 5 Material Form Reynolds Transport Theorem Lecture 6 Reynolds Lemma General Balance Equation Lecture 7 Linear Momentum Balance Global Form Lecture 8 Local Form 2 Overview (cont’d) Angular Momentum Balance Global Spatial Lecture 9 Local Form Mechanical Energy Balance External Mechanical Power Lecture 10 Mechanical Energy Balance External Thermal Power Energy Balance Thermodynamic Concepts Lecture 11 First Law of Thermodynamics Internal Energy Balance in Local and Global Forms Lecture 12 Second Law of Thermodynamics Lecture 13 Reversible and Irreversible Processes Lecture 14 Clausius-Planck Inequality Lecture 15 3 Overview (cont’d) Governing Equations Governing Equations Lecture 16 Constitutive Equations The Uncoupled Thermo-mechanical Problem 4 5.1. Balance Principles Ch.5. Balance Principles 5 Balance Principles The following principles govern the way stress and deformation vary in the neighborhood of a point with time. REMARK The conservation/balance principles: These principles are always Conservation of mass valid, regardless of the type of material and the range of Linear momentum balance principle displacements or deformations. Angular momentum balance principle Energy balance principle or first thermodynamic balance principle The restriction principle: Second thermodynamic law The mathematical expressions of these principles will be given in, Global (or integral) form Local (or strong) form 6 5.2. Convective Flux Ch.5. Balance Principles 7 Convection The term convection is associated to mass transport, i.e., particle movement. Properties associated to mass will be transported with the mass when there is mass transport (particles motion) convective transport Convective flux of an arbitrary property A through a control surface S : amountof crossing S Φ= A S unitof time 8 Convective Flux or Flux by Mass Transport Consider: An arbitrary property of a continuum medium (of any tensor order) A The description of the amount of the property per unit of mass, Ψ () x , t (specific content of the property ) . A The volume of particles dV crossing a differential surface dS during the interval [] t , t + dt is dV=⋅=⋅ dS dhvn dt dS dm=ρρ dV =vn ⋅ dSdt Then, The amount of the property crossing the differential surface per unit of time is: Ψ dm d Φ= =ρ Ψ⋅vn dS S dt 9 Convective Flux or Flux by Mass Transport Consider: An arbitrary property of a A inflow continuum medium (of any tensor order) vn⋅≤0 outflow The specific content of A (the amount vn⋅≥0 of per unit of mass) Ψ x , t . A () Then, The convective flux of through a spatial surface, S , with unit normal n is: A v is velocity Φ()t =ρ Ψ⋅vndS Where: S ∫s ρ is density If the surface is a closed surface, SV = ∂ , the net convective flux is: Φ∂ ()t =ρ Ψ⋅vndS = outflow - inflow V ∫∂V 10 Convective Flux REMARK 1 The convective flux through a material surface is always null. REMARK 2 Non-convective flux (conduction, radiation). Some properties can be transported without being associated to a certain mass of particles. Examples of non-convective transport are: heat transfer by conduction, electric current flow, etc. Non-convective transport of a certain property is characterized by the non- convective flux vector (or tensor) qx () , t : non - convectiveflux =qn⋅=dS ; convectiveflux ρψ vn⋅dS ∫∫ss non-convective flux convective vector flux vector 11 Example Compute the magnitude and the convective flux Φ S which correspond to the following properties: a) volume b) mass c) linear momentum d) kinetic energy 12 Φ =ρ Ψ⋅ S ()t ∫ vndS Example - Solution s a) If the arbitrary property is the volume of the particles: ≡ V A The magnitude “property content per unit of mass” is volume per unit of mass, i.e., the inverse of density: V 1 Ψ= = M ρ The convective flux of the volume of the particles V through the surface S is: 1 Φ=ρ vn ⋅dS = vn ⋅ dS VOLUME FLUX S ∫∫ssρ 13 Φ =ρ Ψ⋅ S ()t ∫ vndS Example - Solution s b) If the arbitrary property is the mass of the particles: ≡ M A The magnitude “property per unit of mass” is mass per unit of mass, i.e., the unit value: M Ψ= =1 M The convective flux of the mass of the particles M through the surface S is: Φ=ρρ1 vn ⋅dS = vn ⋅ dS MASS FLUX S ∫∫ss 14 Φ =ρ Ψ⋅ S ()t ∫ vndS Example - Solution s c) If the arbitrary property is the linear momentum of the particles: ≡ M v A The magnitude “property per unit of mass” is mass times velocity per unit of mass, i.e., velocity: M v Ψ = = v M The convective flux of the linear momentum of the particles M v through the surface S is: Φ =ρ vvn() ⋅ dS MOMENTUM FLUX S ∫s 15 Φ =ρ Ψ⋅ S ()t ∫ vndS Example - Solution s d) If the arbitrary property is the kinetic energy of the particles: 1 ≡ M v2 A 2 The magnitude “property per unit of mass” is kinetic energy per unit of mass, i.e.: 1 M v2 1 Ψ=2 = v2 M 2 1 The convective flux of the kinetic energy of the particles M v 2 through the surface S is: 2 1 Φ=ρ v2 () vn ⋅ dS KINETIC ENERGY FLUX S ∫s 2 16 5.3. Local and Material Derivative of a Volume Integral Ch.5. Balance Principles 17 Derivative of a Volume Integral Consider: An arbitrary property of a continuum medium (of any tensor order) A The description of the amount of the property per unit of volume (density of the property ), µ () x , t A REMARK and Ψ are related The total amount of the property through . in an arbitrary volume, V , is: Q()t Q()() t= ∫ µ x, t dV V Q()tt+∆ The time derivative of this volume integral is: Qt()()+∆ t − Qt Qt′() = lim ∆→t 0 ∆t 18 Local Derivative of a Volume Integral Consider: Q()t The volume integral Q()() t= ∫ µ x, t dV V Q()tt+∆ Control Volume, V The local derivative of Qt () is: µµ()()xx,,t+∆ t dV − t dV local not ∂ ∫∫ REMARK = µ ()x,t dV = lim VV ∫ ∆→t 0 derivative ∂∆t V t The volume is fixed in space (control volume). It can be computed as: µµ()()xx,t+∆ t dV − , t dV ∂ Qt()()+∆ t − Qt ∫∫ µ ()x,t dV = lim = lim VV= ∫ ∆→tt00∆→ ∂∆ttV ∆ t [µµ()()xx ,t+∆ t − ,] t dV ∫ µµ()()xx,,tt+∆ − t∂ µ() x , t = lim V = lim dV = dV ∆→tt00∆∫∫∆→ ∆∂ tVV tt ∂µx,t ∂t 19 Material Derivative of a Volume Integral Consider: The volume integral Q()() t= ∫ µ x, t dV V Q()tt+∆ The material derivative of Qt () is: Q()t material not d derivative = ∫ µ ()x,t dV = dt ≡ VVt µµ()()xx,,t+∆ t dV − t dV REMARK ∫∫Vt()+∆ t Vt() = lim The volume is mobile in space ∆→t 0 ∆ t and can move, rotate and It can be proven that: deform (material volume). dd∂∂µµ µ()x,t dV =µ dV + ∇⋅() µ vv dV = +∇⋅() µµ dV = + ∇⋅ v dV dt ∫∂∂t ∫∫ ∫t ∫dt VVt ≡ V V V V material local convective derivative of derivative of derivative of the integral the integral the integral 20 5.4. Conservation of Mass Ch.5. Balance Principles 21 Principle of Mass Conservation It is postulated that during a motion there are neither mass sources nor mass sinks, so the mass of a continuum body is a conserved quantity (for any part of the body). The total mass () t of the system satisfies:M ()()t= tt +∆ >0 MM Where: ()()t= ρ x, t dV∀∆ V ⊂ V ∆ tt M ∫ Vt ()()tt+∆ =ρ x, ttdVV +∆ ∀∆ ⊂ V ∆ tt+∆ tt +∆ M ∫ Vtt+∆ 22 Conservation of Mass in Spatial Form Conservation of mass requires that the material time derivative of the mass () t be zero for any region of a material volume, M ()()tt+∆ − t d ′()t = lim MM= ρdV=0 ∀∆ V ⊂ V , ∀ t ∫∆⊂≡ M ∆→t 0 ∆t dt Vtt VV The global or integral spatial form of mass conservation principle: ddµ µµ()xv,t dV =( + ∇⋅ ) dV dt ∫∫dt VVt ≡ V ddρ ρρ(,)xvt dV = + ∇ ⋅dV =0 ∀∆ V ⊂ V , ∀ t ∫∫∆ ⊂≡ ∆⊂ dt Vtt VV VVdt By a localization process we obtain the local or differential spatial form of mass conservation principle: for∆→ V dV(,)x t (localization process) CONTINUITY EQUATION dtρρ(,)xx∂ (,)t +(ρρ∇∇ ⋅vx )( ,t ) = + ⋅(vx )( ,t ) = 0 ∀∈ x Vt , ∀ dt ∂t 23 Conservation of Mass in Material Form d F 1 d F =Fv∇ ⋅ ()∇ ⋅=v Consider the relations: dt F dt dV= F dV0 The global or integral material form of mass conservation principle can be rewritten as: dtF ∂ FX( ,) ddρρ 1 ∂ ρ(X ,) t ()(+ρρ ∇⋅vdV = + ) dV = (FX ( ,t ) +ρ) dV0 ∫∫V dt dtF dt ∫∂∂t t ∆VV0 ∂ρ F dV0 (X ,)t ∂ [ρ |FX |]( ,t ) ∂ t ∂t ∂ → ρ = ∀∆ ⊂ ∀ ∫ FX(),t dV00 V00 V , t ∂t ∆VV00⊂ The local material form of mass conservation principle reads : ∂ ρ ρ = ρρ= 0 FX()(),0t tt=0 XFtt=0 () X F() X ρ = ∀∈X Vt, ∀ ∂t t F 0 =1 t 24 5.5. Reynolds Transport Theorem Ch.5. Balance Principles 25 dρ +ρ∇ ⋅=v 0 Reynolds Lemma dt Consider: An arbitrary property of a continuum medium (of any tensor order) A The spatial description of the amount of the property per unit of mass, ψ () x , t (specific contents of ) A The amount of the property in the continuum body at time t A for an arbitrary material volume is: Q() t= ∫ ρψ dV VVt = Using the material time derivative leads to, d d ddψρ Q′() t =ρψdV =()() ρψ + ρψ ∇∇ ⋅vvdV =ρ + ψ( +⋅ ρ ) dV dt ∫∫dt ∫dt dt VVt ≡ V V ddψρ =ρψ + Thus, dt dt =0 (continuity equation) ddψ REYNOLDS ρψdV= ρ dV dt ∫∫dt LEMMA VVt ≡ V 26 d ∂ µ()xv,t dV= µµ dV + ∇⋅() dV dt ∫∂t ∫∫ VVt ≡ V V Reynolds Transport Theorem The amount of the property in the continuum body at time t for A an arbitrary fixed control volume is: Q() t= ∫ ρψ dV V Using the material time derivative leads to, d ∂ ()ρψ ρψ dV = dV +⋅∇ ()ρ ψ; v dV dt ∫∫∂t ∫ VVt ≡ V V dψ dψ = ∫ ρ dV = ∫ nv⋅()ρψ dV ρ V dt ∂V dt And, introducing the Reynolds Lemma ∂V and Divergence Theorem: dV dψ ∂ ()ρψ ∫∫ρ dV = dV +⋅ ∫ρψ vn dS VVdt ∂t ∂ V eˆ ρψ REMARK 3 ∫ dV eˆ 2 V The Divergence Theorem: eˆ1 ∫ ∇⋅vdV = ∫∫ nv ⋅ dS = vn ⋅ dS V ∂∂ VV 27 dψ ∂ (ρψ ) ρ dV = dV +⋅ρψ vn dS ∫∫dt ∂t ∫ Reynolds Transport TheoremVV ∂ V The eq.
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