Fluid Mechanics 2 Notes PDF

MASS CONSERVATION FOR A FLOW SYSTEM Derivation of continuity equation in differential form: 𝜕𝜌 + ∇ 𝜌𝒗 = 0 𝜕𝑡 Describe the transport of a quantity (mass) Its derivation from an infinitesimal control volume. DIFFERENTIAL FORM OF MASS CONSERVATION...

MASS CONSERVATION FOR A FLOW SYSTEM Derivation of continuity equation in differential form: 𝜕𝜌 + ∇ 𝜌𝒗 = 0 𝜕𝑡 Describe the transport of a quantity (mass) Its derivation from an infinitesimal control volume. DIFFERENTIAL FORM OF MASS CONSERVATION Infinitesimal control volume From the general form of law of mass conservation for a control volume: 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 + 𝑜𝑢𝑡 𝑜𝑓 𝑐. 𝑣. − =0 𝑖𝑛𝑡𝑜 𝑐. 𝑣. 𝑤𝑖𝑡ℎ𝑖𝑛 𝑐. 𝑣. 𝜕 𝜕𝜌 𝑑𝑦 𝜌𝑑𝑉 = 𝑑𝑥𝑑𝑦𝑑𝑧 𝜕𝑡 𝜕𝑡 𝑦 𝑑𝑧 𝑦 𝑑𝑥 𝑥 𝑥 𝑧 𝑑𝑉 = 𝑑𝑥𝑑𝑦𝑑𝑧 DIFFERENTIAL FORM OF MASS CONSERVATION Infinitesimal control volume Mass flowrate: 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 × 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 × 𝑎𝑟𝑒𝑎 𝑑𝑦 𝑑𝑧 𝑑𝑥 At x: 𝜌 𝑣𝑥 ቚ 𝑑𝑦𝑑𝑧 = (𝜌𝑣𝑥 )𝒙 𝑑𝑦𝑑𝑧 𝒙 − 𝒔𝒖𝒓𝒇𝒂𝒄𝒆𝒔 𝒙 At x + dx: 𝜌 𝑣𝑥 ቚ 𝑑𝑦𝑑𝑧 = (𝜌𝑣𝑥 )𝒙+𝒅𝒙 𝑑𝑦𝑑𝑧 𝒙+𝒅𝒙 𝑑𝑧 𝑑𝑦 Net outflow of mass from control volume: 𝑣𝑥 𝑣𝑥 x-direction : (𝜌𝑣𝑥 )𝒙+𝒅𝒙 𝑑𝑦𝑑𝑧 − (𝜌𝑣𝑥 )𝒙 𝑑𝑦𝑑𝑧 𝑑𝑥 y-direction : (𝜌𝑣𝑦 )𝒚+𝒅𝒚 𝑑𝑥𝑑𝑧 − (𝜌𝑣𝑦 )𝒚 𝑑𝑥𝑑𝑧 𝑥 𝑥 + 𝑑𝑥 𝑦 z-direction : (𝜌𝑣𝑧 )𝒛+𝒅𝒛 𝑑𝑥𝑑𝑦 − (𝜌𝑣𝑧 )𝒛 𝑑𝑥𝑑𝑦 𝑥 𝑧 DIFFERENTIAL FORM OF MASS CONSERVATION Infinitesimal control volume The inflow and outflow of mass towards 𝑑𝑦 y-directions through y-surfaces: 𝑑𝑧 𝑑𝑥 𝒛 − 𝒔𝒖𝒓𝒇𝒂𝒄𝒆𝒔 𝜌 𝑣𝑦 ቚ 𝑑𝑥𝑑𝑧 = (𝜌𝑣𝑦 )𝒚 𝑑𝑥𝑑𝑧 𝒚 𝒚 − 𝒔𝒖𝒓𝒇𝒂𝒄𝒆𝒔 𝑑𝑥 𝑣𝑧 𝜌 𝑣𝑦 ቚ 𝑑𝑥𝑑𝑧 = (𝜌𝑣𝑦 )𝒚+𝒅𝒚 𝑑𝑥𝑑𝑧 𝑣𝑦 𝒚+𝒅𝒚 𝑑𝑥 The inflow and outflow of mass towards z-directions 𝑑𝑧 through z-surfaces: 𝑧 𝑦 + 𝑑𝑦 𝑑𝑦 𝑑𝑧 𝜌 𝑣𝑧 ቚ 𝑑𝑥𝑑𝑦 = (𝜌𝑣𝑧 )𝒛 𝑑𝑥𝑑𝑦 𝑑𝑦 𝒛 𝑦 𝑣𝑧 𝑧 + 𝑑𝑧 𝑣𝑦 𝑦 𝜌 𝑣𝑧 ቚ 𝑑𝑥𝑑𝑦 = (𝜌𝑣𝑧 )𝒛+𝒅𝒛 𝑑𝑥𝑑𝑦 𝑥 𝒛+𝒅𝒛 𝑧 CONTINUITY EQUATION The sum of net outflow of mass flowrate within c.v.: 𝑑𝑦 𝑑𝑧 𝜌𝑣𝑥 𝒙+𝒅𝒙 − 𝜌𝑣𝑥 𝒙 𝑑𝑦𝑑𝑧 + 𝜌𝑣𝑦 − 𝜌𝑣𝑦 𝑑𝑥𝑑𝑧 + 𝜌𝑣𝑧 𝒛+𝒅𝒛 − 𝜌𝑣𝑧 𝒛 𝑑𝑥𝑑𝑦 𝑑𝑥 𝒚+𝒅𝒚 𝒚 From the general form, the law of mass conservation in differential form: 𝜕𝜌 𝑑𝑥𝑑𝑦𝑑𝑧 + 𝜌𝑣𝑥 𝒙+𝒅𝒙 − 𝜌𝑣𝑥 𝒙 𝑑𝑦𝑑𝑧 + 𝜌𝑣𝑦 − 𝜌𝑣𝑦 𝑑𝑥𝑑𝑧 + 𝜌𝑣𝑧 𝒛+𝒅𝒛 − 𝜌𝑣𝑧 𝒛 𝑑𝑥𝑑𝑦 = 0 𝜕𝑡 𝒚+𝒅𝒚 𝒚 Dividing by 𝑑𝑥𝑑𝑦𝑑𝑧 and taking the limit of 𝑑𝑥 → 0, 𝑑𝑦 → 0 and 𝑑𝑧 → 0 : 𝑪𝒐𝒏𝒕𝒊𝒏𝒖𝒊𝒕𝒚 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝜕𝜌 𝜕 𝜕 𝜕 𝜕𝜌 + (𝜌𝑣𝑥 ) + (𝜌𝑣𝑦 ) + (𝜌𝑣𝑧 ) = 0 + ∇ (𝜌𝒗) = 0 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑡  Infinitesimal control volume for CONTINUITY EQUATION cylindrical polar coordinate. CYLINDRICAL COORDINATE SYSTEM 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒇𝒓𝒐𝒎 𝒐𝒓𝒊𝒈𝒊𝒏 𝑦 𝑣𝜃 𝑣𝑟 𝑦 𝑟 𝑦 𝑣𝜃 𝑣𝑟 𝜃 𝑥 Coordinate transformation 𝑥 𝑟 𝑣𝑍 𝑥 𝒂𝒏𝒈𝒍𝒆 𝒇𝒓𝒐𝒎 𝒙 − 𝒂𝒙𝒊𝒔 𝑟= 𝑥2 + 𝑦2 𝑦 = 𝑟 sin 𝜃 𝜃 𝑧 𝑦 2-DIMENSIONAL 𝑥 = 𝑟 cos 𝜃 𝜃 = tan−1 𝑥 3-DIMENSIONAL CONTINUITY EQUATION CYLINDRICAL COORDINATE SYSTEM Infinitesimal control volume for cylindrical polar coordinate. 𝜕𝜌 Rate of mass accumulation: 𝜕𝑡 Net outflow of mass flowrate: 1 𝜕 1 𝜕 𝜕 ∇ 𝜌𝒗 = (𝑟𝜌𝑣𝑟 ) + (𝜌𝑣𝜃 ) + (𝜌𝑣𝑧 ) 𝑟 𝜕𝑟 𝑟 𝜕𝜃 𝜕𝑧 Continuity equation for cylindrical polar coordinate: 𝜕𝜌 1 𝜕 1 𝜕 𝜕 + 𝑟𝜌𝑣𝑟 + 𝜌𝑣𝜃 + 𝜌𝑣𝑧 = 0 𝜕𝜌 𝜕𝑡 𝑟 𝜕𝑟 𝑟 𝜕𝜃 𝜕𝑧 + ∇ (𝜌𝒗) = 0 𝜕𝑡 𝒄𝒐𝒎𝒑𝒂𝒄𝒕 𝒇𝒐𝒓𝒎 CONTINUITY EQUATION COMMON FLOW CASES 1. Steady flow: No change of properties with time. 0 𝜕𝜌 𝜕 𝜕 𝜕 𝜕 𝜕 𝜕 + (𝜌𝑣𝑥 ) + (𝜌𝑣𝑦 ) + (𝜌𝑣𝑧 ) = 0 (𝜌𝑣𝑥 ) + (𝜌𝑣𝑦 ) + (𝜌𝑣𝑧 ) = 0 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑥 𝜕𝑦 𝜕𝑧 2. Steady and incompressible flow: Steady with constant density. common form 𝜕 𝜕 𝜕 𝜕 𝜕 𝜕 (𝜌𝑣𝑥 ) + (𝜌𝑣𝑦 ) + (𝜌𝑣𝑧 ) = 0 (𝑣𝑥 ) + (𝑣𝑦 ) + (𝑣𝑧 ) = 0 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑥 𝜕𝑦 𝜕𝑧 Cylindrical polar coordinate form 1 𝜕 1 𝜕 𝜕 𝑟𝑣𝑟 + 𝑣𝜃 + 𝑣 =0 𝑟 𝜕𝑟 𝑟 𝜕𝜃 𝜕𝑧 𝑧 CONTINUITY EQUATION SUMMARY In analyzing fluid problems, mathematical equations are derived from the laws of physics: 𝜕𝜌 Law of mass conservation: Continuity equation. + ∇ (𝜌𝒗) = 0 𝜕𝑡 𝜕𝒗 Newton’s second law of motion: Equations of motion. 𝜌 + 𝒗 ∇𝒗 = 𝜌𝒈 − ∇𝑃 + ∇ τ Cauchy’s equation 𝜕𝑡 𝑑𝒗 Navier-Stokes equation 𝜌 = 𝜌𝒈 − ∇𝑃 + 𝜇∇2 𝒗 𝑑𝑡