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The Reynolds Transport Theorem Ú Correlation between System (Lagrangian) concept ↔ Control-volume (Eulerian) concept for comprehensive understanding of fluid motion? y Reynolds Transport Theorem Let’s set a fundamental equation of physical parameters B = mb where B: Fluid property which is proportional to amount of mass (Extensive property) b: B per unit mass (Independent to the mass) (Intensive property) r r e.g. a) If B = mV (Linear momentum): Extensive property r r then, b = V (Velocity): Intensive property 1 b) If B = mV 2 (Kinetic energy): Extensive property 2 1 then, b = V 2 : Intensive property 2 i. B of a system Bsys at a given instant, ∑ bi ( ρiδVi ) = ∫sys ρbdV Bsys = lim δV → 0 i δmi for ith fluid particle in the system where δVi : Volume of ith fluid particle And Time rate of change of Bsys, dBsys dt = ( d ∫ sys ρbdV ) dt ii. B of fluid in a control volume Bcv Bcv = lim ∑ bi ( ρiδVi ) = ∫cv ρbdV δV → 0 i and d (∫cv ρbdV ) dBcv = dt dt Ú Relationship between dBsys dt and Only difference from B of a system dBcv : Reynolds Transport Theorem dt Derivation of the Reynolds Transport Theorem Consider 1-D flow through a fixed control volume shown Fixed control surface at t (coincide with a system boundary) System boundary at t + δt a) At time t, Control volume (CV) & System (SYS): Coincide b) At t + δt (after δt ), CV: fixed & SYS: Move slightly ) Fluid particles at section (1): Move a distance dl1 = V1δt ) Fluid particles at section (2): Move a distance dl2 = V2δt ) I : Volume of Inflow (entering CV) ) II : Volume of Outflow (leaving CV) That is, SYS (at time t) = CV SYS (at time t + δt ) = CV – I + II Or if B: Extensive fluid property, then Bsys(t) = Bcv(t) (at time t) Bsys (t + δt ) = Bcv (t + δt ) − BI (t + δt ) + BII (t + δt ) (at time t + δt ) Then, Time rate of change in B can be; δBsys Bsys (t + δt ) − Bsys (t ) = Bcv(t), at time t = δt δt Bcv (t + δt ) − BI (t + δt ) + BII (t + δt ) − Bsys (t ) = δt B (t + δt ) − Bcv (t ) BI (t + δt ) BII (t + δt ) − + = cv δt δt δt In the limit δt → 0 , Left-side: δBsys DBsys = δt Dt (according to Lagrangian Concept) ⎛ ⎞ ∂⎜ ∫ ρbdV ⎟ B ( t + δ t ) − B ( t ) ∂ B ⎝ ⎠ cv = cv = cv 1st term on Right-side: lim cv δt → 0 ∂t δt ∂t B (t + δt ) 2nd term on Right-side: B&in = lim I = ρ1 A1V1b1 δt → 0 δt (4.13) because BI (t + δt ) = ( ρ1δV1 )b1 = ρ1 A1V1b1δt where A1 : Area at section (1) V1 : Velocity at section (1) B (t + δt ) 3rd term on Right-side: B& out = lim II = ρ 2 A2V2b2 δt → 0 δt (4.12) because BII (t + δt ) = ( ρ 2δV2 )b2 = ρ 2 A2V2b2δt Relationship between the time rate of change of Bsys and Bcv ∴ DBsys Dt = ∂Bcv & ∂B + Bout − B&in = cv + ρ 2 A2V2b2 − ρ1 A1V1b1 ∂t ∂t : Special version of Reynolds transport theorem - Fixed CV with one inlet and one outlet - Velocity normal to Sec. (1) and (2) y General expression of Reynolds Transport Theorem Consider a general flow shown At time t, CV & SYS: Coincide At time t + δt , CV: Fixed & SYS: Move slightly ∂Bcv & + Bout − B&in Dt ∂t ) Still valid, but B& out & B&in : Different DBsys = Ú What are B& out & B&in ? 1) B& out : Net flowrate of B leaving CV (Outflow) across the control surface between II and CV ( CS out ) B across the area element δA on CSout δB = bρδV = bρ (V cos θδt )δA where δV (Fluid volume leaving CV across δA = δlnδA = δl cos θδA = (Vδt cos θ )δA Then, the time rate of B across δA ρbδV ( ρbV cos θδt )δA = lim = ρbV cos θδA δt → 0 δt δt → 0 δt δB& out = lim By integrating over the entire CS out , B& out = ∫CS out dB& out = ∫CS out ρbV cos θ dA = ∫CS out r ρbV ⋅ nˆ dA 2) B& out : Net flowrate of B entering CV (Inflow) across the control surface between I and CV ( CSin ) By the similar manner, r B&in = − ∫CS ρbV cos θ dA = − ∫CS ρbV ⋅ nˆ dA in (because π 2 <θ < 3π ) 2 in Finally, Net flowrate of B across the entire CS ( = CSin + CS out ) B& out − B&in = ∫CS r out r ρbV ⋅ nˆdA − (− ∫CS ρbV ⋅ nˆ dA) in r = ∫CS ρbV ⋅ nˆ dA DBsys Dt = r r ∂Bcv ∂ + ∫CS ρbV ⋅ nˆ dA = ∫cv ρbdV + ∫CS ρbV ⋅ nˆ dA ∂t ∂t : General expression of Reynolds Transport Theorem PHYSICAL INTERPRETATION y DBsys Dt : Time rate of change of an extensive B of a system ) Lagrangian concept ∂ ∫ ρbdV : Time rate of change of B within a control volume ∂t cv ) Eulerian concept r y ∫CS ρbV ⋅ nˆ dA : Net flowrate of B across the entire control surface y ) Correlation term – Motion of a fluid c.f. Comparison with the definition of Material Derivative D( ) ∂( ) ∂( ) ∂( ) ∂( ) ∂( ) r = +u +v +w = + (V ⋅ ∇)( Dt ∂t ∂x ∂y ∂z ∂t y y ) D( ) Dt : Time rate of change of a property of fluid particle ∂( ) ∂t : Time rate of change of a property at a local space r y (V ⋅ ∇) ) Lagrangian concept ) Eulerian concept: Unsteady effect : Change of a property due to the fluid motion ) Correlation term – Convective effect Ú Reynolds Transport Theorem ) Transfer from Lagrangian viewpoint to Eulerian one (Finite size) Ú Special cases DBsys 1. Steady Effects. Dt = r ∂ ρ bd V ρ b V + ⋅ nˆ dA ∫ ∫ cv CS ∂t ) Any change in property B of a system = Net difference in flowrates B& entering CV and leaving CV 2. Unsteady Effects. ∂ ∫ ρbdV ≠ 0 ∂t CV ) Any change in property B of a system = Change in B within CV + Net difference in flowrates B& entering and leaving CV e.g. For 1-D flow r V = V0 (t )iˆ ρ = Constant r r r r r Choose B = mV (Momentum), and thus b = B / m = V = V0 (t )iˆ r r ∫CS ρbV ⋅ nˆdA = ∫CS ρ (V0iˆ)V ⋅ nˆdA = ∫(1) ρ (V0iˆ )(−V0 )dA + ∫( 2) ρ (V0iˆ )(V0 )dA + ∫side ρ (V0iˆ )(V0 cos 90o )dA = − ρV0 2 Aiˆ + ρV0 2 Aiˆ = 0 (Inflow of B = Outflow of B) ∴ DBsys Dt = ∂ ∫ ρbdV ∂t CV : No convective effect Reynolds Transport Theorem for a moving control volume DBsys Dt = r ∂ ρ bd V ρ b V + ⋅ nˆ dA ∫ ∫ cv CS ∂t : Valid for a stationary CV In case of moving control volume as shown, r Consider a constant velocity of CV = Vcv Ú Reynolds transport theorem : Relation between a system and CV, (Neglect the surrounding) ) Velocity of a system: Defined w.r.t. the motion of CV r r r ) Relative velocity of a system: W = V − VCV r where V : Absolute velocity of a system Finally, r ∂ = ∫cv ρbdV + ∫CS ρbW ⋅ nˆ dA Dt ∂t DBsys r : Valid for a stationary or moving CV with constant Vcv