Xem mẫu

20 BASIC CONCEPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 F = m dt = d(mV) (1.64) where mV is the momentum. Equation (1.64) is the statement of the conservation of momentum principle. Note that the conservation of momentum principle is stated in terms of the proper-ties of particles and not in terms of the properties of a field. To derive the momentum theorem, a region in a fluid confined by the control surface S1, shown in Fig. 1.7, is employed. The surface S1 contains a definite and fixed number of particles at time t1. At time t2, these particles will have moved to a region bounded by the control surface S2, which is shown as a dashed curve to distinguish it from S1. The control surfaces S1 and S2 enclose three separate and distinct regions, desig-nated by a,b, and c. Let the momentum in the three regions be Pa,Pb, and Pc, re-spectively.Attimet1 theparticleswithinsurfaceS1 willpossessmomentumPa+Pb1. At time t2 these particles will have momentum, Pb2 + Pc because they have moved into the region enclosed by surface S2. Hence the momentum change during the time interval t2 −t1 may be described by (Pb2 +Pc)−(Pb1 +Pa) = (Pb2 −Pb1)+(Pc −Pa) and the time rate of change of momentum will be tlim t2 −t1b1 + tc −t1a (1.65) As t2 approaches t1 as a limit, the control surface S2 will coincide with S1. The first term in eq. (1.65) is therefore the time rate of change of momentum of the fluid containedwithinregion1,R1,containedwithinS1.Thismaybewrittenastheintegral over R1. Because the mass of fluid contained in R1 is ZZZ ρdR1 R1 the time rate of change of momentum of the fluid contained within region 1 will be ZZZ ∂t ρVdR1 R1 The second term in eq. (1.65) is the momentum efflux through the control surface S1.Ifthefluxintheoutwarddirectionistakenaspositive,thiseffluxcanbeexpressed by the integral ZZ ρVVn dS1 S1 where Vn is the component of velocity normal to S1. [20] Lin 0.9 —— Nor PgE [20] MOMENTUM AND THE MOMENTUM THEOREM 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 S 1 a b c Figure 1.7 Regions bounded by control surfaces used for the development of the momentum theorem. The conservation of momentum principle then becomes ˆ ZZZ ZZ F = m dt = ∂t ρV dR1 + ρVVn dS1 (1.66) R1 S1 or, by rearrangement of terms, ZZZ ZZ ∂t ρVx dR1 = Fx − ρVx ˆn dS1 (1.67a) R S ZZZ ZZ ∂t ρVy dR1 = Fy − ρVyVn dS1 (1.67b) ZZZ Z1 ∂t ρVz dR1 = Fz − ρVx ˆn dS1 (1.67c) R1 S1 in the three rectangular coordinate directions. The foregoing development leads to the statement of the momentum theorem: The time rate of increase of momentum of a fluid within a fixed control volume R will be equal to the rate at which momentum flows into R through its confining surface S, plus the net force acting on the fluid within R. When the flow is incompressible, the viscosity is constant, and the flow is laminar, the Navier–Stokes equations result. In Cartesian coordinates, with Fx, Fy, and Fz taken as the components of the body force per unit volume, the Navier–Stokes equations are ! ∂Vx ∂Vx ∂Vx ∂Vz ∂t x ∂x y ∂y z ∂z [21 Lin 0.4 —— No PgE [21 22 BASIC CONCEPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 2 ˆ 2 ˆ 2 ˆ ! = −∂x +µ ∂x2 + ∂y2 + ∂z2 +Fx (1.68a) ! ρ ∂ty +Vx ∂x +Vy ∂y +Vz ∂z 2 ˆ 2 ˆ 2 ˆ ! = −∂y +µ ∂x2 + ∂y2 + ∂z2 +Fy (1.68b) ! ∂Vz ∂Vx ∂Vz ∂Vz ∂t z ∂x y ∂y z ∂z 2 ˆ 2 ˆ 2 ˆ ! = −∂z +µ ∂x2 + ∂y2 + ∂z2 +Fz (1.68c) In cylindrical coordinates with Fr, Fθ, and Fz taken as the components of the body force per unit volume, the Navier–Stokes equations are ! ρ ∂tr +Vr ∂rr + rθ ∂θ − r +Vz ∂z = −∂r +µ ∂rˆr + r ∂rr − rr + r2 ∂2Vr − r2 ∂θ + ∂zˆr !+Fr (1.69a) ! ∂Vθ ∂Vθ Vθ ∂Vθ VrVθ ∂Vθ 1 ∂P ∂t r ∂r r ∂θ r z ∂z r ∂θ +µ ∂r2θ + 1 ∂rθ − rθ + r2 ∂ θˆθ + r2 ∂θ + ∂z2θ !+Fθ (1.69b) ! ρ ∂tz +Vr ∂rz + rθ ∂θ +Vz ∂z = −dz +µ ∂rˆz + r ∂rz + r2 ∂2 ˆz + ∂zˆz !+Fz (1.69c) Finally, in spherical coordinates with Fr, Fθ, and Fφ taken as the components of the body force per unit volume and with ˆ ˆ DT = dt +Vr ∂r + r ∂φ + r sinφ ∂θ ∇2 = r2 ∂r r2 ∂r+ r2 sinφ ∂φ sinφ∂φ+ r2 sin2 φ ∂ 2 [22] Lin 1.6 —— Lon * PgE [22] CONSERVATION OF ENERGY 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 the Navier–Stokes equations are DVr V2 +V2 ! ∂P Dt r ∂r ! +µ ∇2Vr − r2r − r2 ∂φ − 2Vθr otφ − r2 sinφ ∂θ +Fr (1.70a) ! DVφ VrVφ Vθ cotφ 1 ∂P Dt r r r ∂φ ! 2 ∂Vr Vφ 2cosφ ∂Vθ φ r2 ∂φ r2 sinφ r2 sin2 φ ∂θ φ DVθ ˆθ ˆr ˆφ ˆθ cotφ! ˆθ ∂P Dt r r r sinφ ∂θ ˆ ˆ ˆ ! +µ ∇ Vθ − r2 sin2 φ + r2 sinφ ∂θ + r2 sin2 φ ∂θ +Fθ (1.70c) 1.5 CONSERVATION OF ENERGY In Fig. 1.8, an imaginary two-dimensional control volume of finite size ∆x ∆y with flow velocity V = exVx +eyVy, heat flux q00 = exq00 +eyq00, specific internal energy u, and rate of internal heat generation q000, the first law of thermodynamics requires that rate of energy ! accumulation within the control volume net transfer ! net heat ! = of energy by + transfer by fluid flow conduction !  net work   transfer from the  control volume to the environment Four of the five terms indicated do not involve work transfer from the control volume to the environment. • The rate of energy accumulated in the control volume is ∆x ∆y∂t(ρu) (1.72a) [23 Lin 2.0 —— Lon * PgE [23 24 BASIC CONCEPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 ρV u1 ∂y (ρV u) Dy Dx ρV u1 ∂ (ρV u) Dx Dy ρV u y ∂(ρe)Dx Dy y ρˆ u x x (q0 1 ∂q0 Dy)Dx q0 Dy q-Dx Dy (q0 1 ∂q0 Dx)Dy q0 Dx Figure 1.8 First law of thermodynamics applied to an imaginary control volume in two-dimensional flow. • The net transfer of energy by fluid flow is −(∆x ∆y)∂x ρVxu+ ∂y ρVyu (1.72b) • The net heat transfer by conduction is 00 00 −(∆x ∆y) ∂x + ∂y (1.72c) [24] Lin 2.9 —— Lon * PgE [24] ... - tailieumienphi.vn