Basiceqs3
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The document outlines the basic equations of fluid mechanics, beginning with an introduction. It then summarizes the continuity equation in 3 parts - the differential formulation, integral formulation of continuity equation, and Reynolds transport theorem. Finally, it discusses the Navier-Stokes equation by outlining the balance of forces, constitutive relations of Stokes, and differential and integral formulations of momentum equations.
![Outline
Introduction Differential formulation
Continuity Equation Integral formulation of Continuity Equation
Navier-Stokes Equation
Three distinct regions are:
I - System at time t
II- System passing
III- System at time t+∆ t
dN (NIII + NII )t+∆t − (NI + NII )t
= Lim∆t→0 (6)
dt ∆t
dN [(NII )t+∆t − (NII )t ] + (NIII )t+∆t − (NI )t
= Lim∆t→0
dt ∆t
(7)
[(NII )t+∆t − (NII )t ] ∂
Lim∆t→0 = ηρdϑ (8)
∆t ∂t CV
Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te
Basic Equations of Fluid Mechanics](https://image.slidesharecdn.com/basiceqs3-130212014248-phpapp02/85/Basiceqs3-9-320.jpg)
Basiceqs3
- 1. Outline Introduction Continuity Equation Navier-Stokes Equation Basic Equations of Fluid Mechanics Dr.-Ing. Naseem Uddin Mechanical Engineering Department NED University of Engineering & Technology Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 2. Outline Introduction Continuity Equation Navier-Stokes Equation 1 Introduction 2 Continuity Equation Differential formulation Integral formulation of Continuity Equation 3 Navier-Stokes Equation Balance of forces Constitutive Relations of Stokes Differential formulation Integral formulation of momentum equations Chapter 5(RTT) and 6 Munson etal. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 3. Outline Introduction Continuity Equation Navier-Stokes Equation Basic laws of Physics 1 Law of conservation of mass ⇒ Continuity Equation 2 Law of conservation of momentum ⇒ Navier-Stokes Equation 3 Law of conservation of energy ⇒ Energy Equation Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 4. Outline Introduction Differential formulation Continuity Equation Integral formulation of Continuity Equation Navier-Stokes Equation Mass cannot be created or destroyed. This laws for an infinitesimal control volume or region is called Continuity equation. Figure shows the mass balance for a fixed volume element in 2-dimensions (x,y plane). Depth is unity. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 5. Outline Introduction Differential formulation Continuity Equation Integral formulation of Continuity Equation Navier-Stokes Equation Velocity in x direction −→ u = u(x,y,z,t) Velocity in y direction −→ v= v(x,y,z,t) Velocity in z direction −→ w= w(x,y,z,t) Density ρ = ρ (x,y,z,t) Size: dx & dy Mass flow rate (m)= velocity normal to surface × surface area ˙ mx /unit area= ρ .u ˙ my /unit area= ρ .v ˙ mz /unit area= ρ .w ˙ Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 6. Outline Introduction Differential formulation Continuity Equation Integral formulation of Continuity Equation Navier-Stokes Equation Net mass flow rate = rate of mass accumulation/decrease in CV . Using Taylor series (neglecting higher order terms) we have: Net mass out-flow rate = 1 ∂ρu 1 ∂ρu ρu + dx dy − ρu − dx dy 2 ∂x 2 ∂x 1 ∂ρv 1 ∂ρv + ρv + dy dx − ρv − dy dx (1) 2 ∂y 2 ∂y Mass decrease in CV = ∂ρ − dxdy (2) ∂t Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 7. Outline Introduction Differential formulation Continuity Equation Integral formulation of Continuity Equation Navier-Stokes Equation Simplify and we get: ∂ρ ∂(ρu) ∂(ρv) + + =0 (3) ∂t ∂x ∂y For three dimensions: ∂ρ ∂(ρu) ∂(ρv) ∂(ρw) + + + =0 (4) ∂t ∂x ∂y ∂z ∂ρ + .v = 0 (5) ∂t where v = ui + v j + wk Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 8. Outline Introduction Differential formulation Continuity Equation Integral formulation of Continuity Equation Navier-Stokes Equation Reynolds Transport Theorem A system of fluid is passing through a flow field V(x,y,z,t). The streamlines are shown in figure and system takes a new position in time t+∆ t. An extensive sytem property N is changing in the system. η is N per unit mass. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 9. Outline Introduction Differential formulation Continuity Equation Integral formulation of Continuity Equation Navier-Stokes Equation Three distinct regions are: I - System at time t II- System passing III- System at time t+∆ t dN (NIII + NII )t+∆t − (NI + NII )t = Lim∆t→0 (6) dt ∆t dN [(NII )t+∆t − (NII )t ] + (NIII )t+∆t − (NI )t = Lim∆t→0 dt ∆t (7) [(NII )t+∆t − (NII )t ] ∂ Lim∆t→0 = ηρdϑ (8) ∆t ∂t CV Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 10. Outline Introduction Differential formulation Continuity Equation Integral formulation of Continuity Equation Navier-Stokes Equation Net efflux rate= (NIII )t+∆t − (NI )t Lim∆t→0 (9) ∆t The volume of fluid that swept out of area dA in time dt is: dϑ = V dt dA cos α or dϑ = (V.dA) Efflux rate through control surface = ηρV.dA (10) CS Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 11. Outline Introduction Differential formulation Continuity Equation Integral formulation of Continuity Equation Navier-Stokes Equation dN ∂ = ηρV.dA + ηρdϑ (11) dt CS ∂t CV Now as η = N/mass we have η=1 (in case of mass). dM ∂ =m= ˙ ρV.dA + ρdϑ (12) dt CS ∂t CV for incompressible flows: m= ˙ ρV.dA (13) CS Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 12. Outline Balance of forces Introduction Constitutive Relations of Stokes Continuity Equation Differential formulation Navier-Stokes Equation Integral formulation of momentum equations 1 Body forces fb (gravitational, magnetic etc.) 2 Surface forces fs (pressure, forces due to shear & normal stresses) F = m.a (14) Du F = m. (15) Dt Du F = ρ.ϑ. (16) Dt F Du = fb + fs = ρ (17) ϑ Dt Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 13. Outline Balance of forces Introduction Constitutive Relations of Stokes Continuity Equation Differential formulation Navier-Stokes Equation Integral formulation of momentum equations There are two types of stresses, Normal strsses (σ) and shear stresses (τ ). Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 14. Outline Balance of forces Introduction Constitutive Relations of Stokes Continuity Equation Differential formulation Navier-Stokes Equation Integral formulation of momentum equations Forces due to surface stresses Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 15. Outline Balance of forces Introduction Constitutive Relations of Stokes Continuity Equation Differential formulation Navier-Stokes Equation Integral formulation of momentum equations Net force in x direction= ∂(σxx ) ∂(τyx ) dxdy + dxdy (18) ∂x ∂y Du ∂(σxx ) ∂(τyx ) ∂(τzx ) ρ = ρfx + + + (19) Dt ∂x ∂y ∂z for y direction: Dv ∂(σyy ) ∂(τxy ) ∂(τzy ) ρ = ρfy + + + (20) Dt ∂y ∂x ∂z for z direction: Dw ∂(σzz ) ∂(τyz ) ∂(τxz ) ρ = ρfz + + + (21) Dt ∂z ∂y ∂x Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 16. Outline Balance of forces Introduction Constitutive Relations of Stokes Continuity Equation Differential formulation Navier-Stokes Equation Integral formulation of momentum equations Relation between stresses and velocity gradients 2 ∂u σxx = λ − µ Θ + 2µ +A ∂v ∂u 3 ∂x τxy = + ∂x ∂y 2 ∂v σyy = λ − µ Θ + 2µ +A ∂w ∂u 3 ∂y τxz = + ∂x ∂z 2 ∂w σzz = λ − µ Θ + 2µ +A ∂w ∂v 3 ∂z τyz = + ∂y ∂z ∂u ∂v ∂w Θ= + + ∂x ∂y ∂z Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 17. Outline Balance of forces Introduction Constitutive Relations of Stokes Continuity Equation Differential formulation Navier-Stokes Equation Integral formulation of momentum equations For simple flows A= - thermodynamic pressure = - Pthermo Average pressure = 1 P = − (σxx + σyy + σzz ) (22) 3 It is related to thermodynamic pressure as: P = −λΘ + Pthermo (23) For constant density flows Θ is zero! 2 ∂u σxx = −P − µΘ + 2µ 3 ∂x 2 ∂v σyy = −P − µΘ + 2µ 3 ∂y 2 ∂w σzz = −P − µΘ + 2µ 3 ∂z Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 18. Outline Balance of forces Introduction Constitutive Relations of Stokes Continuity Equation Differential formulation Navier-Stokes Equation Integral formulation of momentum equations Navier-Stokes Equations Du ∂u ∂u ∂u ∂u 1 ∂P 2 = +u +v +w =− + u + fx (24) Dt ∂t ∂x ∂y ∂z ρ ∂x Dv ∂v ∂v ∂v ∂v 1 ∂P 2 = +u +v +w =− + v + fy (25) Dt ∂t ∂x ∂y ∂z ρ ∂y Dw ∂w ∂w ∂w ∂w 1 ∂P 2 = +u +v +w =− + w + fz (26) Dt ∂t ∂x ∂y ∂z ρ ∂z Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics
- 19. Outline Balance of forces Introduction Constitutive Relations of Stokes Continuity Equation Differential formulation Navier-Stokes Equation Integral formulation of momentum equations dN ∂ = ηρV.dA + ηρdϑ (27) dt CS ∂t CV Now as η = N/mass we have η=V (in case of momentum). dM omentum ∂ = F orce = VρV.dA + Vρdϑ (28) dt CS ∂t CV Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering & Te Basic Equations of Fluid Mechanics