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.

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