2. Fundamentals
โข Three pilers of fluid dynamics
โข Continuity : Mass is Conserved
โข Momentum : Newtonโs Second law
โข Energy : Energy is conserved
Fluid dynamics is based on the mathematical statements of these three
physical principles.
3. Mass balance in a flowing fluid โ
Continuity
โ Fluid of density แฟค and velocity u
The equation of mass balance is
(Rate of Mass flow in) โ (Rate of Mass flow out) = Rate of
Accumulation
4. Balance Equations
1. Mass flux in x direction at the face x = ๐๐ข ๐ฅ
2. Mass flux in x direction at the face (x+dx) = ๐๐ข ๐ฅ+๐๐ฅ
Flux is defined as the rate of flow of quantity per unit area
Hence
3. Mass flow rate of the fluid entering the fluid element in x โ direction =
๐๐ข ๐ฅโ๐ฆโ๐ง
4. Mass flow rate of the fluid leaving the fluid element in x โ direction =
๐๐ข ๐ฅ+๐๐ฅโ๐ฆโ๐ง
5. Rate of Accumulation of momentum in the volume element = โ๐ฅ โ๐ฆ โ๐ง
๐๐
๐๐ก
Similarly taking mass balance in โyโ and โzโ directions and substituting in general Mass
balance equation we will get
6. For incompressible fluids
๐ซ๐
๐ซ๐
= โ ๐
๐๐ข
๐๐ฅ
+
๐๐ฃ
๐๐ฆ
+
๐๐ค
๐๐ง
= โ๐ ๐ป. ๐ฝ
๐ป. ๐ฝ =
๐๐ข
๐๐ฅ
+
๐๐ฃ
๐๐ฆ
+
๐๐ค
๐๐ง
= 0
Stream Line
โข Imaginary line drawn in a force field that tangent drawn at any point indicates
the direction of velocity vector at that point and at that time.
โข Properties
โข There cannot be any movement of fluid across the streamlines
โข Streamlines never intersect nor two of them cross each other
โข Converging streamlines indicate the accelerating nature of fluid flow in a
direction.
7. โข Tube of large or small cross section and of any convenient shape entirely
bounded by streamlines.
โข No net flow through the walls of the stream tube.
โข Mass flowrate through differential c/s area dS is denoted by
๐๐ = ๐ ๐ข ๐๐
โข For conduit of cross sectional area S
๐ = ๐
๐
๐ข ๐๐
โข Average velocity of the stream (แนผ) flowing through the c/s area is given by
แนผ =
๐
๐๐
=
1
๐ ๐
๐ข ๐๐
แนผ = volumetric flow rate per unit c. s area of condiut
แนผ =
๐
๐
q = volumetric flow rate
Stream tube
8. Mass Velocity
ฯแนผ =
๐
๐
= ๐บ โก
๐๐
๐2 ๐ ๐๐
โข Advantage of using G is that it is independent of T and P at a steady flow (m) through
a constant cross section.
โข Mass velocity can also be designated as the mass flux which represents the quantity
of material passing through unit area in unit time.
9. Differential momentum balance: Equations of motion
(Rate of Momentum Entering) โ (Rate of Momentum leaving) + ( Sum of forces
acting on the system ) = Rate of Momentum Accumulation
1. Rate at which x component of momentum enters face x = ๐๐ข๐ข ๐ฅ โ๐ฆโ๐ง
2. Rate at which x component of momentum leaves face x+dx = ๐๐ข๐ข ๐ฅ+โ๐ฅ โ๐ฆโ๐ง
3. Rate at which x component of momentum enters face y = ๐๐ฃ๐ข ๐ฆ โ๐ฅโ๐ง
Equations 1, 2 , 3 correspond to convective flow
10. ๐๐ข๐ข ๐ฅ โ ๐๐ข๐ข ๐ฅ+โ๐ฅ โ๐ฆโ๐ง + ๐๐ฃ๐ข ๐ฆ โ ๐๐ฃ๐ข ๐ฆ+โ๐ฆ โ๐ฅโ๐ง
+ ๐๐ค๐ข ๐ง โ ๐๐ค๐ข ๐ง+โ๐ง โ๐ฅโ๐ฆ + ฯ๐ฅ๐ฅ ๐ฅ โ ฯ๐ฅ๐ฅ ๐ฅ+โ๐ฅ โ๐ฆโ๐ง
+ ฯ๐ฆ๐ฅ ๐ฆ
โ ฯ๐ฆ๐ฅ ๐ฆ+โ๐ฆ
โ๐ฅโ๐ง + ฯ๐ง๐ฅ ๐ง โ ฯ๐ง๐ฅ ๐ง+โ๐ง โ๐ฆโ๐ฅ + โ๐ฆโ๐ง ๐๐ฅ โ ๐๐ฅ+โ๐ฅ
+ ๐๐๐ฅโ๐ฅโ๐ฆโ๐ง
=
๐(๐๐ข)
๐๐ก
โ๐ฅโ๐ฆโ๐ง
Dividing by ฮx ฮy ฮz gives and taking the corresponding limit of ฮx ฮy ฮz tend to
zero
Writing similar expressions for convective flow and mass transport of x momentum through all the six faces & substituting
in Momentum balance equation
4. Rate at which x component of momentum enters face x by molecular transport
= ฯ๐ฅ๐ฅ ๐ฅ โ๐ฆโ๐ง
5. Rate at which x component of momentum leaves face x+dx by molecular transport
= ฯ๐ฅ๐ฅ ๐ฅ+โ๐ฅ โ๐ฆโ๐ง
6. Rate at which x component of momentum enters face y by molecular transport = ฯ๐ฆ๐ฅ ๐ฆ
โ๐ฆโ๐ง
12. C. ๐
๐๐ค
๐๐ก
+ ๐ข
๐๐ค
๐๐ฅ
+ ๐ฃ
๐๐ค
๐๐ฆ
+ ๐ค
๐๐ค
๐๐ง
= ฮผ
๐2๐ค
๐๐ฅ2
+
๐2๐ค
๐๐ฆ2
+
๐2๐ค
๐๐ง2
โ
๐๐
๐๐ง
+ ๐๐๐ง
๐
๐ซ๐ฝ
๐ซ๐
= โ๐ป๐ + ฮผ ๐ป2๐ฝ + ๐g
Navier โ Stokes Equation
Vector form of Navier โ Stokes Equation
Eulerโs Equation
๐
๐ซ๐ฝ
๐ซ๐
= โ๐ป๐ + ๐g
Assumptions
โข Fluid is incompressible
โข Fluid is Inviscid (ideal)
โข Fluid has zero viscosity
โข Streamline and irrotational
flow
โข Flow is steady
13. Macroscopic Momentum Balances
Mass flow rate = ; Velocity = u
Momentum flow rate =
Momentum carried by the fluid through c/s area dS in unit time
Variation in instantaneous velocity (u) along the flow section changes the
momentum flow rate estimations.
Momentum rate, estimated even from average velocity brings errors in
Momentum estimations
Momentum correction factor is being used in momentum balance equation;
estimated from principle of momentum flux.
15. Steady and Unsteady flow
Flow parameters such as velocity, pressure, density does not change with
time in a steady flow.
๐๐
๐๐ก
= 0;
๐๐
๐๐ก
= 0;
๐๐
๐๐ก
= 0 For steady flow
๐๐
๐๐ก
โ 0;
๐๐
๐๐ก
โ 0;
๐๐
๐๐ก
โ 0 For unsteady flow
Uniform and non-uniform flow
Type of flow in which Velocity, pressure, density does not with respect to
spatial co-ordinates.
๐๐
๐๐
= 0;
๐๐
๐๐
= 0;
๐๐
๐๐
= 0 For uniform flow
๐๐
๐๐
โ 0;
๐๐
๐๐
โ 0;
๐๐
๐๐
โ 0 For non-uniform flow
Steady uniform flow Steady non-uniform flow
Unsteady uniform flow
Unsteady non-uniform flow
Ex: Flow through a channel at constant discharge Ex: Constant flow through expanding or diverging
section
Non practicable situation
Flow in a pipe through a valve.
16. Energy Equation for potential flow โ
Bernoulliโs equation w/o friction
๐
๐๐ข
๐๐ก
+ ๐ข
๐๐ข
๐๐ฅ
+ ๐ฃ
๐๐ข
๐๐ฆ
+ ๐ค
๐๐ข
๐๐ง
= โ
๐๐
๐๐ฅ
+ ๐๐๐ฅ
For unidirectional flow component of velocity in y(v) and z(w) are zero
Multiplying remaining terms with u gives
๐๐ข
๐๐ข
๐๐ก
+ ๐ข
๐๐ข
๐๐ฅ
= โ ๐ข
๐๐
๐๐ฅ
+ ๐๐ข๐๐ฅ
๐
๐(
๐ข2
2 )
๐๐ก
+ ๐ข
๐(
๐ข2
2 )
๐๐ฅ
= โ ๐ข
๐๐
๐๐ฅ
+ ๐๐ข๐๐ฅ
Mechanical energy equation for unidirectional potential flow of fluids
of constant density and when flow rate varies with time
18. ๐๐
๐
+ ๐๐๐ +
๐ข2
๐
2
=
๐๐
๐
+ ๐๐๐ +
๐ข2
๐
2
Bernoulliโs equation without friction
๐
๐
=
๐
๐2 ๐๐
๐3
=
๐ ๐
๐๐
๐ข2
2
=
๐2
๐ 2
=
๐2
๐ 2
๐๐
๐๐
=
๐๐
๐๐ ๐๐ =
๐ ๐
๐ 2
=
๐2
๐ 2
๐๐
๐๐
=
๐๐
๐๐
Total energy per unit mass of fluid at every point in a flow is constant
๐๐
๐๐
+ ๐๐ +
๐ข2
๐
2๐
=
๐๐
๐๐
+ ๐๐ +
๐ข2
๐
2๐
๐
๐๐
=
๐
๐2 ๐๐
๐3
๐
๐ 2
=
๐
๐๐
๐ 2
=
๐๐
๐๐๐
๐ 2
=
๐ ๐
๐
๐ข2
2๐
=
๐2
๐ 2 ๐
๐ 2
= ๐ =
๐๐
๐
๐ = m =
๐๐
๐
Total energy per unit weight of fluid at every point in a flow is constant
๐๐ + ๐๐๐๐ +
๐๐ข2
๐
2
= ๐๐ + ๐๐๐๐ +
๐๐ข2
๐
2
๐ =
๐
๐2
=
๐ ๐
๐2๐
=
๐ ๐
๐3
๐๐ข2
2
=
๐๐ ๐2
๐3๐ 2 =
๐ ๐
๐3
๐๐๐ =
๐๐ ๐ ๐
๐3๐ 2 =
๐ ๐
๐3
Total energy per unit volume of fluid at every point in a flow is constant
19. Bernoulliโs equation: Corrections for effects of solid boundaries
โข Problems in engineering involve streams which are influenced by solid
boundary
โข Flow of fluid through a pipe where entire stream is in boundary layer
flow
โข Practical Situation involves correction terms in Bernoulliโs equation
for
1. Kinetic energy
2. Existence of fluid friction
20. Kinetic energy
Element of c/s area ds
Mass flow rate through the c/s = ฯu ds
Energy flow rate through c/s area
๐ร๐ = ๐๐ข ๐๐ โ
๐ข2
2
Total rate of flow of Kinetic Energy through c/s S is
ร๐ =
๐
2
โ
๐
๐ ๐ข3๐๐
Term which replaces u2/2 in Bernoulliโs equation is
ร๐
๐
= ๐
๐ ๐ข3
๐๐
2 ๐
๐ ๐ข ๐๐
= ๐
๐ ๐ข3
๐๐
2 แนผ๐
21. K.E correction factor
โ แนผ2
2
=
ร๐
๐
= ๐
๐ข3
๐๐
2 แนผ๐
โ = ๐
๐ข3
๐๐
แนผ3๐
Correction factor for fluid friction
๐๐
๐
+ ๐๐๐ +
๐ผ๐๐2
๐
2
=
๐๐
๐
+ ๐๐๐ +
๐ผ๐๐2
๐
2
+ โ๐
โข Whenever flow occurs friction is generated and hf denote the
friction generated per unit mass of fluid (conversion of
mechanical energy into heat) that occurs when fluid flows
between two stations)
โข hf is positive value; it will be zero for potential flow
โข Skin friction โ friction generated in unseparated boundary layers
โข Form friction โ boundary layers separate and form wakes,
additional energy dissipation appears in the wake
22. Pump work in Bernoulli Equation
๐
๐ โ โ๐๐ = ฮท๐
๐
๐
๐- work done by pump per unit mass of fluid
โ๐๐- friction generated in the pump per unit mass of fluid
ฮท < 1
๐๐
๐
+ ๐๐๐ +
๐ผ๐๐2
๐
2
+ ฮท๐
๐ =
๐๐
๐
+ ๐๐๐ +
๐ผ๐๐2
๐
2
+ โ๐