Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
CE 204
FLUID MECHANICS
Onur AKAY
Assistant Professor
Onur Akay, Ph.D. CE 204 Fluid Mechanics 1
Assistant Professor
Okan University
Department of Civil Engineering
Akfırat Campus
34959 Tuzla-Istanbul/TURKEY
Phone: +90-216-677-1630 ext.1974
Fax: +90-216-677-1486
E-mail: [email protected]
Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
Mass Flow Rate (Review):
The mass flow rate, , is the mass of fluid passing through a cross-sectional area per unit
time [M/T].
Onur Akay, Ph.D. CE 204 Fluid Mechanics 2
(velocity vector is aligned with the area vector.)
*The equations for discharge and
mass flow rate are summarized in
Table F.2.
Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
Reynolds Transport Theorem (Review):
Onur Akay, Ph.D. CE 204 Fluid Mechanics 3
Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
General Form of the Continuity Equation (Review):
- The continuity equation derives from the conservation of mass (dm / dt = 0):
Onur Akay, Ph.D. CE 204 Fluid Mechanics 4
- The continuity equation derives from the conservation of mass (dmsys / dt = 0):
Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
Momentum Equation: Derivation
Newton’s second law:
Force is equal to the time derivative of
momentum.
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For a fluid system composed of a group of particles:
Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
Momentum Equation: Derivation
Recall the Reynolds transport theorem:
Bsys = Momsys
b = v
How can we write this equation in Eulerian form?
Onur Akay, Ph.D. CE 204 Fluid Mechanics 6
b = v
Remember that the
momentum equation is a
vector equation!
Momentum Equation:
Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
Momentum Equation: Derivation
- If the flow crossing the CS occurs through a series of inlet and outlet ports,
- and the velocity v is uniformly distributed across each port:
Onur Akay, Ph.D. CE 204 Fluid Mechanics 7
The three components of the Momentum Equation for the Cartesian coordinate system:
Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
Force Diagram:
In most cases we will use force diagrams (FD) to determine forces acting on the matter in
the CV.
Onur Akay, Ph.D. CE 204 Fluid Mechanics 8
Pipe
schematic
CV inside
the pipe
CV surrounding
the pipe
Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
Force Diagram:
CV inside
the pipe
Body force: Acts on mass elements within the
body (gravitational forces).
Surface force: Acts at the control surface
(pressure and shear forces).
Onur Akay, Ph.D. CE 204 Fluid Mechanics 9
CV surrounding
the pipe
Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
Momentum Diagram:
Momentum diagram is created by sketching a CV and then drawing a vector to represent
momentum flow at each section.
Onur Akay, Ph.D. CE 204 Fluid Mechanics 10
Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
Application of the Momentum Equation:
Fluid Jets: A fluid jet is created by a high-speed stream of fluid leaving a nozzle.
pB = pC = ps (for subsonic jets)
Onur Akay, Ph.D. CE 204 Fluid Mechanics 11
Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
Application of the Momentum Equation:
Nozzles: Flow devices used to accelerate a fluid stream by reducing the cross-sectional
area of the flow.
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Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
Application of the Momentum Equation:
Vanes: Used to turn a fluid jet.
Onur Akay, Ph.D. CE 204 Fluid Mechanics 13
Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
Application of the Momentum Equation:
Pipe Bends: Calculating the force on pipe bends is important in engineering applications
using large pipes to design the support system.
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Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
Navier-Stokes Equation: Differential equation for momentum at a point in the flow.
Claude-Louis Navier
Born: 10 February 1785, Dijon
Died: 21 August 1836, Paris
Sir George Gabriel Stokes
Born: 13 August 1819, Ireland
Died: 1 February 1903, England
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Chapter 6 Momentum Equation
Derivation and Application of the Momentum Equation, Navier-Stokes Eq.
Navier-Stokes Equation:
Onur Akay, Ph.D. CE 204 Fluid Mechanics 16
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