ce 204 fluid mechanics - okan university · chapter 6momentum equation derivation and application...

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]



Mass Flow Rate (Review):

The mass flow rate, , is the mass of fluid passing through a cross-sectional area per unit

time [M/T].


(velocity vector is aligned with the area vector.)

*The equations for discharge and

mass flow rate are summarized in

Table F.2.



Reynolds Transport Theorem (Review):




General Form of the Continuity Equation (Review):

- The continuity equation derives from the conservation of mass (dm / dt = 0):


- The continuity equation derives from the conservation of mass (dmsys / dt = 0):



Momentum Equation: Derivation

Newton’s second law:

Force is equal to the time derivative of

momentum.


For a fluid system composed of a group of particles:




Recall the Reynolds transport theorem:

Bsys = Momsys

b = v

How can we write this equation in Eulerian form?


b = v

Remember that the

momentum equation is a

vector equation!

Momentum Equation:




- 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:


The three components of the Momentum Equation for the Cartesian coordinate system:



Force Diagram:

In most cases we will use force diagrams (FD) to determine forces acting on the matter in

the CV.


Pipe

schematic

CV inside

the pipe

CV surrounding

the pipe



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).


CV surrounding

the pipe



Momentum Diagram:

Momentum diagram is created by sketching a CV and then drawing a vector to represent

momentum flow at each section.




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)





Nozzles: Flow devices used to accelerate a fluid stream by reducing the cross-sectional

area of the flow.





Vanes: Used to turn a fluid jet.





Pipe Bends: Calculating the force on pipe bends is important in engineering applications

using large pipes to design the support system.




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




Navier-Stokes Equation:


ce 204 fluid mechanics - okan university · chapter 6momentum equation derivation and application...

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