Author:  Muhammad  Valiallah  Date:  10  October  2012                                                          Subject:  Transport  Phenomena  Adapted:  Transport  Phenomena  (Bird  Steward  and  Lightfoot)  

Derivation  of  Bernoulli  Equation,  Transport  Phenomena  Chap  3  Page  86  The  equation  of  motion  in  the  substantial  time  derivative  form  is:    

!!!!" = −!! − ! ⋅ ! + !!  

 For  inviscid  fluids  – ! ⋅ ! = 0,  the  equation  of  motion  becomes:    

!!!!" = −!! + !!  

For  steady  flow  the  time  derivative  disappears    

!!Dt =

!!" !+ ! ⋅ !"  

 ! ! ⋅ !" = −!! − ! ⋅ ! + !!  

 Substituting  the  vector  identity  into  the  above  equation:    

! ⋅ !" =12! ! ⋅ ! − !× !×!  

 

!12! ! ⋅ ! − !× !×! = −!! + !!  

 but    

! = −!!    

! = !ℎ    

!12! ! ⋅ ! − !× !×! = −!! − !"!ℎ  

 Divide  through  by  !    

12! ! ⋅ ! − !× !×! = −

1!!! + !!ℎ  

Let  !  be  the  unit  vector  in  the  flow  direction    

! =!!  

     Take  the  dot  product  of  !  with  the  above  equation:    

Author:  Muhammad  Valiallah  Date:  10  October  2012                                                          Subject:  Transport  Phenomena  Adapted:  Transport  Phenomena  (Bird  Steward  and  Lightfoot)  

! ⋅12! ! ⋅ ! − ! ⋅ !× !×! = −! ⋅

1!!! − ! ⋅ !!ℎ  

 

=12 ! ⋅ ! ! ⋅ ! − ! ⋅ !× !×! = −! ⋅

1!!! − ! ⋅ !!ℎ  

=12 ! ⋅ ! !! − ! ⋅ !× !×! = −! ⋅

1!!! − ! ⋅ !!ℎ  

=12 ! ⋅ ! !! − ! ⋅ !× !×! = −! ⋅

1!!! − ! ⋅ !!ℎ  

 12 ! ⋅ ! !! − ! ⋅ !× !×! = − ! ⋅ !

1!! − (! ⋅ !)!ℎ  

 12!!" !! − ! ⋅ !× !×! = −

!!"1!! −

!!" !ℎ  

 

!×! =

!! !! !!!!"

!!"

!!"

!! !! !!

= !!!!" !! −

!!" !! − !!

!!" !! −

!!" !! + !!

!!" !! −

!!" !!  

 

! ⋅ ! =!!! + !!! + !!!

!!! + !!! + !!!= !  

 

! ⋅ !×!×! =

!! !! !!!! !! !!

!!" !! −

!!" !!

!!" !! −

!!" !!

!!" !! −

!!" !!

 

 

= !! !!!!" !! −

!!" !! − !!

!!" !! −

!!" !! − !! !!

!!" !! −

!!" !! − !!

!!" !! −

!!" !!

+ !! !!!!" !! −

!!" !! − !!

!!" !! −

!!" !!  

 

=1! !!!!

!!" !! −

!!" !! − !!!!

!!" !! −

!!" !!

+ −!!!!!!" !! −

!!" !! + !!!!

!!" !! −

!!" !!

+ !!!!!!" !! −

!!" !! − !!!!

!!" !! −

!!" !!  

 ! ⋅ !×!×! = 0  

12!!" !! = −

!!"1!! −

!!" !ℎ  

 Integrating  along  stream  line  which  is  in  the  direction  of  flow  from  point  1  to  2    

Author:  Muhammad  Valiallah  Date:  10  October  2012                                                          Subject:  Transport  Phenomena  Adapted:  Transport  Phenomena  (Bird  Steward  and  Lightfoot)  

12!!

! = −!"! − !"ℎ  

 12 !!!

!

!

= −!"!

!

!

− !"ℎ!

!

 

 

12 !!! − !!! = −

1! !"

!!

!!

− ! ℎ! − ℎ!  

 

∴12 !!! − !!! +

1! !"

!!

!!

+ ! ℎ! − ℎ! = 0