Fluid kinematics 1 - Derivation of continuity equation and basic concepts PDF

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Derivation of continuity equation and basic concepts...

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FLUID KINEMATICS Derivation of the continuity equation Let’s take a 3 dimensional differential element of a fluid such that dx, dy, dz are the lengths of the 3 sides of the element is x, y, z directions respectively. The mass flow rate is given by 𝑚󰇗, mass flow rate means the rate of change of mass with respect to time 𝑚󰇗=

 

()

=



= 𝜌

() 

=𝜌

() 

= 𝜌𝐴

 

= 𝜌𝐴𝑣

Therefore, 𝑚󰇗 = 𝜌𝐴𝑣 𝑚=mass, 𝑉=volume, 𝜌=density, 𝐴=Area, 𝑣=velocity

The principal for deriving the continuity equation is: 𝑚󰇗in - 𝑚󰇗out = 𝑚󰇗stored 𝑚󰇗in = mass flow rate entering the element 𝑚󰇗out = mass flow rate leaving the element 𝑚󰇗stored = mass flow rate stored inside the element Let 𝑚󰇗x be the mass flow rate entering the element in x-direction 𝑚󰇗y be the mass flow rate entering the element in y-direction 𝑚󰇗z be the mass flow rate entering the element in z-direction 󰇗 

(𝑚󰇗x +

  󰇗

(𝑚󰇗y + (𝑚󰇗z +



󰇗 

𝑑𝑥) be the mass flow rate leaving the element in x-direction 𝑑𝑦) be the mass flow rate leaving the element in y-direction 𝑑𝑧) be the mass flow rate leaving the element in z-direction

󰇗 

𝑚󰇗in - 𝑚󰇗out = 𝑚󰇗x + 𝑚󰇗y + 𝑚󰇗z - (𝑚󰇗 x + =-( 𝑚󰇗stored =

󰇗

=



󰇗  

󰇗 

𝑑𝑥 +



𝑑𝑦 +



󰇗 

󰇗

𝑑𝑥 + 𝑚󰇗y +



󰇗  

𝑑𝑦 + 𝑚󰇗z +

𝑑𝑧)

𝑑𝑧)



() 

Therefore, 𝑚󰇗in - 𝑚󰇗out = 𝑚󰇗stored Or, - ( Or, Or, Or, Or,

󰇗  

 󰇗 

𝑑𝑥 +



󰇗 

𝑑𝑥 +



()  ()

󰇗 

𝑑𝑦 +

()



()  () 

()

𝑑𝑧) =

󰇗  𝑑𝑧 



𝑑𝑧𝑑𝑦𝑑𝑥 +



󰇗 

()

𝑑𝑥 +

𝑑𝑧𝑑𝑦𝑑𝑥 +



𝑑𝑦 +

= −

𝑑𝑦 +

 () 

() 

𝑑𝑥𝑑𝑧𝑑𝑦 + 𝑑𝑥𝑑𝑧𝑑𝑦 +

()  () 

𝑑𝑧 = −

() 

𝑑𝑥𝑑𝑦𝑑𝑧 = − 𝑑𝑥𝑑𝑦𝑑𝑧 = −

()  () 

𝑑𝑉

Since, 𝑑𝑉 = 𝑑𝑥𝑑𝑦𝑑𝑧 Therefore dividing both sides by 𝑑𝑥𝑑𝑦𝑑𝑧 We get, Or,

() 

𝝏(𝝆𝒖)

+

𝝏𝒙

+

()

𝝏(𝝆𝒗) 𝝏𝒚

For steady flow,

+



+

()

𝝏(𝝆𝒘) 𝝏𝒛

() 



+

= −

𝝏(𝝆) 𝝏𝒕

() 

=0

=0

The continuity equation stands as: 𝝏(𝝆𝒖) 𝝏𝒙

+

𝝏(𝝆𝒗) 𝝏𝒚

+

𝝏(𝝆𝒘) 𝝏𝒛

=0

If flow is steady as well as incompressible (𝜌 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) The continuity equation stand as: 𝝏(𝒖) 𝝏𝒙

+

𝝏(𝒗) 𝝏𝒚

+

𝝏(𝒘) 𝝏𝒛

=0

The above continuity equations is for 3 dimension. For one dimensional flow the continuity equation stands as: 𝝆𝑨𝒗 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 Note: 𝑢 , 𝑣, 𝑤 are the x, y, z components of velocity....