Equation of continuity in euler' s form.

     

 let ρ be the density of the fluid at p(x,y,z)  and let q=(u,v,w)be the velocity. 

Construct a small parallelopiped taking the point P as centre with edges of length h,k and l parallel to  the coordinate axes which is occupied by fluid.

the mass of fluid that flows across the unit area parallel to yz plane through  P in time δt                                                                        =ρuδt

                                                  =f(x,y,z)δt….say

                                                                         

      

we know that ,mass=volume×density                        and here volume=Area×distance                     velocity= →distance=velocity×δt=uδt and area=1 unit               ∴volume=1×uδt=uδt , so mass=uδt×ρ=ρuδt      since ρ and u depend upon the variable of x,y and z so they both can be considered as f(x,y,z)

                      

     let P₁ and P₂  be the projection of P at the faces kl near to and away from the origin respectively where the coordinate of

it' s because P is the centre of the parallelopiped and the length along x axis is h and following figure

             

   due to only projection of P ,P₁&P₂ do not change the coordinate of y and z it only change x coordinate here P₁ lies on faces kl near to origin and P₂  lies on faces kl but away from origin

                      the x coordinate of P₁=ON-MN

                                                        

                 the x coordinate of P₂=ON+NA

                                                

To find the flow across the face kl of the parallelopiped nearest to origin ,take a point          

 on the surface having area dη.dξ at this point 

then the mass of fluid that flows across the area dη.dξ in time δt 

   [∵using Taylor^' s expansion for several variables neglecting higher order]      

    The mass of the fluid that flows across the faces kl near to the origin in x↑direction in time δt, 

The mass of the fluid that flows across the faces kl away from the origin in x↑direction  in time δt, 

 

therefore the increase in mass inside the parallelopiped in time δt due to the flow across these two faces in x↑direction is

similarly the excess of mass of fluid that flow in across the hl faces near to the origin over the mass that flows out across the hl faces away from the origin in y↑direction in time δt 

 and the excess of mass of fluid that flow in across the kh faces near to origin over the mass flows out across the kh faces away from the origin in z↑direction in time δt,

  

hence the total mass of fliud that flows in the parallelopiped over the mass that flows out of it in time δt,

on the other hand ,the initial mass of the fluid inside the parallelopiped=ρhkl 

and the increase in mass of fluid inside the parallelopiped in time δt

 now by principal of continuity,(1)and (2)are equal. i.e

    which is the required equation of continuity in euler' s form.