## anonymous one year ago ques

1. anonymous

|dw:1440220440808:dw| This is a volume element of in an imcompressible fluid flowing at a distance x,y,z from the respective axes In the book it is given coordinates of P as x,y,z but shouldn't they be (x+dx),(y+dy),(z+dz) or (dx),(dy),(dz) if we consider the element at origin Let $\vec V=V_{x}\hat i+V_{y}\hat j+V_{z}\hat k$ be the velocity of fluid at a point P(x,y,z) Mass of fluid flowing through face ABCD in unit time is $=V_{x}(dy.dz)$ Now the problem is right in starting, this equation is dimensionally incorrect, the left side is having a unit of mass but right side is jut made up of velocity and length elements

2. anonymous

Maybe what they mean by P(x,y,z) is $\vec V \equiv \vec V(x,y,z)$ A vector point function $\vec V=V_{x}(x,y,z)\hat i+V_{y}(x,y,z)\hat j+V_{z}(x,y,z)\hat k$ But using a notation like this makes u think the coordinates of P are x,y,z but that's impossible if the length is dx,dy,dz how can it have coordinates x,y,z It must have either (dx,dy,0) or (x+dx,y+dy,0) Depending on where we taking our volume element (point P lies in the x-y plane, so z=0)

3. anonymous

@ganeshie8 @IrishBoy123

4. ganeshie8

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5. anonymous

,

6. IrishBoy123

In the Gospel of Mary, V is defined as $$\vec V = \rho \vec v$$ with units $$\frac{kg}{m^2 \ s}$$ and so flow rate through "unit area" of surface with normal $$\hat n$$ is $$\vec V \bullet \hat n$$ which has units $$\frac{kg}{m^2 \ s}.\frac{m^2}{1}=\frac{kg}{s}$$ http://www.amazon.co.uk/Mathematical-Methods-Physical-Sciences-Mary/dp/0471365807 it's only 3-4 pages so i'll copy them if you are using a different text. [and as it will come with a recommendation to buy, i reckon the copyright lawyers will forgive me.!]

7. anonymous

and how does $V_{x}((x+dx),y,z)=V_{x}+\frac{\partial V_{x}}{\partial x}.dx$

8. IrishBoy123

does this help?

9. anonymous

thanks, that was a lot helpful