Having dificulty understanding the meaning of continuity equation
 

I am not getting the meaning of continuity equation, mainly the gradient term that is there when flow leaves the control volume, please explain with the example..

I can explain it to you but to not to confuse you more, I prefer understanding what exactly confuses you 'in a more clear way' before I give some analogies

OK,
(Flow rate in=flow rate out+ rate of change of flowrate )#through control volume.
So,what that rate of change of flow rate term is doing at the outlet?

I suggest starting from the meaning of the Reynolds theorem, then you can use the Eulerian formulation to say that:

d /dt Int [v] rho dV + Int [BV] n.(v*rho) dS = 0

For steady flow you have the contranint that the surface integral of the mass flux must be zero.

Yeah that's true.
But how one can physically signify it by considering real life examples related to this particular term and the whole x-direction equation?

along x-direction, quasi 1D flow:

(rho*u*A)inflow -(rho*u*A)outflow = 0

the product (rho*u*A) MUST be the same at any location along the x-axis

Please find attachment and answer..

the answer is in what I already wrote...I strongly suggest to study any basic textbook of fluid dynamics

Yes. I read. Things got pretty much cleared. Thanks FMDerano:)