Incompressibility in boundary layer (Fluid Dynamics)

[Mandeep Deka]

I have started studying fluid mechanics recently and seems to be a very basic conceptual question that is bugging me and unfortunately I am unable to find a reasonable explanation for it. Your help would be more than appreciated.

The mathematical definition for incompressiblility in fluid dynamics is that the divergence of the velocity vector field must be zero at all points in the flow region. Now, when we study the development of boundary layer in pipe flow, we notice that as the boundary layer develops, the velocity of the inner core is said to increase. The justification to this is provided that because the fluid considered is incompressible, the net mass flow out of the cross section of the pipe must be constant, and since the velocity near the pipe wall decreases, there must be some compensation of the mass loss, hence an increment of velocity at the inner core. But, here's a paradox to the argument: if the velocity of the core is increasing as the fluid moves forward (in the region where BL develops), the spatial variation of velocity would suggest that the velocity vector field is a function of the distance along the pipe, and hence, that would mean that the divergence must not be zero, which in turn would mean that the flow isn't incompressible.

So, where is it that i am getting the concept wrong?

 

[RandomGuy88]

Why do you think that the spatial variation in the velocity field implies that the divergence is non zero?

Zero divergence means that the sum of the diagonal elements of the velocity gradient tensor is zero. If the flow is accelerating in the direction parallel to the pipe axis, the x-axis let's say, then du/dx would be positive. Incompressiblity then requires at least one of the other two derivatives dv/dy or dw/dz to be negative and for the sum of all three to be zero.