Pressure in a flowing fluid If a fluid ( ideal) flows, say from left to right, then is the pressure at any point inside the fluid still independent of direction and has the same value no matter how I orient a surface element at that position?
If the above is true then, I am lead to an paradoxical condition. Say the fluid flows in cuboidal tube. If I consider a small cubical volume element then clearly the pressure on the left face of the cube is different from the right face of the cube but again since the volume element is in vertical equilibrium the variation of pressure with depth formula holds, so it means if the pressure on the open surface be P then the pressure at the two faces of the volume element are both (P + rdg) (where r is density, d is depth of that end and g is gravitational acceleration) implying they both have the same pressure.
Where is it that I am making mistake? Should I be considering the stress tensor instead of pressure? 
 A: 
I consider a small cubical volume element then clearly the pressure on the left face of the cube is different from the right face of the cube...

I think that in the idealized case that you're presenting including a fluid which is an ideal liquid (i.e., an incompressible fluid with no viscosity) that the pressure on the left face of the cube should be equal to that on the right side of the cube. You can think of this in terms of Newton's F=ma. If a cubical volume element of fluid is just coasting along and flowing at constant speed with no acceleration, then the net force acting on it should be zero. So the pressures on both the left and right sides of the cube should be the same.
As for a stress tensor, I don't see why that would be needed here since the fluid is an ideal liquid with no viscosity. There will be no off-diagonal shear components to any stress tensor in the fluid. 
(P.S.: Of course any real liquid will have some viscosity and so the pressures on both the left and right sides will not be exactly equal. Some pressure gradient will be required to keep a real fluid flowing at constant speed through a pipe. But, again, your question is for the limit of an ideal liquid.)
