# 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?

• The equation you wrote applies to the case of a free surface (as you said), and your conclusion is correct if the free surface is horizontal. But, if the free surface is not horizontal, then the pressures on the right and left faces of the cuboid would not be equal, and there would be flow from the higher pressure side to the lower pressure side. Apr 9, 2019 at 19:42
• But in the cuboidal tube of flow I consider, the free surface is still horizontal but yet there is pressure differences at the faces of some volume element, yet the vertical equilibrium condition leads to the contradiction that the pressure is same at the two faces of the volume element. What is going on? Apr 9, 2019 at 20:45
• If the free surface is horizontal, then there is no pressure difference between the faces of your volume element. So, you have no flow. Apr 9, 2019 at 20:47