# Pressure generation due to opposing flows

If two fluids are flowing at unequal velocities towards each other in a circular pipe, will a pressure be generated at the intersection? If yes, what will the direction of this pressure generated and why is it generated. If possible, do explain how to calculate the pressure.

• Are the fluids supposed to be incompressible? – Chet Miller Sep 29 '16 at 13:20
• Yes they are incompressible. – Spranav94 Sep 29 '16 at 15:04
• How can they possibly be flowing in opposite directions within the same pipe? After they crash, where does the fluid go? – Chet Miller Sep 29 '16 at 18:09
• That's correct. Assuming there is another pipe connected at the intersection of these two flows, can you explain how the pressure at the intersection can be calculated? – Spranav94 Sep 30 '16 at 2:55
• So, you have two flows flowing into a T? – Chet Miller Oct 1 '16 at 1:31

In this case, the two opposing flows join to form the flow in the center leg of the T. To get the pressure in the center leg after the flow has had sufficient distance to become uniform, you can apply the Bernoulli equation by weighting in terms of the inlet and outlet flows:

$$\dot{m_1}\left(\frac{p_1}{\rho}+\frac{v_1^2}{2}\right)+\dot{m_2}\left(\frac{p_2}{\rho}+\frac{v_2^2}{2}\right)=\dot{m_3}\left(\frac{p_3}{\rho}+\frac{v_3^2}{2}\right)$$ where the $\dot{m}'s$ are the flows in the legs of the T.

Getting the pressure distribution in the intersection is very complicated, because the flow within the intersection region is non-uniform, and you need to solve the detailed Euler equations (or Navier Stokes equations if you want to include viscous effects).

I would agree with @ChetMiller on his answer.

But just for the sake of strictness of the physical model, I would like to add that Bernoulli's equation seems not be applicable here since it's written for two points on one streamline of a flow (it can also be written for two point on different streamlines, but again the streamlines should belong to one flow).

Instead, conservation of energy equation written for control volume is applicable here. Still, conservation of energy equation will be written in the exactly same way as in the @ChetMiller's answer.

I would also add that one can place border number 3 of the control volume at any position in the central leg - not necessarily far from the beginning. It is because we'll get cross section averaged quantities anyway.

Also, one will most likely want to combine conservation of mass with conservation of energy.

For transient flow (when mixing has just begun) both conservation of mass and energy should be rewritten appropriately.

For detailed pressure and velocity distribution in $$T$$ region, direct numerical simulation is necessary.