# Why liquid in a duct flows when there is a pressure difference?

Liquid flows in a pipe when a pressure difference is applied on the ends. But can a liquid flow even if there is no pressure difference? Like in case the liquid (ideal) is flowing with uniform velocity through a horizontal pipe, the pressure along the horizontal direction remains same.

Please explain this in a detailed way considering the liquid to be ideal.

• Welcome to Physics SE. Please, take into account that this site policy does not encourage homework questions. Some more context would avoid your question could be considered homework. Commented Jul 7, 2021 at 8:33

No. When a pressure difference exists between two points, the fluid will start to move from the higher pressure point to the low pressure point. This happens to minimize the pressure difference.

Without such a pressure differential, the fluid is stationary, and the system is absent of any flow.

But if we consider a long horizontal pipe, and consider a small segment of this pipe, Bernoulli's equation tells us that since $$p_1 +\rho gh_1 + \frac{1}{2}\rho v_1^2 = p_2 +\rho gh_2 + \frac{1}{2}\rho v_2^2$$ then since $$h_1=h_2$$ $$p_1 + \frac{1}{2}\rho v_1^2 = p_2 + \frac{1}{2}\rho v_2^2$$ Now since we are considering a small section, we get that $$p_1-p_2\approx 0$$ which will yield $$v_1=v_2$$ meaning that we can have fluid flow over a small pressure differential.

But again, for large sections, the pressure at one end of the pipe must be larger than the pressure at the other end for there to be flow.

• what if the liquid was flowing in the beginning . It will not need any pressure difference in that case to continue moving ... Commented Jul 7, 2021 at 8:50
• There must have been a pressure difference to start it flowing. I guess once the pressure difference is equalized, what's left is possibly the fluids inertia, but even that will be momentary. Commented Jul 7, 2021 at 8:54
• Even if when the liquid is in hydrostatic condition, there is pressure difference between the free surface and bottom of the container, though the liquid doesn't flow in this case. How can we explain this then? Commented Jul 7, 2021 at 9:49
• You can have a pressure difference and no flow, like in your example of static equilibrium, but you cannot have flow without a pressure difference. Commented Jul 7, 2021 at 9:56
• You can show using Bernoulli's equation, that along a small cross-section of pipe, that if at two close points $p_1 =p_2$ then $v_1=v_2\ne 0$, so in this sense, and under small distances of flow, you can have it this way. But remember that to get the flow beginning, there must have been a pressure difference further on either ends of the flow. Cheers and good luck with your studies. Commented Jul 7, 2021 at 10:08

Considering the motion of a perfect fluid is like considering a mass that slides without friction: it can remain at constant speed indefinitely without any external force. It is the same with a perfect fluid that can flow indefinitely without a pressure differential. In a sense, it is what happens for a superfluid of zero viscosity (But the subject is complicated!).

When there is no potential energy term (same height) Bernoulli's theorem simply tells us that, for a perfect fluid, a pressure difference causes a change in speed. This is equivalent to Newton's law: a force produces a change in speed. In contrast, for a viscous fluid, the pressure differential compensates the viscous forces and maintains a constant velocity.