Consider a horizontal pipe with uniform area of cross section at all points kept on the ground.I send water through one end of the pipe at a pressure of around 2atm.Since the pipe is of uniform area throughout, according to the equation of continuity velocity is same throughout.The pressure at the other end is 1 atm because of the atmosphere and due to pressure difference there is water flow but when I use Bernoulli's equation and put P1=2atm and input values of V1 and V2 i get P2 as 2 atm as well.How is this possible? Shouldn't i get 1atm? Is the Bernoulli's equation wrong for this case it is my thinking wrong?Please consider only streamline,steady and non viscous flow
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1$\begingroup$ your question already has an answer here physics.stackexchange.com/q/244699 $\endgroup$– Apoorv MishraCommented Apr 7, 2022 at 16:43
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2$\begingroup$ Does this answer your question? Does continuity equation hold if the flow is accelerated? $\endgroup$– Apoorv MishraCommented Apr 7, 2022 at 16:45
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$\begingroup$ The link doesn't answer my question as to why the pressure is equal when I input the values in Bernoulli's equation.Mathematically it doesn't make sense that though the two ends have different pressures the equation tells us that pressure is same! $\endgroup$– AJknightCommented Apr 7, 2022 at 16:49
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$\begingroup$ You seem to have a notion that the continuity equation is just $$A_1.v_1 = A_2.v_2$$ But the continuity eqn is the mass balance at two points that is, $$d_1.A_1.v_1 = d_2.A_2.v_2$$ where d is density. So for velocities to be equal at the two points, the densities shall be equal or the flow, incompressible. An incompressible fluid cannot here be subjected to different pressures at the ends of a constant area duct. And hence the flow must be compressible or the velocities must be different. But bernoulli's equation does hold true so instead put pressure in it and you'll get velocities different. $\endgroup$– Apoorv MishraCommented Apr 7, 2022 at 17:06
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$\begingroup$ The first form of continuity is correct only when the densities are equal $\endgroup$– Apoorv MishraCommented Apr 7, 2022 at 17:17
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I'll reproduce my comments to this answer here, as they seem to answer your question:
It is perfectly consistent for the fluid to be moving without any change in pressure from one end of the pipe to the other. When the pressures and the areas are the same on either side of a "slice" of water in the pipe, this means that the forces from either side cancel out; but this just means that the water is moving at a constant velocity, not necessarily zero velocity. (Newton's First Law!) You're right that you would need a difference in pressure initially to get the flow started; but once it's moving, no difference in pressure is needed to maintain the flow.
Bernoulli's equation and the continuity equation implicitly assume that nothing is depending on time, i.e., the velocity at a given point in the pipe isn't changing. So you can't really use either one to address how the flow gets started; they only really deal with the steady-state flow, once everything is moving nice and smoothly throughout the system.
In the "real world", fluids have viscosity, which causes an extra force between the pipe walls and the fluid. If you want to take this into account, you end up with something like Poiseuille's equation instead, where a pressure difference is required to maintain a steady flow. Bernoulli's equation is really an approximation in which we treat the viscosity as negligible. It's not unlike how we talk about "frictionless tables" in introductory mechanics — such a thing doesn't really exist, but in many situations the effects are negligible.
answered Apr 8, 2022 at 19:16