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Consider the example shown above. Ideal fluid is flowing along the flat tube of uniform cross section area located in the horizontal plane and bent as shown. The flow is steady. My text book says following two things:

  1. Pressure at point 1 will be greater than the pressure at point 2 and velocity at point 1 will be less than the velocity at point 2.

  2. Streamlines will be higher in density near point 2 than point 1.

I understand that point 1 implies point 2, as velocity is directly proportional to the number of streamlines crossing a unit area.

What I fail to understand are the following points:

  1. How do we intuitively understand the idea that pressure at point 1 will be greater than the pressure at point 2?

  2. My textbook uses Bernoulli's equation and says that since pressure at point 1 is greater than pressure at point 2, so velocity at point 1 will be less than the velocity at point 2. I completely fail to understand the distribution of velocity at the turning point in the appratus above. Mathematically I am convinced that Bernoulli's equation gives a certain answer but Intuitively I totally fail to understand this answer.

  3. Can somebody help me to understand this phenomena just on the basis of forces and not on the basis of Bernoulli's equation?

  4. How do we understand the distribution of velocity intuitively?

  5. Apart from the bending/turning, in the straight portions, will the streamlines be equidistant? or they will follow the same kind of distribution they are following at the turning? And why? If we assume the streamlines to be equidistant in the straight portions then it means that their distribution changes at the turning point and then again it resets back to what it was earlier, how is this all happening?

  6. Can somebody help me to understand the velocity distribution, and the pressure distribution in the above case intuitively, without using Bernoulli's equation?

Thanks in advance.

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    $\begingroup$ @Chet Miller's response, along with the discussion in the comments, makes the answer complete. $\endgroup$ Jan 24, 2020 at 14:14

2 Answers 2

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Laplace's equation in terms of the stream function is based on Euler's equation (which takes into account the inertial forces involved). The solution to Euler's equation will show that the streamlines are closer together at point 2 than at point 1. The pressure has to be higher at point 1 than point 2 because it is on the outside of the curve, so that this region must support the centripetal acceleration of the fluid nearer the center of curvature.

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  • $\begingroup$ Yes Sir. I agree about the pressure, what you mentioned. Your explanation of Pressure is what I call "Intuitive". Thanks for that. My question is, how do we "INTUITIVELY" (without mathematics) understand the idea that velocity at point 1 will be less than the velocity at point 2? With mathematical equations, we can justify that the distribution of velocity, but I am demanding an intuitive explanation. Kindly help. $\endgroup$ Jan 23, 2020 at 16:26
  • $\begingroup$ You’re asking for an explanation of why, when velocity increases, pressure decreases along a streamline, right? $\endgroup$ Jan 23, 2020 at 18:51
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    $\begingroup$ No Sir. Points 1 and 2 are not on the same streamline so we cannot apply Bernoulli's equation there. $\endgroup$ Jan 23, 2020 at 18:53
  • $\begingroup$ I am asking the intuitive explanation behind the velocity distribution at the turning point and specifically how do we know intuitively that V1 will be less than V2. $\endgroup$ Jan 23, 2020 at 18:54
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    $\begingroup$ You can apply Bernoulli's equation to each of the two streamlines individually, referencing them both to the upstream conditions. For the fluid parcels traveling along a given streamline to accelerate, the upstream force (as characterized by the upstream pressure) must be higher than the downstream force (as characterized by the downstream pressure). Apparently, the pressure gradients along the two streamlines are such that the acceleration along streamline 2 results in a higher velocity at point 2 than the acceleration along streamline 1 produces at point 1. $\endgroup$ Jan 23, 2020 at 20:55
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Bennoulis principle has a condition that the flow must be streamline. The curve will create Eddies and Bernoulli's principle will not apply

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  • $\begingroup$ It says the flow is steady. $\endgroup$
    – JMac
    Jan 23, 2020 at 16:22
  • $\begingroup$ In case of steady flow, the Bernoulli's Equation can be applied. $\endgroup$ Jan 23, 2020 at 16:48

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