the derivation for poiseuille flow has an ansatz but i googled that and it says that it is a guess and I don’t knwo how I know which guess is right
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
In the context of deriving Poiseuille flow, an “ansatz” is indeed a form of educated guess or assumption about the form of the solution. It is not arbitrary; rather, it is based on the physical understanding of the problem and the simplifications that can be made given the conditions of the flow.
For Poiseuille flow, the ansatz involves assuming that the flow is steady, fully developed, and axisymmetric. Specifically, the assumptions are:
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Steady Flow: The flow does not change with time, so the time derivative in the Navier-Stokes equations is zero.
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Fully Developed Flow: The velocity profile does not change along the length of the pipe, meaning that the velocity gradients in the flow direction (z) are zero.
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Axisymmetric Flow: The flow is symmetric around the axis of the pipe, so there are no variations in the circumferential direction (\theta).
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Flow in the z Direction Only: The radial and circumferential velocity components are zero ({u}_{\mathcal{r}} = 0, {u}_\theta = 0), and the flow is only in the z direction ({u}_z).
These assumptions simplify the Navier-Stokes equations significantly, allowing us to focus on the balance between the pressure gradient and viscous forces. The resulting equation is:
\frac{\partial p}{\partial z} = \eta \frac{1}{\mathcal{r}} \frac{\partial}{\partial \mathcal{r}} \left(\mathcal{r} \frac{\partial {u}_z}{\partial \mathcal{r}}\right)
This equation is then solved to find the velocity distribution, which is parabolic for Poiseuille flow. The assumptions are validated by the physical setup of the problem and the conditions under which Poiseuille flow is observed, such as low Reynolds numbers indicating laminar flow.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 11.3: Poiseuille Flow Part 1 (HIGHLY RELEVANT)
- Chapter 11.3: Poiseuille Flow Part 3 (HIGHLY RELEVANT)
- Chapter 9.3: Why We Use Irrotational Flow Analysis Part 5 (HIGHLY RELEVANT)
- Chapter 9.4: Finding Analytical Solutions for \psi and \phi Part 3 Part 1 (HIGHLY RELEVANT)
- Chapter 11.3: Poiseuille Flow Part 2 (HIGHLY RELEVANT)
Disclaimer:
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This ansatz is correct only if the Reynolds number is below 1000 or so. so the parabolic solution mentioned is only observed at those low reynolds numbers. The parabolic solution you get with this ansatz is a solution to the governing equation, but that solution is not unique – there are an infinite number of solutions. So the question is, of the infinite solutions possible, which are observed. For Reynolds number above 4000, you will observe one of the infinite number of turbulent solutions but you will never observe the parabolic laminar solution that PK Bot mentions.
how do you know that an ansatz is correct?
- if you make an ansatz in the hopes of simplifying and solving the system of equations, but you get something unsolvable, the ansatz was wrong.
- if you make an ansatz, get a system that you can mathematically solve, but the solution you get doesn’t match what is observed in the lab, then probably the system has some aspect that is not captured by the mathematical equations (in the case of turbulent flow, our steady mathematical equations do not capture that there are always infinitesimal perturbations whose stability or instability we have to consider – at high Re the laminar flow solution correctly solves the equations but is not stable and thus never observed. well technically if you are really careful you can observe laminar flow for a while at high Re but it takes a sophisticated lab setup).
- if the ansatz leads to an equation, you solve it, and you get an answer that matches the physical world, then TADA your ansatz helped you figure out the system