FrontierScience / e1b0b588-915d-46ad-990e-ba5aa554053e / In a fluid flow, a vortex forms around a central axis, which is a straight line.…

In a fluid flow, a vortex forms around a central axis, which is a straight line. The fluid in the vortex flows purely in circles around the central axis of the vortex. The flow has cylindrical symmetry and symmetry of translation along the direction of the axis of the vortex. Gravity can be ignored.

Let `r `be the distance of any point in the vortex from the central axis of the vortex. The speed `v` of fluid flow depends only on `r` and time `t`. We can write it as

`v=(Γ)/(2π)g(r,t)`

where `g` is some unknown function of `r` and `t` and \\(\\Gamma\\) is a constant particular to the vortex.

The fluid is incompressible and has density `ρ`. The fluid is subject to internal stresses from viscosity. Assume the fluid is Newtonian and that the viscosity is `η`.

Find `(∂g)/(∂t)` in terms of `r,t,Γ,ρ,η` and `g,(∂g)/(∂r),(∂²g)/(∂r²)`.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.

In a fluid flow, a vortex forms around a central axis, which is a straight line. The fluid in the vortex flows purely in circles around the central axis of the vortex. The flow has cylindrical symmetry and symmetry of translation along the direction of the axis of the vortex. Gravity can be ignored.

Let `\(r\) `be the distance of any point in the vortex from the central axis of the vortex. The speed `\(v \)` of fluid flow depends only on `\(r \)` and time `\(t \)`. We can write it as

`\(v = \frac{\Gamma}{2\pi} g(r,t) \)`

where `\(g \)` is some unknown function of `\(r \)` and `\(t \)` and \\(\\Gamma\\) is a constant particular to the vortex.

The fluid is incompressible and has density `\(\rho \)`. The fluid is subject to internal stresses from viscosity. Assume the fluid is Newtonian and that the viscosity is `\(\eta \)`.

Find `\(\frac{\partial g}{\partial t} \)` in terms of `\(r, t, \Gamma, \rho, \eta \)` and `\(g, \frac{\partial g}{\partial r}, \frac{\partial^2 g}{\partial r^2} \)`.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.