Obtain the governing equation for a viscous unidirectional flow in a circular pipe. Use the navier-stokes equations and employ the cylindrical coordinates.
direction of the flow: z
cylinder radius: r
angle: a
You have got to be kidding :)
That is far too much for me to produce here but below is a link to the subject worked out:
http://faculty.kfupm.edu.sa/CHE/usamah/CHE204/CHE204-HD22%20-%20Flow%20Through%20Circular%20Pipe.pdf
by the way, results are not a function of angle a (theta). All partial derivatives wrt theta are assumed zero.
Yea i assumed that you would provide a link to a similar subject that's why i asked here:)
Anyway i just assumed the instructor was asking how to produce the Hagen–Poiseuille eq from navier-stokes so got the solution from wikipedia. Thanks anyways Damon :)
To obtain the governing equation for a viscous unidirectional flow in a circular pipe using the Navier-Stokes equations and cylindrical coordinates, we need to consider the conservation equations for mass and momentum.
In cylindrical coordinates, the velocity field can be expressed as:
V = ur(r, a, z) + uθ(r, a, z), where u is the radial velocity component, uθ is the tangential velocity component, and V is the total velocity vector.
Now, let's break down the Navier-Stokes equations in cylindrical coordinates.
1. Conservation of Mass:
The continuity equation in cylindrical coordinates can be written as follows:
∂ρ/∂t + ∂(ρu)/∂r + (1/r) ∂(rρuθ)/∂.a + ∂(ρuz)/∂z = 0,
where ρ is the fluid density, t is time, r is the radial coordinate, a is the azimuthal (angular) coordinate, and z is the axial coordinate.
2. Conservation of Momentum:
The momentum equation in the z-direction (flow direction) can be written as:
∂(ρuz)/∂t + ∂(ρuzu)/∂r + (1/r) ∂(rρuθuz)/∂a + ∂(ρu^2z)/∂z = -∂P/∂z + (μ/r) ∂/∂r(r ∂uz/∂r) + μ ∂^2uz/∂z^2,
where P is the pressure, μ is the dynamic viscosity of the fluid, and u^2z represents the z-component of the velocity squared (u^2z = uz^2).
Note that the other momentum equations in the r and a directions also exist, but since the flow is unidirectional along z, those terms cancel out.
By solving these governing equations, you will be able to obtain the required equation for a viscous unidirectional flow in a circular pipe, using the Navier-Stokes equations and cylindrical coordinates.