Finding Expressions for dv/dz and dsigma/dz in Cylindrical Pipe Flow?

[Hugheberdt]

Homework Statement

A full problem description can be found in the attachment (observe the misprint in eq 1). Here follows an outline:

We are to study the flow of a fluid through a cylindrical pipe.

Two components of Navier Stokes in cylindrical coordinates, with some simplifications due to assumptions given in the problem:

u*du/dz = my*(d^2u/dr^2 + (1/r)*du/dr) - v*du/dr - (1/rho)*dp/dz (1)
du/dz + (1/r)*d(r*v)/dr = 0 (2)

where r is the radial coordinate, z the lengthwise distance, u is the z-wise component of the velocity field and v is the radial component. rho is the density, and p is the pressure.
In the problem (1/rho)*dp/dz is to be substituted with sigma.

Some BC:s and limits of (1) and (2) are also given, but are omitted here.

The objective is to numerically find the velocity components using the Method of Lines (MoL). (1) and (2) are to be discretized in the r-direction (with certain difference approximations for r derivatives) yielding a system of equations to be solved (the FDM).

The system of equations will (is to) consist of 2n variables (u1,u2,...,un,sigma,v2,v3,...,vn). v1=0 due to BC:s.

The z dependence is then found by solving a resultant ODE in 2n equations using Euler Implicit. The ODE should look something like du_bold/dz=f(du_bold,r,z), where u_bold = [u1,u2,...,un,sigma,v2,v3,...,vn], and f is a 2n*1 vector function. Since f will be non-linear, it is approximated with J*du_bold, where J is the Jacobian of f with respect to u_bold.

Our problem (we think), is to find explicit expressions for dv/dz and dsigma/dz, which are required in order to formulate the ODE.

Homework Equations

u*du/dz = my*(d^2u/dr^2 + (1/r)*du/dr) - v*du/dr - (1/rho)*dp/dz (1)
du/dz + (1/r)*d(r*v)/dr = 0 (2)

The Attempt at a Solution

We took the simple approach of solving (1) and (2) for du/dz, discretizing and taking them as f(u_bold). This way however,
we only get du/dz, not dsigma/dz or dv/dz, which we require in order to form the ODE system.

Our main problem here is finding expressions for dsigma/dz and dv/dz. We have considered solving (1) and (2) for either
variable and taking partial derivatives in z, but we don't see how we can from just 2 equations find expressions for
all three of u,v and sigma.

We have considered using other compenents of Navier Stokes, or other fluid mechanics reations, but we haven't succeeded
in that, and are doubtful to that approach since all relevant information for the solution fo the problem should be
included in the problem description.

Could someone please help us find a way to get expressions for dv/dz and dsigma/dz or explain why we don't need them?

Thanks!

 

[mighty2000]

Thank you for posting your problem and providing a detailed description of your approach. It seems like you are on the right track with your solution, but you are missing a key component in your equations.

In order to solve for the velocity components using the Method of Lines, you will need to incorporate the continuity equation into your system of equations. This equation states that the rate of change of density with respect to time is equal to the divergence of the velocity field.

In cylindrical coordinates, this can be written as:

(1/r)*(d(r*rho*u)/dr) + du/dz = 0

This equation can be discretized in the same manner as equations (1) and (2) to form a system of equations with the same number of variables. This will allow you to solve for all three variables (u, v, and sigma) simultaneously.

Once you have solved for the velocity components, you can then use the continuity equation to solve for the pressure (p).

I hope this helps you in your solution. Good luck!