代做MATHS 3021 Capstone Project in Mathematical Sciences III APPLIED MATHEMATICS ASSIGNMENT 2帮做R程序

日期：2024-08-27 07:10

MATHS 3021 Capstone Project in Mathematical Sciences III

APPLIED MATHEMATICS ASSIGNMENT 2

For this assignment, you must use LATEX to type up your solutions and generate a pdf file for submission.

[FB:1-14] A thin wire of radius a is extruded at a fixed velocity V0 through a die (see Figure below, taken from Fulford & Broadbridge, 2002). Let x = 0 be the position of the die exit at which the wire temperature is a fixed value u0. The wire then passes through the air having temperature ua for some distance before it is rolled onto a large spool at a large distance from the die exit. We wish to investigate the relationship between the wire velocity and the distance between the extrusion nozzle and roll for specific values of V0.

1. Assuming that the temperature is independent of radius, i.e. is constant in a cross-section of the wire, derive the PDE for the temperature u(x, t) of the wire:

You should consider a section of the wire between x and x + δx, start with a word equation, and define all additional symbols that you introduce. Assume that all material properties are constant values. (Note that much of this has been outlined in workshops but addition of advection to the heat conduction equation has not been discussed in detail. In adding advection to the derivation you may find it helpful to refer to what was done in workshops for the diffusion equation.) (6 marks)

2. Write down the boundary conditions for the problem and explain why it is reasonable to neglect time dependence. Hence, write down the equation to be solved and the associated boundary conditions for the equilibrium temperature u(x). (3 marks)

3. Scale the steady-state equation, using u0 as the temperature scale to obtain an equation of the form. (using hats to denote scaled variables)

where you must define the parameters ∈1, ∈2, and in terms of the physical parameters in the problem. Also give the scaled boundary conditions. (4 marks)

4. Define the length scale L such that ∈2 = 1. Check the definition of L to make sure it has the correct dimensions and give the new definition of ∈1. (3 marks)

5. Solve the scaled steady-state equation

(6 marks)

6. Convert your dimensionless solution obtained in 5 to a dimensional solution. (2 marks)