The Stefan Problem

The problem we solve is illustrated in Figure 4–29. The flow has solidified to the depth y = ym(t). We assume that molten material of uniform temperature Tm lies everywhere below the growing surface layer. The fact that the molten region does not extend infinitely far below the surface is of no consequence to the solution. We must solve the heat conduction equation (4–68) in the space 0 ≤ y ≤ ym(t) subject to the conditions T = T0 at y = 0, T = Tm at y = ym(t), and ym = 0 at t = 0. The position of the solidification boundary is an a priori unknown function of time. As in the case of the sudden heating, or cooling, of a semi-infinite half-space, there is no length scale in this problem. For this reason, we once again introduce the dimensionless coordinate η = y/2 √ κt as in Equation (4–96); it is also convenient to introduce the dimensionless temperature θ = (T − T0)/(Tm − T0) as in Equation (4–93).

The dimensionless coordinate η is obtained by scaling the depth with the thermal diffusion length √ κt because there is no other length scale in the problem. Similarly, the depth of the solidification interface m must also scale with the thermal diffusion length in such a way that ym/ √ κt is a constant.

Don't use plagiarized sources. Get Your Custom Essay on
The Stefan Problem
Just from $13/Page
Order Essay

In other words, the depth of the solidification boundary increases with time proportionately with the square root of time. We have used dimensional arguments to determine the functional form of the dependence of ym on t, a nontrivial result. Because η = y/2 √ κt and ym is proportional to √ κt, the

solidification boundary corresponds to a constant value ηm = ym/2 √ κt of the similarity coordinate η. We denote this constant value by ηm = λ1. Thus we have ym = 2λ1 √ κt. (4.136) With our definitions of θ and η, the heat conduction equation for θ(η) is clearly identical to Equation (4–100), whose solution we already know to be proportional to erf(η). This form of solution automatically satisfies the condition θ = 0(T = T0) on η = 0(y = 0). To satisfy the remaining condition that θ = 1(T = Tm) at η = ηm(y = ym) = λ1, we need simply choose the constant of proportionality appropriately. 

Essay Writing Pros
Calculate your paper price
Pages (550 words)
Approximate price: -

Why Work with Us

Top Quality and Well-Researched Papers

Our writers have been trained on how to handle papers placed by our clients. The writer must read and understand before embarking on writing the papers. In case of any issue that needs clarification, writers are encouraged to ask the client or support.

Professional and Experienced Academic Writers

Our team comprises of the best writers and editors. We do thorough vetting during recruitment to make sure that our writers have the knowledge and experience we aspire in the team.

Free Unlimited Revisions

Our aim is to give the client the best outcome. If for some reason you are not satisfied with the wok done, you can ask the paper to be revised or rewritten. This will be done to your satisfaction with no extra charges.

Prompt Delivery and 100% Money-Back-Guarantee

We have writers who work round the clock. This helps in making sure that all our clients’ papers are delivered on time. If we have issues with the deadline, we will ask for extension. If its not possible, the money is fully refunded.

Original & Confidential

Our clients’ confidentiality is highly respected. We can never disclose our clients’ details to third parties. In the same regard, we strive to give our clients 100% original papers. We do not tolerate plagiarism from our writers.

24/7 Customer Support

Clients can reach us any time of the day, and any day of the week. There is a live chat, email or phone numbers to help in ease of communication.

Try it now!

Calculate the price of your order

Total price:

How it works?

Follow these simple steps to get your paper done

Place your order

Fill in the order form and provide all details of your assignment.

Proceed with the payment

Choose the payment system that suits you most.

Receive the final file

Once your paper is ready, we will email it to you.

Our Services

You should never be worried about your papers even in the middle of the night. Our team will work round the clock to deliver.


Essay Writing Service

We have an able team that can deliver your work in the shortest time possible. The academic level or the type of work should never be a hindrance. Our highly competent support team is always around (24/7) to give you any assistance you may need.


Admission Essays & Business Writing Help

Do you need to be admitted in your dream institution but find it challenging to write an admission essay? Our team is in a position to write the best letter that will guarantee you an admission. We do as well write the best business proposals and reports.


Editing Support

Writing can be fun and enjoyable when everything has been done right. Writing is not just enough without proper editing and proofreading. We have a team of editors that ensure everything falls in place, whether its issues to do with grammar or referencing styles.


Revision Support

Once the paper has been done and submitted, that is not the end of it. You can always ask for amendment or improvement if you feel something has not been done right. Our team of writers and editors will gladly assist you to your satisfaction. Revision is free of charge.