Penetration of Potential in a Semi-Infinite Region

This Demonstration shows the penetration in a semi-infinite region of a dimensionless potential described by the model , , with the three boundary conditions: , , and , where is dimensionless distance, is dimensionless time, and Bi is the Biot number.
These classical problems are considered in many textbooks and in the booklet by M. D. Mikhailov and M. N. Özisik, Heat Transfer Solver, Englewood Cliffs, NJ: Prentice Hall, 1991. The potential could be temperature, concentration, etc., as described by M. D. Mikhailov and M. N. Özisik, Unified Analysis and Solutions of Heat and Mass Diffusion, New York: Dover, 1994.
For the case , the first animation was presented by Prof. U. Grigull at the opening of the Seventh International Heat Transfer Conference, Munich, FRG, September 6–12, 1982.
In contrast to the existing animations, the frame ticks change with time , which conveniently lets you observe the potential distribution during penetration.
On the same plot are shown the dimensionless potentials (orange curve) and (blue curve) for two different materials with conductivities and related by the ratio .
For a fixed time , the slider determines black points on the two curves. The plot label gives the numerical values of followed by the numerical values of . Thus for any and the precise values of the potential are available.
The slider for the Biot number Bi acts only in the case .



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


The problems
, , , ,
, , , ,
, , ,
have the following solutions:
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+