Recent content by umby

  1. U

    Heat conduction from an isotherm spherical cap

    I refer to the first post. The cap is not meant to be a hemisphere. In normalization, length dimensions were re-scaled by means of ##R## (the radius of the sphere from which the spherical cap is derived), this means that ##p##, the depth of the cap below the surface, can vary from 0 to 1 (0 = no...
  2. U

    Heat conduction from an isotherm spherical cap

    I try to upload an image of the surface of the semi-infinite solid. The spherical cap is isotherm while the rest of the surface is adiabatic. The semi-infinite solid is beneath the surface. I apologize for the misunderstandings.
  3. U

    Heat conduction from an isotherm spherical cap

    Returning to the spherical cap, in a spherical coordinate system centered at the "center" of the cap, the PDE is: ##\frac{\partial }{\partial r}\left(r^2\frac{\partial T}{\partial r}\right)+\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial T}{\partial...
  4. U

    Heat conduction from an isotherm spherical cap

    Semi-infinite solid.
  5. U

    Heat conduction from an isotherm spherical cap

    You have a steady-state if nothing changes with time.
  6. U

    Heat conduction from an isotherm spherical cap

    At the SL interface temperature is the melting temperature.
  7. U

    Heat conduction from an isotherm spherical cap

    In the case of the sphere, one independent variable: ##r##, but for the spherical cap both ##r## and ##\theta##.
  8. U

    Heat conduction from an isotherm spherical cap

    I apologize for the bad exposition of the problem. I will try to explain my thoughts with another example. Let's suppose there is a heat point source of power ##q## in an infinite solid initially at temperature ##T_0##. This source melts part of the solid forming a spherical melt pool; the...
  9. U

    Heat conduction from an isotherm spherical cap

    On the surface of a semi-infinite solid, a point heat source releases a power ##q##; apart from this, the surface of the solid is adiabatic. The heat melts the solid so that a molten pool forms and grows. Let's hypothesize that the pool temperature is homogeneously equal to the melting...
  10. U

    I Approximation of a function with another function

    Dear Office_Shredder, you are right. Both functions go to zero as ##x->0##, or ##y->0##, or both go to zero. Then, it is obvious that, under the same condition, their difference goes to zero as well. Actually, I was looking for a way to solve this integral: ##\int_0^{\infty } \frac{\exp...
  11. U

    I Approximation of a function with another function

    I will enjoy reading your proof anyway, if not more. :smile:
  12. U

    I Approximation of a function with another function

    Thank for your message. Let's call ##f1## the function: ## \pmb{\text{f1}=\frac{\text{Exp}\left[-\frac{1 }{2 \epsilon ^2} \left(\frac{\left(x-\epsilon ^2 t \right)^2+y^2}{t +1}\right)\right]}{\sqrt{t} (t +1)}}\\## And ##f2## the function: ##\pmb{\text{f2}=\frac{\text{Exp}\left[-\frac{1 }{2...
  13. U

    I Approximation of a function with another function

    You are very kind. ##\int_0^{\infty } \frac{\text{Exp}\left[-\frac{1 }{2 \eta ^2} \left(\frac{\left(\xi -\eta ^2 \tau \right)^2+\psi ^2}{(\tau +1)}+\frac{\zeta^2}{\tau }\right)\right]}{\sqrt{\tau } (\tau +1)} \, d\tau##
  14. U

    I Approximation of a function with another function

    ##\int^{\infty }_0{\frac{e^{-\ \ \frac{1}{2{\varepsilon }^2\ }\left[\frac{{(x-{\varepsilon }^2t)}^2+y^2}{t+1}\right]}}{\sqrt{t}(t+1)}}dt## thank you, it was helpful
  15. U

    I Approximation of a function with another function

    You are full right! It was my fault. Unfortunately, I cannot write formulae correctly in this forum, so I did a picture of my integrals, but the resolution was very bad. The correct integrals are: and . I lost the minus in the conversion process. You can interpret x and y as related by the...
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