# Recent content by umby

1. ### 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. ### 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. ### 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. ### Heat conduction from an isotherm spherical cap

Semi-infinite solid.
5. ### Heat conduction from an isotherm spherical cap

You have a steady-state if nothing changes with time.
6. ### Heat conduction from an isotherm spherical cap

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