Recent content by umby

Exsactly what I was mentioning, thank you for point it out. Coefficients cannot be function of the unknown, only of the indipendent variables, in my case ##x## and ##t##. Can you please tell me more about the matter of its correctness?

You mean because of the coefficient of the derivative of ##T## with respect to ##x##? Maybe differential relation is better? Can you help me please in determining the correctness of this problem?

What is the best way to solve numerically the following equation using Comsol 5.3. ##\frac{\partial T}{\partial t}=\frac{\partial ^2T}{\partial x^2}+\text{St}\left[1+\left(\frac{\partial T}{\partial x}\right)_{x=0}\right]\frac{\partial T}{\partial x}## ##T(0,t)=1## ##T(\infty ,t)=0##...

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...

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.

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...

Semi-infinite solid.

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

At the SL interface temperature is the melting temperature.

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

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...

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...

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...

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

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...