1. ANALYSIS OF ISOTHERMALS AND
HEAT FLOW LINES USING
GEOGEBRA/MATLAB
Submitted by:
Saumya Tiwari
Dheeraj Mangal
Praman Satya
Shubham Sharma
Somesh Sharma
2. COMPLEX POTENTIAL
We are now going to investigate the properties of a complex function the real and imaginary
part of which are conjugate functions. In particular we define the complex potential
In the complex (Argand-Gauss) plane every point is associated with a complex number
3. In general we can then write;
The fact that Cauchy-Riemann conditions hold for both and Ψ or equivalently
that these functions are conjugate, is a necessary and sufficient condition for the
function f to be analytic.
Now, if the function f is analytic, this implies that it is also differentiable,
meaning that the limit
ᶲ
4. is finite and independent of the direction of
If then we pose it follows that
5. LAB EXPERIMENT AND MATERIAL USED:-
Aluminum foil as metal for measuring the conduction with in it.
Temperature at the both the ends is 0 C and 40 C.
For const. temperature we use heating plate on as source of heat and ice bath on the other side
which work as sink for 0 C.
Dimension of the foil and the plotted graph (28*22) cm.
We find the isotherms length wise.
Digital as well as analog thermometer for measuring the temperature.
Marker for marking the temperature.
7. DETERMINATION OF GRAPH EQUATION USING
GEOGABRA:-
Here the lines in black signifies isotherms and the lines in red signifies the heat
lines associated with them equations of isotherm respectively.
9. THEORITICAL PROOF OF HARMONICITY OF THE FUNCTION AND
ITS HARMONIC FUNCTION:-
FOLLOWING IS THE THEORITICAL CALCULATION WHICH WE GET BY TAKING
EQUATION (d) AS EXAMPLE AND PROFFING THAT IT IS HARMONIC OR NOT AND ITS
HARMONIC CONJUGATE
10.
11. PHYSICAL INTERPRETATION:-
When we applied constant temperature on the both end of aluminum with
temperature difference of 40’ C then we see that heat due to this
temperature difference we observe that as we move from higher temp. to
lower temperature i.e. 0 C. we also observe that the isothermal lines are
observed parallel to the source i.e. horizontal and the heat flows
perpendicular to these isotherms
12. PROPERTIES OF THIS FUNCTION:-
This function (si) is harmonic.
It’s Analytical along all complex plain i.e. it is an entire function.
It has two component real and imaginary where real part show the isotherm
and imaginary part show the heat lines flow associated with it