Question

Consider the BVP or the steady-state temperature u(x,y,z) in the airspace between two infinite (in the y and z directions) stone walls that are perpendicular to the x-axis, and are 5 centimeters apart from each other (ie the wall surfaces in contact with the air space are 5 cm. apart), with the first (or left) wall located at x=-1 cm. and the second (or right) wall at x= 4 cm, where ux = -10, and u=100, respectively. Temperature is measure in degrees Centigrade and heat in calories. You may assume that the two walls are very thick relative to the space between them.

(a) Mathematically set up the BVP by formally writing down the PDE and boundary conditions, and corresponding independent variable domains, with as much mathematical detail as possible to completely specify and write down the BVP.

(b) Solve the above BVP via the Method of separation of variables. Show all the details of the procedure, and of your work.

(c) In a labelled diagram of the walls, looking from the direction of the y-axis (front or back), indicate with a vector arrow, the direction of the velocity, V, of heat flow at x = -1 and 4. explain why the heat energy in the airspace does not change.

Answer 1

Consider the BVP or the steady-state temperature u(x,y,z) in the airspace between two infinte (in the y and z directions) stone walls that are perpendicular to the x-axis, and are 5 centemeters apart from each other (i.e. the wall surface in contact with the air space are 5cm.apart), with the first (or left) wall located at x=-1cm. And the second (or right), wall at x=4cm, where ux =-10, and u=100, respectively.