PROBLEM 4.1 KNOWN:Method of separation of variables for two-dimensional, steady-state conduction. FIND:Show that negative or zero values of λ2, the separation constant, result in solutions which annot satisfy the boundary conditions. cSCHEMATIC:ASSUMPTIONS:(1) Two-dimensional, steady-state conduction, (2) Constant properties. ANALYSIS:From Section 4.2, identification of the separation constant λ2leads to the two ordinary differential equations, 4.6 and 4.7, having the forms 22dXdYX0 Y0dxdy+=−=λλ(1,2) and the temperature distribution is ( ) ( ) ( )x,yX xY y .=⋅θ(3) Consider now the situation when λ2= 0. From Eqs. (1), (2), and (3), find that ( ) ( ) ( )1234XCC x, YCC y and x,yCC x CC y .=+=++(4) Evaluate the constants - C1, C2, C3and C4- by substitution of the boundary conditions: ( ) ( )( )()()()()()()123432424x0: 0,yCC0 CCy0 C0yx,00CxCC00 C0xL: L,00CL 0Cy0 C0yW: x,W00 x0CW1 ==+⋅====+⋅+⋅=+⋅+⋅=+⋅+⋅=01==≠The last boundary condition leads to an impossibility (0 ≠1). We therefore conclude that a λ2value of zero will not result in a form of the temperature distribution which will satisfy the boundary
This is the end of the preview. Sign up
access the rest of the document.
Boundary value problem, ORDINARY DIFFERENTIAL EQUATIONS, Boundary conditions, C7 cos