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ABSTRACT
A convergence enhancement technique known as the integral balance approach is employed in combination with the Generalized Integral Transform Technique (GITT) for solving diffusion or convection-diffusion problems in physical domains with subregions of markedly different materials properties and/or spatial scales. GITT is employed in the solution of the differential eigenvalue problem with space variable coefficients, by adopting simpler auxiliary eigenproblems for the eigenfunction representation. The examples provided deal with heat conduction in heterogeneous media and forced convection in a microchannel embedded in a substrate. The convergence characteristics of the proposed novel solution are critically compared against the conventional approach through integral transforms without the integral balance enhancement, with the aid of fully converged results from the available exact solutions.
Nomenclature
| d(x) | = | Dissipation operator coefficient, Eq. (1.a) |
| f(x) | = | Initial condition, Eq. (1.a) |
| k(x) | = | Diffusion operator coefficient, Eq. (1.a) |
| M | = | Truncation order of the algebraic eigenvalue problem |
| Ni | = | Normalization integral of the eigenvalue problem, Eq. (7) |
| P(x, t, T) | = | Nonlinear source term appearing in Eq. (1.a) |
| T(x, t) | = | Potential |
| = | Filtering solution, Eq. (2). | |
| t | = | Time variable |
| u | = | Dependent variable in FGM application, Eq. (25) |
| U | = | Dimensionless fluid velocity in conjugated problem |
| x | = | Space variable (one-dimensional problem) |
| x | = | Position vector |
| w(x) | = | Transient operator coefficient, Eq. (1.a) |
| α(x), β(x) | = | Coefficients for the boundary conditions, Eq. (1.c) |
| α0 | = | Reference thermal diffusivity in FGM application, Eq. (23.c) |
| β | = | Coefficient in FGM application, Eq. (23.a,b) |
| λn | = | Eigenvalues of problem (11) |
| = | Eigenvalues of problem (5) | |
| ψi | = | Eigenfunctions of eigenvalue problem (5) |
| Ω | = | Eigenfunction of the auxiliary problem, Eq. (11) |
| ϕ(x, t, T) | = | Nonlinear source term appearing in Eq. (1.c) |
| Subscripts & Superscripts: | = | |
| i, n | = | Order of eigen quantities |
| – | = | Integral transform |
| ∼ | = | Normalized eigenfunction |
| * | = | Filtered temperature field |
| s | = | Quantity corresponding to the solid region (channel walls) |
| f | = | Quantity corresponding to the fluid flow region |
Nomenclature
| d(x) | = | Dissipation operator coefficient, Eq. (1.a) |
| f(x) | = | Initial condition, Eq. (1.a) |
| k(x) | = | Diffusion operator coefficient, Eq. (1.a) |
| M | = | Truncation order of the algebraic eigenvalue problem |
| Ni | = | Normalization integral of the eigenvalue problem, Eq. (7) |
| P(x, t, T) | = | Nonlinear source term appearing in Eq. (1.a) |
| T(x, t) | = | Potential |
| = | Filtering solution, Eq. (2). | |
| t | = | Time variable |
| u | = | Dependent variable in FGM application, Eq. (25) |
| U | = | Dimensionless fluid velocity in conjugated problem |
| x | = | Space variable (one-dimensional problem) |
| x | = | Position vector |
| w(x) | = | Transient operator coefficient, Eq. (1.a) |
| α(x), β(x) | = | Coefficients for the boundary conditions, Eq. (1.c) |
| α0 | = | Reference thermal diffusivity in FGM application, Eq. (23.c) |
| β | = | Coefficient in FGM application, Eq. (23.a,b) |
| λn | = | Eigenvalues of problem (11) |
| = | Eigenvalues of problem (5) | |
| ψi | = | Eigenfunctions of eigenvalue problem (5) |
| Ω | = | Eigenfunction of the auxiliary problem, Eq. (11) |
| ϕ(x, t, T) | = | Nonlinear source term appearing in Eq. (1.c) |
| Subscripts & Superscripts: | = | |
| i, n | = | Order of eigen quantities |
| – | = | Integral transform |
| ∼ | = | Normalized eigenfunction |
| * | = | Filtered temperature field |
| s | = | Quantity corresponding to the solid region (channel walls) |
| f | = | Quantity corresponding to the fluid flow region |