# Unsteady conjugate problem of a dissipative fluid in a horizontal channel with a periodic variation temperature

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s11012-007-9092-4

- Cite this article as:
- Zueco, J. Meccanica (2008) 43: 37. doi:10.1007/s11012-007-9092-4

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## Abstract

The unsteady two-dimensional transient heat transfer problem referring to a fully laminar flow developing in a parallel-plane channel exposed to a periodic variation surface temperature with distance is numerically studied. The effects of channel thickness, Péclet number, wall-to-fluid conductivity ratio, thermal diffusivity ratio, angular frequency and the viscous dissipation parameter are determined in the solutions. The non-linear equations are discretized by means an implicit finite difference scheme and the electric analogy to the resulting system is applied to convert these equations into a network-electrical model that was solved using a computer code (electric circuits simulator). In this scheme, only spatial discretization is necessary, while time remains as a real continuous variable, and its programming does not require manipulation of the sophisticated mathematical software that is inherent in other numerical methods. The network simulation method, which satisfies the conservation law for the heat flux variable and the uniqueness law for temperature, also permits the direct visualization of the local and/or integrated transport variables at any point or section of the medium.

### Keywords

Transient heat conductionConjugate problemViscous dissipationNetwork methodHeat transfer### Abbreviations

*A*Ratio of diffusivities,

*α*_{s}/*α*_{f}*B*Dimensionless angular frequency

*Br*Brinkman number

*c*_{e}Specific heat

*C*Capacitor

*k*Thermal conductivity

*G*Control-voltage current-source

*L*_{0}Half distance between the channel walls

*L*_{1}Thickness of the pipe

*N*Number of cells

*Nu*Nusselt number

*Pe*Péclet number

*q*Heat flux

*R*Resistor

*t*Time

*T*Temperature

*u*Velocity

*U*Dimensionless velocity

*x*Axial co-ordinate

*y*Vertical co-ordinate

### Greek symbols

*α*Diffusivity

*β*Angular frequency

- Δ
*T* Oscillation amplitude temperature

- Δ
*X* Axial thickness of the cell

- Δ
*Y* Vertical thickness of the cell

*Γ*Dimensionless geometric parameter

*θ*Dimensionless temperature

*μ*Dynamic viscosity

*ρ*Density

*τ*Dimensionless time

### Subscripts

*f*Associated to fluid

*i*,*j*Associated with

*i*,*j*nodal point*i*,*i*−Δ,*i*+ΔAssociated to the centre, left and right position on the cell

*m*Medium value

*mea*Associated to measurement

*s*Associated to solid

*w*Solid-fluid interface