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Abstract

This thesis is concerned with steady two-dimensional flow in a rectangular cavity filled with a saturated porous medium governed by Darcy’s law. The flow is driven by differential heating of the upper surface of the cavity, whereas the sides and the bottom of the cavity are thermally insulated. Numerical calculations of the flow and temperature fields are carried out for cases where the temperature profile at the upper surface is a monotonic function of position (either a cosine function or a quadratic function) and results are obtained for a wide range of Darcy-Rayleigh numbers R and aspect ratios L which describe the resulting single-cell circulation. Analytical results are obtained in the limit of small Darcy-Rayleigh number using a perturbation method and are compared with the numerical results. At large Darcy-Rayleigh numbers the flow adopts a boundary-layer structure in which the main variations in temperature and velocity occur near the upper surface. In the case of the quadratic temperature profile at the upper surface, an exact solution of the horizontal boundary layer equations is found which provides useful insight, including a prediction of the almost constant temperature of the fluid in the core region below the horizontal boundary layer. This solution can only be regarded as approximate, however, because it fails to take account of the need for the solution to match with that in a vertical boundary layer at one end of the layer. By considering the properties of this vertical boundary layer it is argued that in the limit of large Darcy-Rayleigh number the entire leading order flow circulation is contained within the combined horizontal and vertical boundary layers near the upper surface. The solution of the combined horizontal and vertical boundary layer system is considered and an asymptotic solution is found at the lower edge of the layers which matches consistently across the layers and with a solution in the core region below. This is used to obtain an improved overall solution of the system in good agreement with the numerical calculations at large Darcy-Rayleigh numbers.

Publication Type:	Thesis (Doctoral)
Subjects:	Q Science > QA Mathematics T Technology > T Technology (General)
Departments:	School of Science & Technology > Mathematics School of Science & Technology > School of Science & Technology Doctoral Theses Doctoral Theses

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