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The first correction to the darcy's law in view of the homogenization theory and experimental resear |
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The first correction to the darcy's law in view of the homogenization theory and experimental research Authors: K. Cieślicki, A. Lasowska
In the paper the non-linear correction term to the Darcy's law laminar flows was determined theoretically and experimentally. The correction term to the Darcy's law is generally assumed to be quadratic in the seepage velocity - the additional term present in the Forschheimer's formula applied since 1901. Basing on the homogenization theory and the research programme, where the experiments were conducted in axisymmetrical tubes with periodic step changes in diameter, it was demonstrated that the first correction term to the Darcy's law will be cubic in the seepage velocity, instead of quadratic. The absence of the quadratic term provides that Darcy's law can be applied asymptotically throughout the range of small seepage velocities. It was found out experimentally that both the applicability range and the magnitude of the coefficient present in the cubic correction term is the function of the geometrical structure of the models. When the seepage velocity [Re number] should exceed a certain value, the correction term becomes quadratic thus marking off the second laminar flow zone. To interpret those correlations in terms of physics the experiments were run which consisted of streamline visualization with marker particles in the selected model segments. The liquid was made optically homogeneous to the model material. The laser sheet was applied to obtain the required visualization planes. It was found that in the zone where the cubic correction to Darcy's law is applicable the geometrical flow patterns change significantly. The main core is thus reduced and the recirculatory flow increases with an increase of the seepage velocity. In the zone to which the quadratic correction term applies no significant changes of the main core were noticed.
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2009