Analytical and numerical solution of the longitudinal porous fin with multiple power‐law‐dependent thermal properties and magnetic effects

Changes in microstructure occur in as-received aluminum alloy (Al-2024-T351) when it is subjected to elevated temperatures (150–260°C). These changes, which are called precipitation hardening, in turn influence the thermal properties, making them time as well as temperature dependent. A computer-assisted transient experimental procedure has been developed to determine the values of thermal conductivity of as-received Al-2024-T351 under the influence of precipitation-hardening. Based on isothermal experimental data and related algebraic modeling of the thermal conductivity, a mathematical model in the form of two differential equations is proposed. Instantaneous values of volume fraction of precipitate and thermal conductivity can be predicted using this model. A method for the simultaneous numerical solution of the partial differential equation of conduction and the proposed differential equations of precipitation are also given. The influence of precipitation—hardening on temperature distribution and on values of thermal conductivity is shown graphically for several cases involving the Al-2024-T351 material.

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Ultrasonic attenuation based on the Roney generalized theory and multiple power-law grain-size distributions

IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control ◽

10.1109/58.265833 ◽

1994 ◽

Vol 41 (1) ◽

pp. 144-149 ◽

Cited By ~ 11

Author(s):

D. Nicoletti ◽

D. Kasper

Keyword(s):

Grain Size ◽

Power Law ◽

Ultrasonic Attenuation ◽

Size Distributions ◽

Grain Size Distributions ◽

Multiple Power

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The Equation of the Limit Replacement Width of Non-Newton Fluid in Eccentric Annulus

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.807-809.2616 ◽

2013 ◽

Vol 807-809 ◽

pp. 2616-2619

Author(s):

Yin Qing Liu ◽

Mei Wei Wang ◽

Hai Qing Cui

Keyword(s):

Numerical Solution ◽

Rheological Properties ◽

Power Law ◽

Solution Method ◽

Cement Slurry ◽

Power Law Fluid ◽

Two Phase ◽

Numerical Solution Method ◽

Eccentric Annulus ◽

The One

The equation of the limit replacement width of the one-dimension two-phase flow of Bingham fluid replacing Power law fluid in eccentric annulus was established, the numerical solution method of the equation mentioned above was given and taking the 3 wells, such as the He 104-27 well etc for examples, the limit replacement widths of cement slurry displacing mud, whose rheological properties can be described as Bingham and Power law modles respectively, were calculated, by using the equation and the numerical solution method mentioned above, and compared with those of cement slurry displacing mud, whose rheological properties are all described as Binghanm modle.

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A semi-infinite hydraulic fracture with leak-off driven by a power-law fluid

Journal of Fluid Mechanics ◽

10.1017/jfm.2017.856 ◽

2017 ◽

Vol 837 ◽

pp. 210-229 ◽

Cited By ~ 12

Author(s):

E. V. Dontsov ◽

O. Kresse

Keyword(s):

Numerical Solution ◽

Power Law ◽

Hydraulic Fracture ◽

Closed Form Solution ◽

Analytic Solutions ◽

Form Solution ◽

Rapid Evaluation ◽

Power Law Fluid ◽

Parametric Space ◽

Fluid Rheology

This study investigates the problem of a semi-infinite hydraulic fracture that propagates steadily in a permeable formation. The fracturing fluid rheology is assumed to follow a power-law behaviour, while the leak-off is modelled by Carter’s model. A non-singular formulation is employed to effectively analyse the problem and to construct a numerical solution. The problem under consideration features three limiting analytic solutions that are associated with dominance of either toughness, leak-off or viscosity. Transitions between all the limiting cases are analysed and the boundaries of applicability of all these limiting solutions are quantified. These bounds allow us to determine the regions in the parametric space, in which these limiting solutions can be used. The problem of a semi-infinite fracture, which is considered in this study, provides the solution for the tip region of a hydraulic fracture and can be used in hydraulic fracturing simulators to facilitate solving the moving fracture boundary problem. To cater for such applications, for which rapid evaluation of the solution is necessary, the last part of this paper constructs an approximate closed form solution for the problem and evaluates its accuracy against the numerical solution inside the parametric space.

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