Derivation of governing equation describing time-dependent penetration length in channel flows driven by non-mechanical forces

2010 ◽  
Vol 666 (1-2) ◽  
pp. 51-54 ◽  
Author(s):  
Sukalyan Bhattacharya ◽  
Dilkumar Gurung
Keyword(s):  
Governing Equation ◽  
Channel Flows ◽  
Time Dependent ◽  
Mechanical Forces ◽  
Penetration Length

Modified Time-Dependent Penetration Length and Inlet Pressure Field in Rectangular and Cylindrical Channel Flows Driven by Non-Mechanical Forces

2011 ◽  
Vol 133 (11) ◽  
Author(s):  
Martin Ndi Azese
Keyword(s):  
Velocity Profile ◽  
Governing Equation ◽  
Control Volume ◽  
Momentum Equation ◽  
Rate Of Change ◽  
Creeping Flow ◽  
Time Dependent ◽  
Penetration Length ◽  
Wide Range ◽  
Accurate Expression

In this paper, we derive the governing equation for the time dependent penetration length of a fluid column in rectangular and cylindrical channels under the action of nonmechanical forces like capillary or electro-osmotic force. For this purpose, first we obtain the velocity profile for unidirectional unsteady flow by satisfying momentum equation in differential form. Then, we relate the rate of change of penetration length with volume flux to obtain the governing equation of the penetration length. As the velocity profile is exact, the analysis is devoid of any mathematical error. As a result, the theoretical results are valid irrespective of the Reynolds number of the system as long as the flow inside the cylindrical or rectangular conduit is laminar. We then use the new expressions of velocity fields of respective conduits to derive a more accurate expression of the entrance pressure by using a hemispherical model for the control volume for finite aspect ratio. As these channels are very common, our governing equations for penetration length will have a wide range of applicability. These applications especially include creeping flow in micro fluidic domain for which we have a simplified version of the derived equation.

scholarly journals Nonlinear Radial Consolidation Analysis of Soft Soil with Vertical Drains under Cyclic Loadings

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Pyol Kim ◽  
Hyong-Sik Kim ◽  
Yong-Gun Kim ◽  
Chung-Hyok Paek ◽  
Song-Nam Oh ◽  
...  
Keyword(s):  
Cyclic Loading ◽  
Governing Equation ◽  
Analytical Solutions ◽  
Soft Soil ◽  
Time Dependent ◽  
Vertical Drains ◽  
Consolidation Analysis ◽  
Consolidation Behavior

This paper presents analytical solutions for nonlinear radial consolidation of soft soil with vertical drains under various cyclic loadings. By considering the nonlinear variations of compressibility and permeability expressed by the logarithm relations (e−log σ′ and e−log kh), the governing equation for nonlinear radial consolidation of the soil under equal strain and time-dependent loading is established. The analytical solutions are derived for nonlinear radial consolidation under haversine cyclic loading, trapezoidal cyclic loading, rectangular cyclic loading, and triangular cyclic loading. The presented solution is verified through the degeneration into the existing solutions for nonlinear radial consolidation under constant and ramp loadings, which shows the solution proposed in this paper is more general for nonlinear radial consolidation under time-dependent loading. The nonlinear radial consolidation behavior of the soil with vertical drains subjected to various cyclic loadings is investigated using the solutions developed. The proposed solutions can be effectively utilized in the analysis of nonlinear radial consolidation under various cyclic loadings.

Peridynamics for Heat Conduction

2020 ◽  
Author(s):  
Yozo Mikata
Keyword(s):  
Heat Conduction ◽  
Constitutive Equation ◽  
Heat Equation ◽  
Governing Equation ◽  
Heat Conduction Equation ◽  
Transient Heat Conduction ◽  
Anisotropic Materials ◽  
Time Dependent ◽  
Heat Sources ◽  
Exact Analytical Solutions

Abstract Peridynamics for transient heat conduction problems in general anisotropic materials is developed. In order to develop a new peridynamic governing equation for heat conduction problems, the microconductivity (or microdiffusivity), which contains equivalent information as the constitutive equation for classical heat conduction, is determined by directly requiring the resulting peridynamic equation to converge to a classical heat conduction equation for anisotropic materials as the generalized material horizon approaches 0. Therefore, the convergence proof is built into the theory from the perspective of the governing equation. For the application of the newly obtained peridynamic governing equation, a time-dependent 3D peridynamic heat equation is analytically solved with two types of heat sources, and the results are discussed. These are believed to be the first exact analytical solutions for peridynamic heat conduction.

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