scholarly journals Stokes Equation in a Semi-Infinite Region: Generalization of the Lamb Solution and Applications to Marangoni Flows

Fluids ◽  
2020 ◽  
Vol 5 (4) ◽  
pp. 249
Author(s):  
Goce Koleski ◽  
Thomas Bickel
Keyword(s):  
General Solution ◽  
Stokes Equation ◽  
Spherical Harmonics ◽  
Spherical Geometry ◽  
Creeping Flow ◽  
Local Source ◽  
Air Interface ◽  
Infinite Region ◽  
The Stokes Equation ◽  
Marangoni Flows

We consider the creeping flow of a Newtonian fluid in a hemispherical region. In a domain with spherical or nearly spherical geometry, the solution of the Stokes equation can be expressed as a series of spherical harmonics. However, the original Lamb solution is not complete when the flow is restricted to a semi-infinite space. The general solution in hemispherical geometry is then constructed explicitly. As an application, we discuss the solutions of Marangoni flows due to a local source at the liquid–air interface.

scholarly journals On the K-th Power of Stokes Operator

2020 ◽  
Vol 63 (1) ◽  
pp. 1-10
Author(s):  
Eleftherios Protopapas ◽  
Keyword(s):  
Stokes Equation ◽  
Differential Operators ◽  
Applied Mathematics ◽  
Sufficient Conditions ◽  
Spherical Geometry ◽  
Stokes Operator ◽  
Creeping Flow ◽  
Elliptic Type ◽  
Spherical Coordinate ◽  
Partial Differential Operators

Stokes operators, are well known partial differential operators of elliptic type, which are often used in Applied Mathematics. Stokes equation describes the irrotational, axisymmetric creeping flow and Stokes bi-stream equation denotes the rotational one, where Necessary and sufficient conditions for the separability and the R-separability of the equation have been proved recently. Moreover, the 0-eigenspace and the generalized 0-eigenspace of the operator have been derived in several coordinate systems. Specifically, the spherical coordinate system is employed in many problems taking into account that in many engineering applications, the solutions in spherical geometry seem to be adequate for solving a problem. In the present manuscript, it is shown that equation admits a solution of the form where are solutions of Stokes equation and r is the radial spherical variable. Additionally, we obtain the kernel of the k-th power of the Stokes operator, in the spherical geometry for every

Exact analytical solutions of the stokes equation for testing the equations of mantle convection with a variable viscosity

2006 ◽  
Vol 42 (7) ◽  
pp. 537-545 ◽  
Author(s):  
V. P. Trubitsyn ◽  
A. A. Baranov ◽  
A. N. Eyseev ◽  
A. P. Trubitsyn
Keyword(s):  
Stokes Equation ◽  
Analytical Solutions ◽  
Mantle Convection ◽  
Variable Viscosity ◽  
Exact Analytical Solutions ◽  
The Stokes Equation

Incorporating attenuation effects into frequency-domain full waveform inversion from zero-offset VSP data from the Stokes equation

2013 ◽  
Vol 10 (3) ◽  
pp. 035004 ◽  
Author(s):  
Cai Liu ◽  
Fengxia Gao ◽  
Xuan Feng ◽  
Yang Liu ◽  
Yuwei Liu
Keyword(s):  
Frequency Domain ◽  
Stokes Equation ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Full Waveform ◽  
Vsp Data ◽  
Zero Offset ◽  
The Stokes Equation

The Permeability of a Uniformly Vuggy Porous Medium

1973 ◽  
Vol 13 (02) ◽  
pp. 69-74 ◽  
Author(s):  
Graham H. Neale ◽  
Walter K. Nader
Keyword(s):  
Porous Medium ◽  
Flow Field ◽  
Stokes Equation ◽  
Spherical Cavity ◽  
Creeping Flow ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Darcy Equation ◽  
Matrix Permeability ◽  
Flow Vector

Abstract Using the creeping Navier Stokes equation within a spherical cavity and the Darcy equation in the surrounding homogeneous and isotropic porous medium, the flow field in the entire system is evaluated. Applying this result to a representative generalizing model of a uniformly vuggy, homogeneous and isotropic porous medium, an engineering estimation of the interdependence of the matrix permeability km, the vug porosity permeability km, the vug porositytotal volume of vug space 0v = ----------------------------total volume of sample and the system permeability ks of the vuggy porous medium is derived. This interdependence can be expressed by the formula: Introduction The objective of this study is the derivation of an engineering formula that shows the interdependence of matrix permeability, km, vug porosity, 0 v, and system permeability, ks, of a uniformly vuggy porous medium. In the first section, with the above porous medium. In the first section, with the above goal in mind and to satisfy more general interests, we shall study and predict the flow field within a single cavity bounded by a sphere, of radius R, and in the surrounding homogeneous and isotropic porous medium. In the second section, we shall porous medium. In the second section, we shall suggest as a generalizing model of a uniformly vuggy, homogeneous and isotropic porous medium a regular cubic array of monosized spherical cavities. Applying the formula for the pressure field near a single spherical cavity, we shall then develop the sought engineering formula. To describe the creeping flow of the incompressible liquid of viscosity, in the spherical cavity, we shall employ the creeping Navier Stokes equation, .............................(1) The Darcy equation, ,...........................(2) will be used to describe the flow of this liquid in the porous medium of permeability k that fills the space outside the cavity. p designates the liquid pressure referred to datum, denotes the flow pressure referred to datum, denotes the flow vector, and * is used to indicate macroscopically averaged quantities pertaining specifically to a porous medium. porous medium. In hydrodynamics, one generally requests continuity of the pressure, of the flow vector, and of the shear tensor throughout the fundamental domain of the problem - in particular, along the boundary surfaces, which separate subdomains. When applying these principles to this problem, one would impose at the spherical boundary that separates the cavity from the porous medium:continuity of the pressure,continuity of the component of u that is orthogonal to the surface,continuity of the other component of u that is tangential to the surface,continuity of the shear component tangential to the surface. Arguments of this nature have lead to the suggestion of a generalization of the Darcy equation, namely, the Brinkman equation, ...............(3) However, both the necessity and the validity of this generalization have been challenged; indeed, it has been shown that a mathematically consistent solution of our problem may be obtained, using Eqs. 1 and 2 within the respective subdomains, provided one abandons the request for continuity of the shear at the wall of the cavity (compare Boundary Condition d above).** SPEJ P. 69

A BEM–FEM overlapping algorithm for the Stokes equation

2006 ◽  
Vol 182 (1) ◽  
pp. 691-710 ◽  
Author(s):  
Víctor Domínguez ◽  
Francisco-Javier Sayas
Keyword(s):  
Stokes Equation ◽  
The Stokes Equation
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