Additional Separated-Variable Solutions of the Biharmonic Equation in Polar Coordinates

Stampouloglou, I. H.; Theotokoglou, Efstathios E.

doi:10.1115/1.3197157

Cited by 8 publications

(16 citation statements)

References 17 publications

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“…To treat Stokes flow singularities at corners [21] and contact lines [22] as well as biharmonic problems in the linear elasticity of wedges [18][19][20]23,24], the similarity solution has been generalized by introducing a logarithmic term in r . As the derivation of general formulas was detailed in [8], we shall here present only the final form of the flow field:…”

Section: Generalized Similarity Solutionmentioning

confidence: 99%

“…Here we shall apply and extend those asymptotic solutions and computational techniques to elucidate the progression of angles Θ = 3π/4, π/2, π/4, emphasizing important ways in which the new cases differ from the old. As before, the classic similarity form for Stokes flow solutions in angular wedges [9][10][11][12][13][14][15][16][17] will need to be generalized with a logarithmic dependence on the radial coordinate r [13,[18][19][20][21][22][23][24]. The extensive literature review in Nitsche and Parthasarathi [8] fleshed out the context of these papers and also provided a survey of theory and computation relevant to flow in channels with porous walls.…”

Section: Introductionmentioning

confidence: 99%

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Stokes flow singularity at a corner joining solid and porous walls at arbitrary angle

Nitsche

2017

J Eng Math

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Motivated by the internal flow geometry of spiral-wound membrane modules with ladder-type spacers, we consider the Stokes flow singularity at a corner that joins porous and solid walls at arbitrary wedge angle Θ. Seepage flux through the porous wall is coupled to the pressure field by Darcy's law; slip is described by a variant of the Beavers-Joseph boundary condition. On a macroscopic, outer length scale, the singularity appears like a jump discontinuity in the normal velocity, characterized by a non-integrable 1/r divergence of the pressure. For arbitrary Θ, we develop an algebraic criterion to determine the admissible radial exponent(s) in a leading, inner similarity solution-which represents a weaker, integrable singularity in the pressure. A complete map of exponent versus Θ is provided for 0 < Θ < 2π : this has an intricate structure with infinitely many solution branches clustering around Θ = π and Θ = 2π . By generalizing the similarity form with a (ln r ) term and iterating on the slip and seepage conditions, we can carry the outer and inner power series to arbitrarily high order. Nevertheless, a numerical splice is required in between. For this purpose, we apply an iterative, numerical-asymptotic patching scheme described by Nitsche and Parthasarathi (J Fluid Mech 713:183-215, 2012). Detailed velocity and pressure profiles are calculated for three wedge angles (Θ = 3π/4, π/2, π/4) and two dimensionless slip lengths (σ = 20, 40). The general trends for decreasing wedge angle are (i) weakening of the pressure singularity, (ii) increasing magnitude of the radial component of velocity, and (iii) movement of the inner-outer transition farther from the corner. Wedges with Θ < π are seen to differ fundamentally from the flat wedge (Θ = π ) previously considered by Nitsche and Parthasarathi (2012).Keywords Low-Reynolds-number flow · Porous walls · Generalized similarity solutions · Boundary integral method Electronic Supplementary material The online version of this article

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Section: Generalized Similarity Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stokes flow singularity at a corner joining solid and porous walls at arbitrary angle

Nitsche

2017

J Eng Math

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“…Expression 24is a more general solution with respect to (20). Taking the value ΑΦ 0 =0 in (24), dependences 20are obtained.…”

Section: The Methods Of Argument Functions Of a Complex Variablementioning

confidence: 99%

“…Consequently, the change of signs in the Cauchy-Riemann relations in expressions (20), (24) leads to a change of signs not only in front of the basic functions CσexpθcosΑΦ but also in the signs of exponents. Taking into account the latter, we can write the following for 20: Let us consider a solution with two exponents having argument functions with opposite signs.…”

Section: The Methods Of Argument Functions Of a Complex Variablementioning

confidence: 99%

“…It was shown in [20] that there are transient conditions for introduction into consideration of additionally separated variables (analogy of the argument functions) when reformatting one type of differential equations into another. The very idea of transition is productive, however, the appearance of additional solutions, in this case, does not mean the determination of conditions for the existence of solutions.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

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Invariant differential generalizations in problems of the elasticity theory as applied to polar coordinates

Chigirinsky¹,

Naumenko²

2020

EEJET

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The method of argument functions has become famous for solving problems of continuum mechanics. The solution of problems of the elasticity theory in polar coordinates was the further development of this method. The same approaches are applied to solving problems of the theory of plasticity, the theory of elasticity, and the theory of dynamic processes. If regularities of the solution are determined correctly, then they should be continued in other fields including the problems of the theory of elasticity in polar coordinates. The proposed approach features finding not the solution itself but the conditions for its existence. These conditions may include differential or integral relations which make it possible to close the solution in a general form. This becomes possible when additional functions are introduced into consideration or the argument functions of coordinates of the deformation zone. Basic dependences that satisfy the boundary or edge conditions as well as the functions that simplify the solution of the problem in general should be the carriers of the proposed argument functions. For various reasons, two basic dependences were used in the solution: trigonometric and exponential. Their arguments are two unknown argument functions. In the process of transformations, a mathematical connection was established between them in a form of the Cauchy-Riemann relations which had a stable tendency to be repeated in problems of the continuum mechanics. From these positions, the flat problem was solved in the most detailed way, tested, and compared with the studies of other authors. By reducing the solution to a particular result, a way to classical solutions was found which confirms its reliability. The result obtained is useful and important since it becomes possible to solve an extensive class of axisymmetric applied problems using the method of argument functions of a complex variable

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Boundary Conditions for Planar Stokes Equations Inducing Vortices Around Concave Corners

Gazzola

Sperone

2019

Milan J. Math.

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Additional Separated-Variable Solutions of the Biharmonic Equation in Polar Coordinates

Cited by 8 publications

References 17 publications

Stokes flow singularity at a corner joining solid and porous walls at arbitrary angle

Stokes flow singularity at a corner joining solid and porous walls at arbitrary angle

Invariant differential generalizations in problems of the elasticity theory as applied to polar coordinates

Boundary Conditions for Planar Stokes Equations Inducing Vortices Around Concave Corners

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