Series solutions of coupled differential equations with one regular singular point

Tomantschger, Kurt

doi:10.1016/s0377-0427(01)00598-2

Cited by 23 publications

(11 citation statements)

References 3 publications

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“…In the paper [16], the author postulated the solution of coupled differential equations (1) and (2) in the form of an infinite mathematical order, introducing the MacDonald function and modified Bessel functions of the second kind of zero and the first order. The final results (Figures 3 and 4) showed good agreement with the solution given by expressions (43)-(50).…”

Section: Resultsmentioning

confidence: 99%

“…The mathematical model of the flow of suspension, describing the velocity field (v) of the movement of the suspension and the velocity field of the microrotation of the suspension (w), depending on the radial coordinate (r) and defined by a coupled system of two ordinary linear differential equations of the second order with variable coefficients, which have the following form [16]:…”

Section: Figure 1 a Simplified Functional Diagram Of The Movement Ofmentioning

confidence: 99%

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Application of the theory of micropolar continuum on the flow suspension in a cylindrical channel

Dedić¹,

Salemović²,

Jovanović³

2017

FME Transactions

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This paper presents an analytical solution of a mathematical model which treats fluid flow suspension in a cylindrical channel. The model is the application of the theory of micropolar continuum on the flow of suspension and it consists of coupled linear differential equations with variable coefficients. The cylindrical channel consists of two cylinders: the internal cylinder was still and the external one rotated with constant velocity. This model enabled us to analyze the motion of a suspension, as heterogeneous mixture of a liquid with solid particles. The solution was found in the form of special Bessel's functions of the zero and the first order. The results were shown on diagrams for some characteristic values, and the good agreement was achieved between the calculated and expected results.

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Section: Resultsmentioning

confidence: 99%

Section: Figure 1 a Simplified Functional Diagram Of The Movement Ofmentioning

confidence: 99%

Application of the theory of micropolar continuum on the flow suspension in a cylindrical channel

Dedić¹,

Salemović²,

Jovanović³

2017

FME Transactions

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“…The mathematical model of the flow of the suspension describes the velocity field of the flow of suspension, describing the velocity field of the movement of the suspension (𝑣) and the velocity field of the microrotation of the suspension (𝑤), depending on the radial coordinate (𝑟) and defined by a coupled system of two ordinary linear differential equations of the second order with variable coefficients, which have the following form [4,7,19]:…”

Section: Figure 1 a Simplified Functional Diagram Of The Movement Of The Suspension Between Two Coaxial Cylindersmentioning

confidence: 99%

Micropolar fluid between two coaxial cylinders (numerical approach)

Salemović

Dedić

Jovanović

2021

Theor appl mech (Belgr)

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The paper describes the flow of a suspension which is a mixture of two phases: liquid and solid granules. The continuum model with microstructure is introduced, which involves two independent kinematic quantities: the velocity vector and the micro-rotation vector. The physical analogy is based on the movement of the suspension between two coaxial cylinders. The inner cylinder is stationary and the outer one rotates with constant angular velocity. This physical analogy enabled a mathematical model in a form of two coupled differential equations with variable coefficients. The aim of the paper is to present the numerical aspect of the solution for this complex mathematical model. It is assumed that the solid granules are identically oriented and that under the influence of the fluid they move translationally or rotate around the symmetry axis but the direction of their symmetry axes does not change. The solution was obtained by the ordinary finite difference method, and then the corresponding sets of points (nodes) were routed by interpolation graphics.

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“…By using the method of separation of variables the model of instant vibration problem is reduced to four second-order coupled ordinary differential equations in radial coordinates. One of the standard techniques to solve ordinary differential equations with variable coefficients is the Frobenius method available in literature (Tomantschger, 2002). The secular equation which governs the three-dimensional vibration of solid cylinder has been derived by using Matrix Frobenius method.…”

Section: Introductionmentioning

confidence: 99%

Modeling of thermoelastic damping and frequency shift of vibrations in a transversely isotropic solid cylinder

Sharma

Singh²,

Sharma³

2011

Multidiscipline Modeling in Materials and Structures

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Purpose -The purpose of this paper is to analyze the free vibrations in a stress free and thermally insulated (or isothermal), homogeneous, transversely isotropic, solid cylinder based on three-dimensional coupled thermoelasticity, which is initially undeformed and kept at uniform temperature. Design/methodology/approach -The displacement potential functions have been introduced in the equations of motion and heat conduction in order to decouple the purely shear and longitudinal motions. The system of governing partial differential equations is reduced to four second-order coupled ordinary differential equations in radial coordinate by using the method of separation of variables. The matrix Frobenius method of extended power series is employed to obtain the solution of coupled ordinary differential equations along the radial coordinate. The convergence analysis of matrix Frobenius method has been successfully carried out. Findings -The purely transverse mode is found to be independent of the rest of the motion and temperature change. The natural frequency, dissipation factor, inverse quality factor and frequency shift of vibrations in a stress free solid cylinder get significantly affected due to thermal variations and thermo-mechanical coupling. Originality/value -A new procedure is used and compared to other methods available in the literature.

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Series solutions of coupled differential equations with one regular singular point

Cited by 23 publications

References 3 publications

Application of the theory of micropolar continuum on the flow suspension in a cylindrical channel

Application of the theory of micropolar continuum on the flow suspension in a cylindrical channel

Micropolar fluid between two coaxial cylinders (numerical approach)

Modeling of thermoelastic damping and frequency shift of vibrations in a transversely isotropic solid cylinder

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