A Self-Similar Solution of Navier–Stokes and Energy Equations for Rotating Flows between a Cone and a Disk

Shevchuk, Igor V.

doi:10.1023/b:hite.0000020097.59838.02

Cited by 19 publications

(39 citation statements)

References 4 publications

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“…In the previous works (Shevchuk, 2004a(Shevchuk, , 2004b(Shevchuk, , 2011(Shevchuk, , 2009(Shevchuk, , 2015, detailed calculations were performed based on equations ( 21)-( 24) and ( 29)-(31) for the cases of a stationary disk and a rotating cone, a rotating disk and a stationary cone, contra-rotating disk and cone, as well as co-rotating disk and cone. In this paper, we will perform an extended simulation of the case of a stationary disk and a rotating cone, for which reliable experimental data from various authors are presented in the literature (Sdougos et al, 1984;Malek et al,1995), and approximate analytical solutions (Sdougos et al, 1984;Buschmann, 2002;Buschmann et al, 2005) mentioned above are also available.…”

Section: Radial Thermal Conductivitymentioning

confidence: 99%

“…In this paper, we will perform an extended simulation of the case of a stationary disk and a rotating cone, for which reliable experimental data from various authors are presented in the literature (Sdougos et al, 1984;Malek et al,1995), and approximate analytical solutions (Sdougos et al, 1984;Buschmann, 2002;Buschmann et al, 2005) mentioned above are also available. The additional extended validation and interpretation of the self-similar solution given below based on systems (21)-( 24) and ( 29)-(31) uses experimental data (Sdougos et al, 1984;Malek et al,1995) and analytical solutions (Buschmann, 2002;Buschmann et al,2005), which were not used in the previous works (Shevchuk, 2004a(Shevchuk, , 2004b(Shevchuk, , 2011(Shevchuk, , 2009(Shevchuk, , 2015. As said in the statement of the problem of this study (Section 1), this will enable us to accurately determine and justify the limits of applicability of the self-similar solution to the problem of fluid flow in conical gaps with small conicity angles.…”

Section: Radial Thermal Conductivitymentioning

confidence: 99%

“…In this case, calculations for heat transfer were made for the Prandtl number equal to 0.71 (air). In Shevchuk (2004aShevchuk ( , 2004b, calculations were also performed for a conicity angle of 45°just to test the functioning of the proposed technique for large conicity angles. In Shevchuk (2011), the model under consideration was used to study the influence of various Prandtl or Schmidt numbers on the Nusselt or Sherwood HFF 33,1 numbers, respectively, in the Pr (or Sc) range from 0.71 to 800.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, a review of previous studies by various authors showed that the question of the possibility of using the self-similar solution proposed in Shevchuk (2004aShevchuk ( , 2004b for geometries with a large conicity angle remains open. From a formal mathematical point of view, calculations using the resulting system of equations can be performed for any conicity angles of the gaps.…”

Section: Introductionmentioning

confidence: 99%

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Concerning the effect of radial thermal conductivity in a self-similar solution for rotating cone-disk systems

Shevchuk

2022

HFF

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Purpose Thus, the purposes of this study are to study the limits of applicability of the self-similar solution to the problem of fluid flow, heat and mass transfer in conical gaps with small conicity angles, to substantiate the impossibility of using a self-similar formulation of the problem in the case of large conicity angles and to substantiate the absence of the need to take into account the radial thermal conductivity in the energy equation in its self-similar formulation for the conicity angles up to 4°. Design/methodology/approach In the present work, an in-depth and extended analysis of the features of fluid flow and heat transfer in a conical gap at small angles of conicity up to 4° is performed. The Couette-type flow arising, in this case, was modeled using a self-similar formulation of the problem. A detailed analysis of fluid flow calculations using a self-similar system of equations showed that they provide the best agreement with experiments than other known approaches. It is confirmed that the self-similar system of flow and heat transfer equations is applicable only to small angles of conicity up to 4°, whereas, at large angles of conicity, this approach becomes unreasonable and leads to significantly inaccurate results. The heat transfer process in a conical gap with small angles of conicity can be modeled using the self-similar energy equation in the boundary layer approximation. It was shown that taking into account the radial thermal conductivity in the self-similar energy equation at small conicity angles up to 4° leads to maximum deviations of the Nusselt number up to 1.5% compared with the energy equation in the boundary layer approximation without taking into account the radial thermal conductivity. Findings It is confirmed that the self-similar system of fluid flow equations is applicable only for small conicity angles up to 4°. The inclusion of radial thermal conductivity in the model unnecessarily complicates the mathematical formulation of the problem and at small conicity angles up to 4° leads to insignificant deviations of the Nusselt number (maximum 1.5%). Heat transfer in a conical gap with small conicity angles up to 4° can be modeled using the self-similar energy equation in the boundary layer approximation. Originality/value This paper investigates the question of the validity of taking into account the radial heat conduction in the energy equation.

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Section: Radial Thermal Conductivitymentioning

confidence: 99%

Section: Radial Thermal Conductivitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Concerning the effect of radial thermal conductivity in a self-similar solution for rotating cone-disk systems

Shevchuk

2022

HFF

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“…Poncet et al [28] proposed a modified Reynolds stress model (RSM) to investigate the mean structure of annular radial outflow in rotor-stator disks, where the one-point statistical model was used for establishing a Reynolds stress tensor to close momenta equations. The hydro-viscous film flows can also be solved with common numerical methods, such as the self-similar method [29] and the SIMPLE family of algorithms [30]; these methods have been widely employed for fluid computation. Generally, these methods require additional complicated mathematical techniques and computation time and are not suitable for quick computations of hydro-viscous film.…”

Section: Introductionmentioning

confidence: 99%

A Numerical Method for Extrication Characteristics of TBM Cutter-Head with the HVC

Gong

Xie

Yang

2019

Chin. J. Mech. Eng.

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Achieving highly efficient extrication of the tunnel boring machine (TBM) cutter-head driving system from the collapsed surrounding rock has become a key problem globally, and significant effort has been directed to improve TBM cutter-head extricating ability. In this study, the characteristics of a hydro-viscous device have been investigated to improve extricating performance of the TBM cutter-head. A numerical method based on an explicit pressure-linked equation is presented for computing the film parameters of the HVC, which is then applied to investigate extrication characteristics of a TBM cutter-head with a hydro-viscous clutch (HVC). The explicit pressure-linked equation is derived from the Navier-Stokes equations and the conservation equation, where boundary conditions are involved. The model of a cutter-head driving system with an HVC is established, and the extrication characteristics of the cutterhead driving system are analyzed and compared with three extrication strategies. The variation in extrication torque shows that the linear strategy or positive parabolic strategy are preferred for their relatively high extrication efficiency and low rigid impact, and the effects of throughflow rate on torque transmission are also investigated. The test rig of the TBM cutter-head driving system was set up to validate the numerical method and the model of a cutter-head driving system, and the feasibility of the proposed numerical method for researching the extrication of the TBM cutter-head is verified.

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Lie group analysis of flow and heat transfer of a nanofluid in cone–disk systems with Hall current and radiative heat flux

Mahanthesh

Bhatta

2023

Math Methods in App Sciences

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A study of the rheological and heat transport characteristics in cone–disk systems finds relevance in many applications such as viscometry, conical diffusers, and medical devices. Therefore, a three‐dimensional axisymmetric flow with heat transport of a magnetized nanofluid in a cone–disk system subjected to Hall current and thermal radiation effects is investigated. The simplified Navier–Stokes (NS) equations for the cone–disk system given by Sdougos et al. [18] Journal of Fluid Mechanics, 138, 379–404 are solved by using the asymptotic expansion method for the four different models, such as rotating cone with static disk (Model I), rotating disk with static cone (Model II), co‐rotating cone and disk (Model III), and counter‐rotating cone and disk (Model IV). The Khanafer–Vafai–Lightstone (KVL) model along with experimental data‐based properties of 37 nm Al2O3–H2O nanofluid is considered. To obtain the transformations leading to self‐similar equations from the Navier–Stokes (NS) and energy conservation equations, the Lie group technique is used. The self‐similar nonlinear problem is solved numerically to examine the effects of physical parameters. There are critical values of the power exponent at which no heat transport from the disk surface occurs. Nanoparticles significantly enhance heat transport when both the cone and disk rotate in the same or opposite directions. The centrifugal force and thermal radiation improve the heat transport in cone–disk systems.

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A Self-Similar Solution of Navier–Stokes and Energy Equations for Rotating Flows between a Cone and a Disk

Cited by 19 publications

References 4 publications

Concerning the effect of radial thermal conductivity in a self-similar solution for rotating cone-disk systems

Concerning the effect of radial thermal conductivity in a self-similar solution for rotating cone-disk systems

A Numerical Method for Extrication Characteristics of TBM Cutter-Head with the HVC

Lie group analysis of flow and heat transfer of a nanofluid in cone–disk systems with Hall current and radiative heat flux

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