Original articles
On the fluid flow and heat transfer between a cone and a disk both stationary or rotating

https://doi.org/10.1016/j.matcom.2020.04.004Get rights and content

Abstract

The present paper investigates the role that the radial component of heat conduction plays in a cone-plate viscometer. The cone and the disk may be taken as stationary or in action; both co-rotating or counter-rotating. Hydrodynamic and thermal fields are resolved by means of computationally simulating the resulting systems of equations. Upon working out the well-documented velocity field in the gap region from the similarity governing equations, the rates of heat transfer at both surfaces are calculated from an amended energy equation by further adding radial diffusive terms, which were missing in the previous data published in the literature. It is shown that addition of such physical streamwise heat conduction terms into the energy equation much influences the well-known results of heat transfer rates, particularly when the conical gap section is not small. The missing heat transfer rates pertaining to the cone wall are also presented here. In particular, it is demonstrated that the best cooling of cone-disk apparatus can be achieved for a rotating disk with a stationary cone, provided that the wall temperatures are kept as uniformly constant. The critical power index of passage from cooling to heating is determined to be 1.54492.

Introduction

Literature shows that the disk-cone apparatus has many technological and practical importance, for instance in the calculation of fluid viscosity via viscosimetry [22] and [18], in stability analysis of the creeping flow of an Oldroyd-B fluid [25], in stationary conical diffuser used in gas turbines to compress air in the cooling system [24] and [30], in medical purposes [4] and [31], in convective diffusion of feeding culture [29], in measuring oxygen concentration and resultant degradation rate [15] and in biomedicine applications [1] and [17]. The present scientific devotion is also concerned with the fluid flow and heat transfer occurring in the conical gap between a cone and a disk; both of which are either stationary or co-counter-rotating at various angular speeds.

Rotating cone has attracted a number of researchers to search for the induced flow and other physical insights. For example, [5] numerically solved the unsteady heat and mass transfer problem from a rotating vertical cone. [33] computed the laminar flow over a rotating cone making use of the semi-analytical homotopy analysis method, successfully reproducing the well-documented results. A series of boundary layer stability analysis was later on fulfilled in the papers [7], [8], [9], digging into the absolute or convective nature of instabilities caused within the rotating cone problem. The MHD nanofluid flow over a rotating cone was then investigated in [23] with the consideration of Brownian motion and thermophoresis effects. Results were reported on the induced magnetic field in Casson fluid flow over a cone with chemical reaction in [26]. Similarity solutions of the laminar compressible boundary-layer flows over a family of rotating cones subject to surface mass flux were presented in [32], after a careful modeling of the mean flow.

Rotating disk problem is still one of the most studied problem in fluid dynamics after formulating the problem in self-similar form by Karman [19]. Rotor–stator system is considered as practical technological application commonly used in industry [38]. Amongst many relevant papers on the hot topic, the recently published ones examining a plenty of physical mechanisms can be listed as [2], [6], [21], [34], [35], [36], [37], and [3]. The interested reader can further refer to publications [10], [11], [12], [13], [16] and [20] for a diversity of physical mechanisms.

As mentioned earlier, one of the practical application is the cone-plate viscometry, which is an experimental way of measuring the viscosity of a Newtonian fluid. Using the spherical coordinates, Giacomin and Gilbert [14] were able to extract an elegant formula representing the Newtonian viscosity as a consequence of determining the torque. This formula was shown to be valid for slightly wider viscometer gaps rather than narrow gaps. They also provided an exact solution for the temperature field and delineated the role of viscous dissipation in the temperature rise, when the walls of cone-plate geometry are kept at isothermal condition. The work of [14] is significant in learning the Newtonian viscosity influenced by temperature when wider gaps are taken into account in the cone-plate instrument. Shevchuk [28] also fulfilled a fruitful research on the cone-disk apparatus problem by detecting similar solutions fairly matching with the experimental observations in [22] and with the theoretical estimates in [27]. Although Shevchuk [28] calculated the heat transfer rate on the disk surface only, the data ignored the radial temperature gradient terms in the energy equation. The motivation here is to restore such terms into the governing equations and recompute the relevant temperature field as well as the heat transfer rate from the disk surface. It is shown that as the apex angle increases, addition of such physical terms plays considerably significant role depending on the form of the radial temperature constraint imposed on the disk surface. Since the flow is not yet effective enough, the temperature field is not so affected though, for the small conical gaps. The heat transfers from the cone surface are also computed with the new energy equation whose conductive terms are incorporated.

Section snippets

Flow and heat equations of cone-disk apparatus

The conical gap between a cone and a disk filled by an incompressible viscous fluid is under consideration with the convective heat transport. Both tools are presumed to be either stationary or rotating with changing angular velocities as shown in Fig. 1. In the publication by Shevchuk [28], the physical phenomenon was successfully applied with a radially varying wall temperature Tw=T+crn on the disk surface, where T is the uniform temperature along the cone wall with c and n being constants.

Flow and heat fields in similar form

Closely pursuing the group scale analysis of [28], we may adopt the following similarity variables η=zr,F=urν,G=vrνH=wrν,P=pr2ρν2,θ=TTTwT,resulting in the reduced system HηF=0,(1+η2)F+(3η+ηFH)F+F2+G2+2P+ηP=0,(1+η2)G+(3η+ηFH)G=0,(1+η2)H+(3η+ηFH)H+(1+F)HP=0,(1+η2)θ+η(12n)θ+n2θPr(nFθ+(HηF)θ)=0,F(0)=0,G(0)=Reω,H(0)=0,θ(0)=1,F(η0)=0,G(η0)=ReΩ,H(η0)=0,θ(η0)=0, where n is the power index of the wall temperature on the disk, Pr is the usual Prandtl number taken to be equal

Results and discussions

The system (6) is simulated numerically here to work out the velocity and temperature fields corresponding to the conical gap in a cone-disk apparatus. The particular interest is with reassessing the heat transfer rate from the disk surface by adding the radial diffusive terms given in (7), which were omitted in [28]. In addition to this, the exploration of heat transfer rate from the cone surface is also of concern. To keep in line with the literature [28], we restrict the computations to

Conclusions

The cone-disk apparatus, used mainly for viscosimetry in real applications is revisited in the present investigation. A power-law radial distribution of temperature prevails on the disk surface with a constant temperature on the cone surface. There is a viscous fluid trapped in a wedge between a flat circular plate and a cone, both of infinite extent. Either can rotate with the other stationary, or they can co-rotate, or they can counter-rotate.

Having established the flow field making use of

References (38)

  • HayatT. et al.

    Irreversibility characterization and investigation of mixed convective reactive flow over a rotating cone

    Comput. Methods Programs Biomed.

    (2020)
  • Phan-ThienN.

    Cone-and-plate flow of the Oldroyd-B fluid is unstable

    J. Non-Newton. Fluid Mech.

    (1985)
  • TurkyilmazogluM.

    On the purely analytic computation of laminar boundary layer flow over a rotating cone

    Internat. J. Engrg. Sci.

    (2009)
  • YuanZ.X. et al.

    Turbulent heat transfer on the stationary disk in a rotor–stator system

    Int. J. Heat Mass Transfer

    (2003)
  • BhattacharyyaA. et al.

    Simulation of Cattaneo–Christov heat flux on the flow of single and multi-walled carbon nanotubes between two stretchable coaxial rotating disks

    J. Therm. Anal. Calorim.

    (2020)
  • BucschmannM.H. et al.

    Analysis of flow in a cone-plate apparatus with respect to spatial and temporal effects on endothelial cells

    Biotechnol. Bioeng.

    (2005)
  • GarrettS.J. et al.

    The crossflow instability of the boundary layer on a rotating cone

    J. Fluid Mech.

    (2009)
  • GarrettS.J. et al.

    Boundary-layer transition on broad cones rotating in an imposed axial flow

    AIAA J.

    (2010)
  • GiacominA.J. et al.

    Exact-solution for cone-plate viscometry

    J. Appl. Phys.

    (2017)
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