Thermo-physical characteristics of liquids and gases near a rotating disk

https://doi.org/10.1016/j.chaos.2020.110304Get rights and content

Abstract

In this study, we consider the thermo-physical properties of Newtonian boundary layer fluid flow near a heated rotating disk. The viscosity is assumed to vary inversely as a linear function of temperature while thermal and concentration diffusion coefficients vary directly as linear functions of temperature and concentration. Temperature dependent viscosity and conductivity are adopted to analyze the flow phenomena. Numerical solution of the governing equations is then performed by using a fifth order Runga-Kutta Fehlberg integration scheme with a shooting method. The effect of viscosity for gases shows an increase behavior of the velocity profiles but for liquids the opposite behavior is noticed. Variable thermal conductivity and diffusivity also increase the temperature and concentration distributions. We determine the mean flow profiles and give a physical interpretation of the problem through figures and tables. Correlations in terms of temperature, concentration and viscosity are plotted graphically. Comparison has been made with available results in the literature, achieving good agreement.

Introduction

The boundary layer flow near a rotating disk having Newtonian flow has received great interest over the past decade. The first theoretical study of the three-dimensional flow problem was presented by Kàrmàn [1]. He analyzed the steady incompressible flow due to rotation of an infinite surface with uniform angular velocity and calculated its analytical solutions. Shoo and Poncet [2] discussed the slip flow arising due to a rotation of non-Newtonian fluid at a very huge distance from a stationary disk. Turkyilmazoglu et al. [3] discussed the instability of inviscid flow over a rotating disk. Griffiths [4] analyzed the boundary layer flow of generalised Newtonian fluid due to a rotating disk. He first utilized a large Reynolds number boundary layer approximation, followed by the initiation of the von-Kàrmàn similarity transformation. The governing partial differential equations are converted into a set of ordinary differential equations. Yin et al. [5] gave a theoretical analysis of the flow and heat transfer of a nanofluid induced by a rotating disk. Three types of nanoparticles are considered with water as a base fluid. Alqarni et al. [6] discussed the steady incompressible generalised Newtonian fluid flow due to a rough rotating disk. Utilizing pseudoplastic fluid and dilatant fluids they determined the steady base profiles under partial slip effects. Turkyilmazoglu [7] calculated an exact analytical solution of the viscous fluid due to a permeable rotating disk. Many authors [8], [9], [10], [11] investigated the boundary layer over a rotating disk by considering different physical aspects.

Viscosity is one of the most essential thermo-physical properties of the fluid. In previous investigations viscous effects in the fluid are assumed to be independent of temperature. The variation of viscosity is more significant in the temperature variation. Many researchers explored flow with temperature dependent viscosity and reported results in complex geometries under different flow conditions. Ramanathan and Muchikel [12] and Khan et al. [13] discussed the temperature based viscosity on a generalized Newtonian fluid flow due to a nonlinear stretching surface. Umavathi et al. [14] investigated the effects of temperature dependent viscosity and mixed convection flow in a vertical channel filled with viscous fluid having isothermal wall conditions. Khan et al. [15] considered the effect of temperature dependent plastic dynamic viscosity in the Maxwell fluid induced by a stretching surface.

The importance of viscous effects on flow has led us to move towards temperature dependent thermal conductivity. Temperature processes play a significant role in controlling heat transfer in quality of products; there is a great literature available relating temperature based conductivity. Salawu and Dada [16] discussed the heat transfer temperature dependent conductivity with inclined magnetic field over a linear stretching surface. Xun et al. [17] studied the temperature dependent viscosity and thermal conductivity in bioconvection flow between two rotating plate surfaces. Here, it was analyzed that the temperature profile increases with increase of variable thermal conductivity parameter. Salawu and Ogunseye [18] analyzed the numerical solution for the thermodynamic second law in a solar radiative Eyring-Powell nanofluid flow with variable electrical conductivity.

Thus, in the present analysis we restrict our attention towards temperature dependent viscosity, thermal and concentration diffusion coefficients. In this case we can use the boundary-layer similarity transformation to give an analytic representation of three dimensional flow for high Reynolds number. We will discuss the following new findings in the present paper:

  • 1)

    We find the numerical solution of laminar flow near a rotating disk for both fluid and gas-type temperature dependent viscosity relationships.

  • 2)

    We also consider the effects of thermal and concentration diffusion coefficients on temperature and concentration profiles.

  • 3)

    No one has ever taken variable thermo-physical features of gases and liquids (especially gases) near a rotating disk.

  • 4)

    For variable viscosity the velocity profile effect has been illustrated for gases and liquids and which phenomenon has a major effect on velocity and which one has a minor effect is determined. Similarly for energy and concentration distributions, variable thermal conductivity and diffusivity phenomena has been analyzed for liquids and gases.

The partial differential equations are initiated in Section 2 and converted into ordinary differential equations by employing a von Kàrmàn similarity transformation and then solved by using a shooting method. In Section 3 we discuss the numerical solution and physical interpretations in the form of figures and tables. The resulting base flow velocity, concentration and temperature profiles are presented and discussed in detail, and finally concluding statements are presented in Section 4.

Section snippets

Formulation

Consider the three-dimensional axisymmetric flow of a viscous fluid flow due to a rotating disk by considering thermal diffusion coefficients. The flow rotates about the z*-axis with an angular velocity Ω* and the boundary is located at z*=0. The geometrical coordinates are presented in Fig. 1. In a rotating frame, the continuity, Navier-Stokes, energy and concentration equations can be expressed as [4], [6], [19], [20]:∇*.u*=0,ρ*[∂u*∂t*+u*.∇*u*+2Ω*×u*+Ω*×(Ω*×r*)]=−∇*p*+∇*.τ*,(ρ*Cp*)(∂T*∂t*+u*.∇

Results and discussion

Fig. 2 shows the radial, azimuthal and normal velocity profiles for different values of λ. As λ is increased, the radial flow profile reduces near the rotating surface. We achieved conflicting behaviors of the radial, azimuthal and normal velocity profiles for positive and negative values of λ. For λ > 0, the radial, azimuthal and normal velocity profiles are concentrated close to the rotating disc surface and so the boundary layer thickness decreases, but for λ < 0, the flow extends fourther

Conclusions

We have analyzed the boundary layer flow near a rotating disk with temperature dependent viscosity and thermal diffusion coefficients. Using a large Reynolds number boundary layer approximation, followed by the Von Kármán similarity transformation, the partial differential equations are converted into ordinary differential equations. These equations are solved by utilizing a fifth order Runga-Kutta Fehlberg intergrator alongside a Newton-Raphson scheme. The main findings are:

  • •

    We analyze that the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper

Cited by (0)

View full text
© 2020 Elsevier Ltd. All rights reserved.