Solidification of circular Couette flow with viscous dissipation

Mackie, Calvin; Hall, Carsie A.; Perkins, Judy

doi:10.1016/s0142-727x(01)00108-4

Cited by 3 publications

(2 citation statements)

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“…In particular, Hall and Mackie 5 have derived a closed‐form expression for the instantaneous location of the solid–liquid interface, the dimensionless Nusselt number, and the dimensionless power density as a function of pertinent dimensionless parameters including viscous dissipation. Furthermore, Mackie et al 6 presented a semianalytical solution for one‐dimensional freezing of laminar, circular Couette flow within a finite annular region with viscous dissipation. Later, Basu et al 7 studied the influence of the imposed pressure gradient on the solidification of liquid in planar Poiseuille–Couette flows with viscous dissipation.…”

Section: Introductionmentioning

confidence: 99%

Solidification of electrically conducting liquid flow driven by shear and pressure gradients subject to viscous and Joule heating

et al. 2022

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Solidification of a liquid in motion driven by shear and pressure gradients occurs in many natural settings and technological applications. When the liquid is electrically conducting, its solidification rates can potentially be modulated by an imposed magnetic field. The shearing motion results in viscous dissipation and the Lorentz force induced by the magnetic field causes Joule heating of the fluid, which can influence the structure of the flow, thermal fields, and thereby the solidification process. In this study, a mathematical model is developed to study the combined effects of shear and pressure gradients in the presence of a magnetic field on the solidification of a liquid between two parallel plates, with one of them being insulated and under constant motion, and the other being cooled convectively and at rest. Under the quasi‐steady assumption, closed‐form semianalytical solutions are obtained for the instantaneous location of the solid–liquid interface, Nusselt number, and dimensionless power density as a function of various characteristic parameters such as the Hartmann number, pressure gradient parameter, Brinkman number, and Biot number. Furthermore, an interesting remelt or steady‐state condition for the interfacial location is derived as arising from the competing effects of the solid side heat flux and viscous dissipation and Joule heating on the liquid side. The newly derived analytical results are shown to reduce to the various classical results in the limiting cases. A detailed systematic study is performed by the numerical solution of the semianalytical formulation, and the effects of different characteristic parameters on the solidification process are discussed.

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Section: Introductionmentioning

confidence: 99%

Solidification of electrically conducting liquid flow driven by shear and pressure gradients subject to viscous and Joule heating

et al. 2022

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“…Some of these analytical methods applied to semi-in nite regions have included a similarity method by Pearson, 1 an integral energy balance method by Grif n, 2 and a combination of similarity and Green's function by Huang. 8 Recently, however, Mackie et al 9 used a semi-analytic method to analyze a solidifying circular 587 Couette ow (rotating outer cylinder) in a nite region, cooled isothermally by a stationary inner cylinder.…”

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Solidification of Couette Flow in an Annulus with Inner Cylinder Rotation

Hall

Mackie

2002

Journal of Thermophysics and Heat Transfer

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A semi-analytic solution is presented for the solidi cation of laminar Couette ow within a one-dimensional annular region with a rotating inner cylinder and stationary outer cylinder. Viscous dissipation in the liquid is taken into account. The inner cylinder is maintained under adiabatic conditions while the outer cylinder is convectively cooled. Analytic expressions for the dimensionless quasi-steady temperature distribution in the solid and liquid regions, Nusselt number at the solid-liquid interface, dimensionless power and torque per unit length, and dimensionless steady-state freeze front location are derived as a function of liquid-to-solid thermal conductivity ratio, Brinkman number, annulus radius ratio, and Stefan number, which is assumed to be small (<0.1) but nonvanishing. The dimensionless instantaneous solid-liquid interface location is determined using numerical integration. In addition, the Brinkman number is related to the Reynolds number, and expressions for the factor increase in transition Reynolds and Brinkman numbers are derived. It is observed that an in nitely large Biot number, corresponding to the isothermal cooling limit, increases the ow stability in the liquid region as the transition Reynolds increases by a factor of up to 11. However, this stability increase comes at the expense of a factor increase in the power per unit length of up to 2.4. NomenclatureBi = Biot number Br = Brinkman number c = speci c heat, J ¢ kg ¡1 ¢ K ¡1 h = convective heat transfer coef cient, W ¢ m ¡2 ¢ K ¡1 h s f = latent heat of fusion, J ¢ kg ¡1 k = thermal conductivity, W ¢ m ¡1 ¢ K ¡1 Nu = Nusselt number P = power, W P 0 = power per unit length, W ¢ m ¡1 P ¤ = dimensionless power per unit length Pr = Prandtl number Re = Reynolds number R i = inner radius of annulus, m R o = outer radius of annulus, m r = radial coordinate location, m Ste = Stefan number T = temperature, K = = torque, kg ¢ m 2 ¢ s ¡2 = 0 = torque per unit length, kg ¢ m ¢ s ¡2 t = time, s U = dimensionless velocity component in azimuthal direction u = velocity component in azimuthal direction, m ¢ s ¡1 ®= thermal diffusivity, m 2 ¢ s ¡1 = annulus radius ratio°= liquid-to-solid thermal conductivity ratio 1 = dimensionless solid-liquid interface location ± = solid-liquid interface location, ḿ = dimensionless radial coordinate location µ = dimensionless temperature ¹ = dynamic viscosity, kg ¢ m ¡1 ¢ s ¡1 º = kinematic viscosity, m 2 ¢ s ¡1 ¿ = dimensionless time Á = dimensionless parameter of Eq. (26) Ã = solid-to-liquid thermal diffusivity ratio ! = angular speed of inner cylinder, 1 ¢ s ¡1 Subscripts = liquid region m = solid-liquid interface conditions s = solid region ss = steady-state conditions tr = transition from laminar to turbulent ow 1 = far-eld or freestream conditions

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