Deceleration of a porous rotating disk in a viscous fluid
TL;DR: In this paper, the Navier-Stokes equations are transformed to non-linear ordinary differential equations using similarity transformations, and the resulting equations are solved numerically using a globally convergent homotopy method.
About: This article is published in International Journal of Engineering Science.The article was published on 1985-01-01 and is currently open access. It has received 16 citations till now. The article focuses on the topics: Flow (mathematics) & Viscous liquid.
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TL;DR: In this article , the authors analyze the unsteady flow over a rotating disk in a hybrid nanofluid with suction and deceleration effects and find that the first and second solutions are stable and physically relevant, whereas the third solution is unstable as time evolves.
Abstract: The present study attempts to analyze the unsteady flow over a rotating disk in a hybrid nanofluid with suction and deceleration effects. The partial derivatives of multivariable differential equations are converted to ordinary differential equations using appropriate transformations. The bvp4c function in MATLAB software is employed to solve the mathematical model. The outcomes show that multiple solutions are verifiable in certain operating parameters. The stability of the multiple solutions over time is investigated. It is discovered that the first and the second solutions are stable and physically relevant, whereas the third solution is unstable as time evolves. Moreover, the stronger deceleration contributes to enhancing the skin friction coefficient in the radial direction Rer1/2Cf and in the azimuthal direction Rer1/2Cg, for the first and third solutions whereas the second solution reduces. The values of Rer1/2Cf and Rer1/2Cg for the third solution enhance in the presence of suction, while the opposite behaviors are observed for the first and second solutions. The enhancement of the local Nusselt number Rer−1/2Nur on all solutions is noticed with the imposition of suction on the surface and stronger deceleration strength.
14 citations
TL;DR: In this article, the nonaxisymmetric unsteady motion produced by a buoyancy-induced cross-flow of an electrically conducting fluid over an infinite rotating disk in a vertical plane and in the presence of an applied magnetic field normal to the disk has been studied.
14 citations
01 Jan 2010
TL;DR: In this paper, the homotopy analysis method is employed to derive closed-form solutions for the unsteady Von Karman swirling flow caused by the transient, laminar, axially-symmetric viscous incompressible flow from an impulsively-started infinite disk rotating at constant angular velocity.
Abstract: The Homotopy Analysis Method (HAM) is employed to derive closed-form solutions for the unsteady Von Karman swirling flow caused by the transient, laminar, axially-symmetric viscous incompressible flow from an impulsively-started infinite disk rotating at constant angular velocity. Complete analytical solutions are presented for the unsteady Von Karman swirling flow which are uniformly valid for all dimensionless time 0 in the entire spatial domain 0 . Excellent agreement is obtained for the special case of steady flow (obtained for the infinite time scenario i.e. ) with the earlier study by Yang and Liao [Comm. Nonlinear Science Numerical Simulation, 11, 83-93 (2006)]. Flow velocity is found to increase with progression of time and reach the steady state after some time has elapsed. The hydrodynamic boundary layer thickness at the disk surface becomes greater initially but then stabilizes at a constant value when the steady state is attained. The present solutions provide an important benchmark for more general numerical studies which have applications in helicopter rotor aerodynamics, chemical engineering rotating electrode disk mechanisms and turbine blade flows.
13 citations
TL;DR: In this article , the authors discuss the impact of unsteady magnetohydrodynamics (MHD) hybrid ferrofluid flow over a stretching/shrinking rotating disk.
Abstract: The flow of fluids over the boundaries of a rotating disc has many practical uses, including boundary-layer control and separation. Therefore, the aim of this study is to discuss the impact of unsteady magnetohydrodynamics (MHD) hybrid ferrofluid flow over a stretching/shrinking rotating disk. The time-dependent mathematical model is transformed into a set of ordinary differential equations (ODE’s) by using similarity variables. The bvp4c method in the MATLAB platform is utilised in order to solve the present model. Since the occurrence of more than one solution is presentable, an analysis of solution stabilities is conducted. Both solutions were surprisingly found to be stable. Meanwhile, the skin friction coefficient, heat transfer rate—in cooperation with velocity—and temperature profile distributions are examined for the progressing parameters. The findings reveal that the unsteadiness parameter causes the boundary layer thickness of the velocity and temperature distribution profile to decrease. A higher value of magnetic and mass flux parameter lowers the skin friction coefficient. In contrast, the addition of the unsteadiness parameter yields a supportive effect on the heat transfer rate. An increment of the magnetic parameter up to 30% reduces the skin friction coefficient by 15.98% and enhances the heat transfer rate approximately up to 1.88%, significantly. In contrast, the heat transfer is rapidly enhanced by improving the mass flux parameter by almost 20%.
12 citations
26 Feb 2020
TL;DR: In this article, temperature-dependent physical properties and convective boundary conditions are taken into account, and the governing coupled nonlinear partial differential equations are transformed into a system of ordinary differential equations by adopting the well-known similarity transformations.
Abstract: In this article, the unsteady magnetohydrodynamic two-dimensional boundary layer flow and heat transfer over a stretchable rotating disk with mass suction/injection is investigated Temperature-dependent physical properties and convective boundary conditions are taken into account The governing coupled nonlinear partial differential equations are transformed into a system of ordinary differential equations by adopting the well-known similarity transformations Further, the solutions are obtained through the semi-analytical method called an Optimal Homotopy Analysis Method (OHAM) The obtained results are discussed graphically to predict the features of the involved key parameters which are monitoring the flow model The skin friction coefficient and Nusselt number are also examined The validation of the present work is verified with the earlier published results and is found to be in excellent agreement It is noticed that an increase in the viscosity parameter leads to decay in momentum boundary layer thickness, and the inverse trend is observed in the case of the temperature profile
9 citations
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TL;DR: This article showed that most existence theorems using degree theory are in principle relatively constructive and showed that the Brouwer fixed point theorem is constructive with probability one, which can be implemented by computer.
Abstract: We illustrate that most existence theorems using degree theory are in principle relatively constructive. The first one presented here is the Brouwer Fixed Point Theorem. Our method is "constructive with probability one" and can be implemented by computer. Other existence theorems are also proved by the same method. The approach is based on a transversality theorem.
390 citations
TL;DR: In this paper, the von Karman problem is extended to the case of flow started impulsively from rest; also, the steady-state problem is solved to a higher degree of accuracy than previously by a simple analytical-numerical method which avoids the matching difficulties in Cochran's (1934) well-known solution.
Abstract: The von Karman (1921) rotating disk problem is extended to the case of flow started impulsively from rest; also, the steady-state problem is solved to a higher degree of accuracy than previously by a simple analytical-numerical method which avoids the matching difficulties in Cochran's (1934) well-known solution. Exact representations of the non-steady velocity field and pressure are given by suitable power-series expansions in the angle of rotation, Ωt, with coefficients that are functions of a similarity variable. The first four equations for velocity coefficient functions are solved exactly in closed form, and the next six by numerical integration. This gives four terms in the series for the primary flow and three terms in each series for the secondary flow.The results indicate that the asymptotic steady state is approached after about 2 radians of the disk's motion and that it can be approximately obtained from the initial-value, time-dependent analysis. Furthermore, the non-steady flow has three phases, the first two of which are accurately and fully described with the terms computed. During the first-half radian (phase 1), the velocity field is essentially similar in time, with boundary-layer thickening the only significant effect. For 0·5 [lsim ] Ωt [lsim ] 1·5 (phase 2), boundary-layer growth continues at a slower rate, but simultaneously the velocity profiles adjust towards the shape of the ultimate steady-state profiles. At about Ωt = 1·5, some flow quantities overshoot the steady-state values by small amounts. In analogy with the ‘Greenspan-Howard problem’ (1963) it is believed that the third phase (Ωt > 1·5) consists of a small amplitude decaying oscillation about the steady-state solution.
378 citations
TL;DR: In this article, an exact solution of the Navier-Stokes equation for unsteady flow is a semi-infinite contracting or expanding circular pipe is calculated and reveals the following characteristics of this type of flow.
Abstract: Physiological pumps produce flows by alternate contraction and expansion of the vessel. When muscles start to squeeze its wall the valve at the upstream end is closed and that at the downstream end is opened, and the fluid is pumped out in the downstream direction. These systems can be modelled by a semi-infinite pipe with one end closed by a compliant membrane which prevents only axial motion of the fluid, leaving radial motion completely unrestricted. In the present paper an exact similar solution of the Navier–Stokes equation for unsteady flow is a semi-infinite contracting or expanding circular pipe is calculated and reveals the following characteristics of this type of flow. In a contracting pipe the effects of viscosity are limited to a thin boundary layer attached to the wall, which becomes thinner for higher Reynolds numbers. In an expanding pipe the flow adjacent to the wall is highly retarded and eventually reverses at Reynolds numbers above a critical value. The pressure gradient along the axis of pipe is favourable for a contracting wall, while it is adverse for an expanding wall in most cases. These solutions are valid down to the state of a completely collapsed pipe, since the nonlinearity is retained in full. The results of the present theory may be applied to the unsteady flow produced by a certain class of forced contractions and expansions of a valved vein or a thin bronchial tube.
204 citations
TL;DR: Chow, Mallet-Paret, and Yorke have recently proposed an algorithm for computing Brouwer fixed points of C^2 maps as discussed by the authors, and a numerical implementation of that algorithm is presented here.
158 citations
110 citations