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Research Papers: Fundamental Issues and Canonical Flows

Unsteady Flow of Fluids With Arbitrarily Time-Dependent Rheological Behavior

[+] Author and Article Information
Irene Daprà

DICAM,
University of Bologna,
2 via Risorgimento,
Bologna 40136, Italy
e-mail: irene.dapra@unibo.it

Giambattista Scarpi

DICAM,
University of Bologna,
2 via Risorgimento,
Bologna 40136, Italy
e-mail: giambattista.scarpi@unibo.it

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received December 3, 2015; final manuscript received December 21, 2016; published online March 16, 2017. Assoc. Editor: D. Keith Walters.

J. Fluids Eng 139(5), 051202 (Mar 16, 2017) (6 pages) Paper No: FE-15-1887; doi: 10.1115/1.4035637 History: Received December 03, 2015; Revised December 21, 2016

This paper presents an analytical solution of the momentum equation for the unsteady motion of fluids in circular pipes, in which the kinematic viscosity is allowed to change arbitrarily in time. Velocity and flow rate are expressed as a series expansion of Bessel and Kelvin functions of the radial variable, whereas the dependence on time is expressed as Fourierlike series. The analytical solution for the velocity is compared with the direct numerical solution of the momentum equation in a particular case, verifying that the difference between analytical and numerical values of axial velocity is less than 1%, except near the discontinuity of the applied pressure gradient, where the typical behavior due to the Gibbs phenomenon is to be noted.

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Figures

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Fig. 1

Pressure gradient, kinematic viscosity, and ψ versus time

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Fig. 2

Response to a rectangular pressure gradient pulse: axial velocity as a function of time—numerical results and analytical values for a constant and linearly varying viscosity

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Fig. 3

Axial velocity as a function of time: numerical results and analytical values using 30 and 50 terms of the first series

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Fig. 4

Enlargement near (0,0) of Fig. 2

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Fig. 5

Percent differences between the numerical and analytical values of axial velocity

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Fig. 6

Velocity profiles at t = 0.8 and t = 8; analytical values are obtained using 50 terms of the first series

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Fig. 7

Discharge as a function of time: numerical and analytical values using 50 terms of the first series

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Fig. 8

Percent differences between the numerical and analytical values of flow rate using 50 terms of the first series

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