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Suddenly expanding recirculating and non-recirculating viscoplastic non-newtonian flows

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Abstract

Viscoplastic non-Newtonian flows through axisymmetric 1:5 and 1:2 sudden expansions have been studied numerically by solving the mass and momentum conservation equations and the Bingham constitutive equation. Two distinct viscoplastic flow regimes are observed. For a given Reynolds number, increasing the yield number, a non-dimensional yield stress results in transitioning from a recirculating to a non-recirculating viscoplastic flow regime. For Newtonian fluids, a typical separating and reattaching flow field with a large corner recirculation region and a reattachment point is observed. For a low yield number viscoplastic non-Newtonian flow, the incoming flow separates, creating a weak recirculation region, and unyielded attached-to-the wall, corner and impingement regions. For a high yield number, a non-separating viscoplastic flow field forms, resulting in the elimination of flow recirculation and the formation of significant unyielded region behind the expansion plane. For Newtonian and low yield number viscoplastic flows, axisymmetric 1:5 expansion flows result in more intense recirculation, larger reattachment, and flow redevelopment lengths in comparison with axisymmetric 1:2 expansion flows.

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Abbreviations

d, D :

Diameters of small and large pipes

e i :

Relative recirculation intensity, ratio of the maximum amount of backflow in the recirculation region to the inlet flow rate \({{ - \psi_{\hbox{min} } } \mathord{\left/ {\vphantom {{ - \psi_{\hbox{min} } } {\psi_{\hbox{max} } }}} \right. \kern-0pt} {\psi_{\hbox{max} } }}\)

ER:

Expansion ratio, R o/R or D/d

L r :

Reattachment length

P :

Non-dimensional pressure, \(P^{*} /\rho u_{b}^{2}\)

r :

Non-dimensional radial distance, r */R

R, R o :

Radii of small and large pipes

Re:

Reynolds number, ρdu b /η p

u :

Non-dimensional streamwise velocity, u */u b

u b :

Inflow streamwise bulk velocity, \(2\int\nolimits_{0}^{R} {u^{*} r^{*} dr^{*} /R^{2} }\)

x :

Non-dimensional streamwise distance, x */R

Y :

Yield number, τ y d/η p u b

ψ max :

Maximum value of stream function

ψ min :

Minimum value of stream function

\(\dot{\gamma }_{{_{\text{ij}} }}\) :

Rate of deformation tensor, ∂u i /∂x j  + ∂u j /∂x i

\(\dot{\gamma }_{{_{\text{II}} }}\) :

Second invariant of rate of deformation tensor, \(\dot{\gamma }_{{_{\text{ij}} }}\) \(\dot{\gamma }_{{_{\text{ij}} }}\)

η p :

Plastic viscosity

μ eff :

Non-dimensional effective viscosity, \(\mu_{{_{\text{eff}} }}^{ * } /\eta_{p}\)

ρ :

Density

τ ij :

Stress tensor element

τ y :

Yield stress

*:

Dimensional quantities

b:

Bulk properties of initial jet

c:

Centerline properties

i:

Inflow properties, i.e. properties at x = 0

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Acknowledgments

The author acknowledges the support received through a 2015–2016 CSU-AAUP Research Grant.

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Authors and Affiliations

  1. Department of Engineering, Central Connecticut State University, New Britain, CT, 06050, USA

    Khaled J. Hammad

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Hammad, K.J. Suddenly expanding recirculating and non-recirculating viscoplastic non-newtonian flows. J Vis 18, 655–667 (2015). https://doi.org/10.1007/s12650-015-0279-9

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