### Abstract

Laminar radial flow between two parallel disks is a fundamental nonlinear fluid mechanics problem described by the Navier-Stokes (NS) equation, but is unsolved because (1) an exact solution is not found even with extensive references, and (2) it is unclear why radial flow remains laminar at high Reynolds numbers. This paper first presents exact velocity distribution solutions for developed radial inflows and outflows, proving that both flows are described by brief Jacobi elliptic sine-squared functions but with different characteristics. For inflow, a stable velocity distribution forms; for outflow, the velocity distribution may have an inflection point inducing flow instability or separation. Both velocity distributions become the classic parabolic law at low Reynolds numbers, but uniform (similar to turbulent velocity distributions) at high Reynolds numbers. Furthermore, both pressure and boundary shear stress follow an inverse-square law, but the friction factor is invariant. These results are instructive for studying nonuniform open-channel flow for which nonlinear inertia is of importance.

Original language | English (US) |
---|---|

Article number | 04017026 |

Journal | Journal of Engineering Mechanics |

Volume | 143 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2017 |

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### Keywords

- Exact solution
- Jacobi elliptic function
- Laminar flow
- Navier-Stokes equation
- Nonlinear inertia
- Radial flow
- Velocity distribution

### ASJC Scopus subject areas

- Mechanics of Materials
- Mechanical Engineering

### Cite this

*Journal of Engineering Mechanics*,

*143*(6), [04017026]. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001227

**Exact solution to Navier-Stokes equation for developed radial flow between parallel disks.** / Guo, Junke; Shan, Haoyin; Xie, Zhaoding; Li, Chen; Xu, Haijue; Zhang, Jianmin.

Research output: Contribution to journal › Article

*Journal of Engineering Mechanics*, vol. 143, no. 6, 04017026. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001227

}

TY - JOUR

T1 - Exact solution to Navier-Stokes equation for developed radial flow between parallel disks

AU - Guo, Junke

AU - Shan, Haoyin

AU - Xie, Zhaoding

AU - Li, Chen

AU - Xu, Haijue

AU - Zhang, Jianmin

PY - 2017/6/1

Y1 - 2017/6/1

N2 - Laminar radial flow between two parallel disks is a fundamental nonlinear fluid mechanics problem described by the Navier-Stokes (NS) equation, but is unsolved because (1) an exact solution is not found even with extensive references, and (2) it is unclear why radial flow remains laminar at high Reynolds numbers. This paper first presents exact velocity distribution solutions for developed radial inflows and outflows, proving that both flows are described by brief Jacobi elliptic sine-squared functions but with different characteristics. For inflow, a stable velocity distribution forms; for outflow, the velocity distribution may have an inflection point inducing flow instability or separation. Both velocity distributions become the classic parabolic law at low Reynolds numbers, but uniform (similar to turbulent velocity distributions) at high Reynolds numbers. Furthermore, both pressure and boundary shear stress follow an inverse-square law, but the friction factor is invariant. These results are instructive for studying nonuniform open-channel flow for which nonlinear inertia is of importance.

AB - Laminar radial flow between two parallel disks is a fundamental nonlinear fluid mechanics problem described by the Navier-Stokes (NS) equation, but is unsolved because (1) an exact solution is not found even with extensive references, and (2) it is unclear why radial flow remains laminar at high Reynolds numbers. This paper first presents exact velocity distribution solutions for developed radial inflows and outflows, proving that both flows are described by brief Jacobi elliptic sine-squared functions but with different characteristics. For inflow, a stable velocity distribution forms; for outflow, the velocity distribution may have an inflection point inducing flow instability or separation. Both velocity distributions become the classic parabolic law at low Reynolds numbers, but uniform (similar to turbulent velocity distributions) at high Reynolds numbers. Furthermore, both pressure and boundary shear stress follow an inverse-square law, but the friction factor is invariant. These results are instructive for studying nonuniform open-channel flow for which nonlinear inertia is of importance.

KW - Exact solution

KW - Jacobi elliptic function

KW - Laminar flow

KW - Navier-Stokes equation

KW - Nonlinear inertia

KW - Radial flow

KW - Velocity distribution

UR - http://www.scopus.com/inward/record.url?scp=85017516348&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85017516348&partnerID=8YFLogxK

U2 - 10.1061/(ASCE)EM.1943-7889.0001227

DO - 10.1061/(ASCE)EM.1943-7889.0001227

M3 - Article

AN - SCOPUS:85017516348

VL - 143

JO - Journal of Engineering Mechanics - ASCE

JF - Journal of Engineering Mechanics - ASCE

SN - 0733-9399

IS - 6

M1 - 04017026

ER -