# Parametric analysis of laminar pulsating flow in a rectangular channel

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## Abstract

Pulsating flow has potential for enhanced cooling of future electronics and photonics systems. To better understand the mechanisms underlying any heat transfer enhancement, it is necessary to decouple the mechanical and thermal problems. The current work performs a parametric analysis of the flow hydrodynamics using particle image velocimetry (PIV) measurements, CFD simulations and analytical solutions, reorganised in terms of amplitude and phase values using complex notation. To the best of the authors’ knowledge, the frequency-dependent behaviour of amplitude and phase of wall shear stress has not been studied in a two-dimensional channel. For laminar flow, the amplitudes are directly proportional to pressure. The amplitudes of various local and mean wall shear stress measures are augmented with frequency compared to steady flow, especially near the short walls and corners. The phases of wall shear stress differ at each wall at moderate frequencies – with the bulk-mean values at the short wall leading those at the long wall – and tend to *π*/4 in the limit of high frequency. The amplitudes of pressure gradient increase more significantly than wall shear stress magnitudes due to accelerative forces. The boundaries to the quasi-steady, intermediate and inertia-dominated regimes are estimated at Womersley number *W* *o* = 1.6 and 27.6 in a rectangular channel, based on the contribution of viscous and inertial terms.

## List of symbols

*a*,*b*channel width, height [

*m*]*D*_{h}hydraulic diameter [

*m*]*f*oscillation frequency [

*H**z*]- \(\mathfrak {I}\)
imaginary part of complex number

*i*imaginary unit (=\(\sqrt {-1}\))

*L*channel length [

*m*]*m*,*n*summation indices

*p*pressure [

*P**a*]*Q*flow rate [

*m*^{3}/*s*]- \(\mathfrak {R}\)
real part of complex number

*R**e*Reynolds number (= 〈

*u*〉*D*_{ h }/*ν*)- \(Re_{\delta _{\nu }}\)
*R**e*based on Stokes layer thickness (= 〈*u*〉*δ*_{ ν }/*ν*)*t*time [

*s*]*u*velocity in the axial direction [

*m*/*s*]- 〈
*u*〉 space-averaged velocity [

*m*/*s*]*W**o*Womersley number \(\left (=\frac {1}{2}D_{h}\sqrt {\omega /\nu }\right )\)

*x*axial flow coordinate [

*m*]*y*,*z*coordinates normal to flow direction [

*m*]

## *Greek symbols*

*β*function defined by Eq. 4b

*δ*_{ν}Stokes layer thickness [

*m*]*μ*viscosity [

*k**g*/(*m*⋅*s*)]*ν*kinematic viscosity [

*m*^{2}/*s*]*ρ*density [

*k**g*/*m*^{3}]*τ*wall shear stress [

*P**a*]- 〈
*τ*〉 space-averaged wall shear stress [

*P**a*]- Φ
function defined by Eq. 4a

*ψ*complex functions defined by e.g. Eq. 3a

*ω*angular oscillation frequency [

*r**a**d*/*s*]

## *Subscripts & Superscripts*

- 0
steady flow component

*A*oscillating flow amplitude

*yx**x*component with normal*y**zx**x*component with normal*z*^{′}phase relative to \(\nabla p^{\prime } = \mathfrak {R}[\nabla p_{A} e^{i \omega t}]\)

^{″}phase relative to \(Q^{\prime \prime } = \mathfrak {R}[\psi _{Q}e^{i (\omega t - \phi _{Q} - \pi /2)}]\)

## Notes

### Acknowledgements

The authors would like to acknowledge the financial support of the Irish Research Council (IRC) under grant numbers EPSPG/2013/618 and GOIPD/2016/216.

### Compliance with Ethical Standards

### Conflict of interests

On behalf of all authors, the corresponding author states that there is no conflict of interest.

## References

- 1.Jeffers N, Stafford J, Nolan K, Donnelly B, Enright R, Punch J, Waddell A, Erlich L, O’Connor J, Sexton A, Blythman R, Hernon D (2014) Microfluidic cooling of photonic integrated circuits (PICs). In: 4th Eur. Conf. Microfluidics, LimerickGoogle Scholar
- 2.Kurzweg UH, Zhao L (1984) Heat transfer by high-frequency oscillations: a new hydrodynamic technique for achieving large effective thermal conductivities. Phys Fluids 27(11):2624–2627CrossRefGoogle Scholar
- 3.Wälchli R, Linderman R, Brunschwiler T, Kloter U, Rothuizen H, Bieri N, Poulikakos D, Michel B (2008) Radially oscillating flow hybrid cooling system for low profile electronics applications. In: 24th Annual IEEE, Semi-Therm 2008, San Jose, pp 142–148Google Scholar
- 4.Wälchli R, Brunschwiler T, Michel B, Poulikakos D (2010) Self-contained, oscillating flow liquid cooling system for thin form factor high performance electronics. J Heat Transfer 132(5):051401CrossRefGoogle Scholar
- 5.Walsh TE, Yang KT, Nee VW, Liao QD (1993) Forced convection cooling in microelectronic cabinets via oscillatory flow techniques. Exp Therm Fluid Sci 7(2):140CrossRefGoogle Scholar
- 6.Persoons T, Saenen T, Van Oevelen T, Baelmans M (2012) Effect of flow pulsation on the heat transfer performance of a minichannel heat sink. J Heat Transfer 134(9):091702CrossRefGoogle Scholar
- 7.Womersley JR (1955) Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J Physiol 127(3):553–563CrossRefGoogle Scholar
- 8.Stokes GG (1851) On the effect of the internal friction of fluids on the motion of pendulums. Trans Cambridge Philos Soc 9:8Google Scholar
- 9.Uchida S (1956) The pulsating viscous flow superposed on the steady laminar motion of incompressible fluid in a circular pipe. Z Angew Math Phys 7(5):403–422MathSciNetCrossRefzbMATHGoogle Scholar
- 10.Blythman R, Persoons T, Jeffers N, Nolan KP, Murray DB (2017) Localised dynamics of laminar pulsatile flow in a rectangular channel. Int J Heat Fluid Flow 66:8–17CrossRefGoogle Scholar
- 11.Linford RG, Ryan NW (1965) Pulsatile flow in rigid tubes. J Appl Physiol 20(5):1078–1082CrossRefGoogle Scholar
- 12.Zhao TS, Cheng P (1996) The friction coefficient of a fully developed laminar reciprocating flow in a circular pipe. Int J Heat Fluid Flow 17(2):167–172CrossRefGoogle Scholar
- 13.Shemer L, Wygnanski I, Kit E (1985) Pulsating flow in a pipe. J Fluid Mech 153:313–337CrossRefGoogle Scholar
- 14.He S, Jackson JD (2009) An experimental study of pulsating turbulent flow in a pipe. Eur J Mech B Fluids 28(2):309–320CrossRefzbMATHGoogle Scholar
- 15.Ohmi M, Iguchi M, Usui T (1981) Flow pattern and frictional losses in pulsating pipe flow: Part 5, wall shear stress and flow pattern in a laminar flow. Bull JSME 24(187):75–81CrossRefGoogle Scholar
- 16.Ray S, Ünsal B, Durst F, Ertunc Ö, Bayoumi OA (2005) Mass flow rate controlled fully developed laminar pulsating pipe flows. J Fluids Eng 127(3):405–418CrossRefGoogle Scholar
- 17.Haddad K, Ertunç Ö, Mishra M, Delgado A (2010) Pulsating laminar fully developed channel and pipe flows. Phys Rev E 81(1):016303CrossRefGoogle Scholar
- 18.Gaver DP, Grotberg JB (1986) An experimental investigation of oscillating flow in a tapered channel. J Fluid Mech 172:47–61CrossRefGoogle Scholar
- 19.Godleski DA, Grotberg JB (1988) Convection-diffusion interaction for oscillatory flow in a tapered tube. J Biomech Eng 110(4):283–291CrossRefGoogle Scholar
- 20.Richardson EG, Tyler E (1929) The transverse velocity gradient near the mouths of pipes in which an alternating or continuous flow of air is established. Proc Phys Soc 42(1):1CrossRefzbMATHGoogle Scholar
- 21.Fan C, Chao BT (1965) Unsteady, laminar, incompressible flow through rectangular ducts. Z Angew Math Phys 16(3):351–360CrossRefzbMATHGoogle Scholar
- 22.Blythman R, Jeffers N, Persoons T, Murray DB (2016) Localized and time-resolved velocity measurements of pulsatile flow in a rectangular channel. In: 18th Int. Conf. Fluid Mech. Thermodyn. Rio de Janeiro, BrazilGoogle Scholar
- 23.Blythman R, Persoons T, Jeffers N , Murray DB (2016) Effect of oscillation frequency on wall shear stress and pressure drop in a rectangular channel for heat transfer applications. J Phys Conf Ser 745(3):032044CrossRefGoogle Scholar
- 24.Blythman R, Jeffers N, Persoons T, Murray DB (2017) Wall temperature of laminar pulsating flow in a channel. In: 9th World Conf. Exp. Heat Transfer, Fluid Mech. and Thermodyn. Iguazu Falls, BrazilGoogle Scholar
- 25.Kurzweg UH (1985) Enhanced heat conduction in oscillating viscous flows within parallel-plate channels. J Fluid Mech 156:291–300CrossRefzbMATHGoogle Scholar
- 26.Sergeev SI (1966) Fluid oscillations in pipes at moderate Reynolds numbers. Fluid Dyn 1(1):121–122CrossRefGoogle Scholar
- 27.Alimohammadi S, Murray DB, Persoons T (2015) On the numerical–experimental analysis and scaling of convective heat transfer to pulsating impinging jets. Int J Therm Sci 98:296–311CrossRefGoogle Scholar
- 28.Alimohammadi S, Fanning E, Persoons T, Murray DB (2016) Characterization of flow vectoring phenomenon in adjacent synthetic jets using CFD and PIV. Comput Fluids 140:232–246MathSciNetCrossRefzbMATHGoogle Scholar
- 29.Hughes PE, How TV (1994) Pulsatile velocity distribution and wall shear rate measurement using pulsed doppler ultrasound. J Biomech 27(1):103–110CrossRefGoogle Scholar