# Oscillating pressure-driven slip flow and heat transfer through an elliptical microchannel

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## Abstract

This paper studies the transient slip flow and heat transfer of a fluid driven by the oscillatory pressure gradient in a microchannel of elliptic cross section. The boundary value problem for the thermal-slip flow is formulated based on the assumption that the fluid flow is fully developed. The semi-analytical solutions of velocity and temperature fields are then determined by the Ritz method. These solutions include some existing known examples as special cases. The effects of the slip length and the ratio of minor to major axis of the elliptic cross section on the velocity and temperature distribution in the microchannel are investigated.

## Keywords

Slip flow Heat transfer Oscillatory pressure gradient Microchannel Elliptic cross section## 1 Introduction

*μm*and 1 mm, the flow behaviour in the microchannel has a non-continuum effect. In the literature, a number of experimental and/or mathematical investigations deal with slip flow and/or heat transfer through microchannels, but the microflow phenomenon is not well understood due to contradictions related to drag effect and transition from laminar to turbulent flow. Due to the difficulty in experiments in this area, continuing effort to resolve these problems mathematically is important. Slip flow phenomenon in microducts has traditionally been studied analytically and numerically [7, 8, 9, 13, 17, 18, 19, 20]. The governing equations of the slip flow in microducts include the classical Navier–Stokes equations. Based on the assumption of incompressible Newtonian fluid with constant properties, negligible body forces and hydrodynamically fully developed steady state flow, the Navier–Stokes equations reduce to the Poisson equation

*β*is the slip parameter, \(\beta _{v} = \frac{2-\sigma _{v}}{\sigma _{v}}\) depending on the tangential momentum accommodation coefficient (\(\sigma _{v}\)),

*λ*denotes the molecular mean flow path and

*n*is unit outward normal vector to the boundary.

*q*) in elliptic microchannels with cross section area (

*A*) by the Poisson equation (1) and energy equations (4) in which the force term is defined by \(F=\frac{q}{A}\frac{u}{\bar{u}}\). To the best of authors’ knowledge, little work has been done to study the unsteady slip flow and heat transfer driven by oscillating pressure gradient in microchannels of elliptic cross section.

This paper is to study transient oscillating pressure-driven slip flow and heat transfer in elliptic microchannels. The model is subject to the Navier slip and convective heat flow conditions at the boundary. Constant heat flux is assumed in the microchannel. Semi-analytical solutions of velocity and temperature fields will be obtained by the Ritz method.

## 2 Governing equations

*u*and

*T*represent respectively the flow speed and temperature of fluid, \(\mu, \rho, k\) and \(c_{p}\) denote respectively viscosity, density, thermal conductivity and specific heat of the fluid, \(h_{\infty }\) represents the convective heat transfer coefficient, \(T_{\infty }\) is the tube outer temperature which is assumed to be uniform and \(q_{p}\) is heat flux.

*q*and

*p̄*are respectively a heat flux parameter and the average pressure.

*ω*at time

*t*, i.e.

*ū*is average velocity, \(\alpha = \frac{4 h_{\infty }}{ \rho c_{p} D_{h}}\) and \(\lambda =\sqrt{\frac{h_{\infty } A}{c_{p}}}\) in which

*A*is the area of cross section and \(D_{h}\) is a hydraulic diameter determined by [14]

*F*is a linear form and the vector space

*K*is convex. Our BVP (15)–(17) is thus equivalent to the following system of equations:

_{1}into equation (19) and setting \(\frac{\partial I_{V}}{\partial c_{i}}=0\ (i=1,\ldots, N)\), after some derivation, we obtain the following linear system of equations:

_{1}, the velocity can be obtained by

_{2}into equation (20) and set \(\frac{\partial I_{h}}{ \partial d_{i}}=0\ (i=1,\ldots, N)\). Then we obtain the linear system of equations

_{2}, the temperature

*T*can be determined by

## 3 Numerical example

*a*and semi-minor axis of length

*b*, we use the model parameters as shown in Table 1. For the slip flow control, we set the pressure gradient as \(dp/dz = -5ie^{i\omega t}\) for \(\omega =1.55\). Figure 1 shows variations of the pressure profile over time.

Blood | Tissue | |
---|---|---|

density | 1.05 | 1 |

dynamic viscosity | 0.04 | |

thermal conductivity | 1.43 × 10 | 1.47 × 10 |

specific heat capacity \(c_{p}\) [\(\mathrm{cal}/(\mathrm{g} \cdot {}^{\circ}\mathrm{C})\)] | 0.86 | 0.67 |

initial temperature \(T_{0}\) [ | 37 | 42 |

heat transfer coefficient \(h_{\infty }\) [\(\mathrm{cal}/ (\mathrm{s} \cdot \mathrm{cm}^{2} \cdot {}^{\circ}\mathrm{C})\)] | 5.49 × 10 | 6.45 × 10 |

heat flux parameter | 0.1 | |

average pressure | 100 |

*t*increases when \(dp/dz > 0\), while it increases as

*t*increases when \(dp/dz < 0\). In addition, when \(dp/dz\) approaches zero at \(t=2\) s on the left and \(t=4\) s on the right, the velocity pattern is similar but the fluid moves in the opposite direction.

## 4 Conclusion

This paper presents a mathematical model and its semi-analytical solution for the oscillating pressure-driven flow and heat transfer through an elliptic microchannel under the slip condition using the Ritz method. The results show that the characteristics of the oscillating flow of fluid and heat transfer in microchannels depends on the slip length and the aspect ratio of the microchannel. The results of this research may help in the optimisation of certain bioengineering systems.

## Notes

### Authors’ contributions

The first author formulated a mathematical model, generated the results and wrote the paper. The second author proposed the research idea for finding the results. The third author was responsible for examining results. The last author contributed in editing and revising the manuscript. All authors read and approved the final manuscript

### Funding

The second and third author would like to acknowledge partial financial support from the Centre of Excellence in Mathematics, Commission on Higher Education, Thailand.

### Competing interests

The authors declare that they have no competing interests.

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