Analytical Solutions for Unsteady Pipe Flows with Slip Boundary Condition
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Journal of Applied Fluid Mechanics, Vol. 13, No. 3, pp. 1015-1026, 2020. Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645. DOI: 10.29252/jafm.13.03.30800 Analytical Solutions for Unsteady Pipe Flows with Slip Boundary Condition Anek. V. Pillai and K. V. Manu† Department of Aerospace Engineering, Indian Institute of Space Science and Technology, Thiruvananthapuram 695547, Kerala, India †Corresponding Author Email: [email protected] (Received July 25, 2019; accepted September 17, 2019) ABSTRACT In this paper analytical expressions for time-dependent velocity profiles and pressure gradient are obtained for fully-developed laminar flows with given volume flow-rate conditions in circular pipe flows with slip boundary conditions. The governing equations are solved analytically using the traditional Laplace transform method together with Mellin’s inversion formula. The evolution of velocity profiles and pressure gradient for starting and pulsatile flow with slip boundary conditions are analyzed. New simplified expressions and perspectives on velocity and pressure gradient for no-slip and slip flows are obtained from the analytical results. New scalings in starting and pulsatile flows are proposed for pipe flows with no-slip and slip boundary conditions using non- dimensional numbers. Special attention is paid to the effect of slip factor and pulsatile flow frequency on the time-dependent skin-friction factor. Finally, by using the starting and pulsating flow results, analytical expressions of velocity and pressure for arbitrary inflow are obtained by approximating the arbitrary volume flow-rate by a Fourier series Keywords: Micropipe; Starting flow; Pulsatile flow; Womersley number. NOMENCLATURE c non-dimensional length-scale ratio U0 axial velocity for starting flows Cf friction coefficient Uc axial velocity for cosine flows Kn Knudsen number Us axial velocity for sine flows L mean free path of gas molecule Wo square of Womersley number p pressure x axial distance of the pipe r radial distance from center of pipe x∗ non-dimensional axial distance of the pipe R radius of the pipe Re Reynolds number t time α Womersley number T non-dimensional time scale for oscillatory β viscous slip coefficient flows η navier length u axial component of velocity in pipe ν kinematic viscosity ρ density of the fluid uavg average velocity of axial velocity τ non-dimensional time scale for starting uc axial velocity at the center of the pipe flows ur radial velocity component ω angular frequency of oscillation uθ circumferential velocity component U velocity scale 1 I|NTRODUCTION known applications. One of the most useful and well- known applications is the efficient cooling of chips The scientific and practical interest to micro-pipe using micro heat pipes. Besides these, they have also flows has constantly increased over the past few found applications in bio-engineering systems. Re- years. Micro-energy systems and micro- cent experimental studies at nano and micro level electromechanical systems are few of the well- have shown that the frequently used no-slip condition can conditionally breakdown, depending Anek. V. Pillai and K. V. Manu / JAFM, Vol. 13, No. 3, pp. 1015-1026, 2020. on the interfacial interaction between the fluid and In the first category (Fourier transform approach), solid surfaces. In such cases, the boundary condition the pressure gradient variation is normally assumed can drastically differ from the no-slip boundary a waveform, rep-resented by Fourier series condition. Aforesaid, boundary condition can (Womersley, 1955; Uchida, 1956; Khan, 2007; commonly arise from rarefaction effect and Majdalani, 2008; Hayat et al. 2010). The most superhydrophobicity. influential studies and widely cited works related to the first approach are the pipe flow solutions of It has been observed that for Knudsen number (Kn = Womersley (1955) and Uchida (1956). Majdalani λ/D, where λ is the mean free path and D is the −2 (2008) extended the method proposed by Uchida characteristic length) in the range of 10 ≤ Kn ≤ −1 (1956) to channel flows with no slip boundary 10 , the flow is essentially governed by Navier- condition. Recently, Ray et al. (2005) obtained the Stokes equation with slip flow boundary condition. analytical solution for different types of pulsation by Because of wide modern potential applications in approximating the mass flow-rate by Fourier series. micro and nano-fluidic systems, the slip boundary In the second category (Laplace transform approach condition has regained attention from the fluid (Das & Arakeri, 2000; Brereton, 2000; Muntges & mechanics community. Therefore it is of utmost Majdalani, 2002; Khaled & Vafai, 2004; Chen et al., importance to gain a clear understanding of flow 2008; Hayat et al., 2011; Avramenko et al., 2015), behavior with slip boundary conditions. the pressure gradient is an unknown parameter and The concept of partial-slip boundary condition is as the additional condition to solve the governing old as the Navier-Stokes equation itself. Navier, in equation is obtained from the time-varying mean the middle of the nineteenth century, posited a Robin volume flow-rate. boundary condition by relating the wall-velocity to Das and Arakeri (2000) successfully applied the the shear-rate at the wall. In Navier’s model, the second method to obtain the transient variation of the velocity at the solid boundary (slip velocity) varies velocity profile and pressure gradient for different in- linearly with the magnitude of flow shear-rate at the flow configurations. Muntges and Majdalani (2002) wall as follows: extended the method suggested by Das and Arakeri (2000) for an arbitrary volume flow-rate in channel , 1 flows. In an interesting study, (Brereton (2000) obtained the analytical relation between flow-rate, Where η is the ‘Navier-length’ (commonly called as pressure gradient and wall friction for arbitrary slip-length) defined as the distance into the wall at inflow using Laplace transform methods for channel which the fluid velocity extrapolates to zero. and pipe flows. Maxwell rigorously quantified slip-length for gases Chen et al. (2008) have used the method suggested flowing over a solid wall and the value was found to by Das and Arakeri (2000) to circular pipes with slip be of the order of mean free path of the fluid. The boundary conditions for trapezoidal and starting commonly used no-slip and perfect-slip conditions flows. Using Laplace transform method Khaled and are cases with proportionality constant values equal Vafai (2004) obtained an exact analytical solutions to zero and infinity respectively. of Stokes and Couette flows commenced by Over the past two decades, the theory of starting and sinusoidal motion of a wall with slip boundary pulsating flows with no-slip boundary conditions has conditions. Recently, many works have been devoted been a subject of intense research and the theory of to study the effect of dynamic slip boundary time-dependent laminar flows has been well condition (Kaoullas & Georgiou, 2013; Damianou et documented in books by Batchelor (Batchelor, al., 2014; Kaoullas & Georgiou, 2015) (where the 1967), Schlichting and Gersten (2016) and Jog slip velocity is a function of previous wall shear (2002). The first mathematical solution of stress state) on pressure gradient and velocity profiles unidirectional unsteady flow was given by Stokes in different flows configurations. using similarity method by solving the diffusion In an interesting study Avramenko et al., 2015) equation during the course of study of pendulum obtained an analytical solution for starting flows with friction (Panton, 2013). In Stokes’ study, the slip conditions for pipe and channel flows. To model pressure upstream and downstream of the solid the starting flow, Avramenko et al. (2015) surface is kept constant during the motion. Rayleigh represented the pressure gradient term using a illustrated the boundary layer character of the Stokes Heaviside function and solved using Cauchy’s flow and creatively used the analytical solution of residue theorem in conjunction with the Laplace Stokes to derive an approximate expression for skin- transform method. In this work we have considered friction on a finite length flat-plate. The essence of three unsteady flows in a pipe with known inlet flow- Rayleigh argument was the idea of replacing the time rate (Q (t)) variation than the pressure gradient scale with /U, where is the distance from the variation which is considered in many other previous leading edge. Further mathematical extension of the studies. Defining known time varying volume rate at Stokes problem is the pressure gradient-driven the inlet instead of pressure gradient is beneficial in starting flows in long pipes and channels. In such flow conditions where pressure is more laborious to cases, the expression for velocity and pressure are measure than volume flow rate. Also, from practical generally represented by Fourier/Bessel series. point of view, known inlet flow-rate conditions are Two types of procedures are commonly used for more frequently encountered than the pressure obtaining the analytical solutions for laminar, fully- gradient variation. Here, the transient behavior of developed pulsatory flows with constant properties. 1016 Anek. V. Pillai and K. V. Manu / JAFM, Vol. 13, No. 3, pp. 1015-1026, 2020. Fig. 1. Schematic depiction of the investigated flow-configurations with boundary conditions. velocity and pressure gradient in pipe flows with slip represented by sine or cosine function. The inlet flow conditions for starting, oscillatory and arbitrary conditions for single-frequency oscillatory flows inflows are analytically investigated. We use the starting from rest are as follows: classical Laplace transform method to obtain the explicit expressions between the flow variables. Due uavg(t) = 0 for t ≤ 0, (3) to the captivating applications of hydrophobic = Uc cosωt for t > 0 (for cosine flows), surfaces associated with the drag reduction, special attention is given to the skin-friction factor. = Us sinωt for t > 0 (for sine flows). This paper is organized as follows. The mathematical Here, uavg is the mean velocity at the inlet of the pipe details used for solving the governing equations are and ω is the angular frequency of the oscillation.