An existence of trapezoidal waveform of oscillating motion from triangular pressure waveforms: The case of an Oldroyd-B model Rozaini Roslan, Mohammed Abdulhameed and Maselan Ali Centre for Research in Computational Mathematics Universiti Tun Hussein Onn Malaysia 86400 Parit Raja, Batu Pahat, Johor, Malaysia Abstract—Oscillating motion of Oldroyd-B flow in a capillary tube at a triangular sine and cosine pressure waveform is considered. An analytical solution of velocity field is obtained for an oscillating laminar flow, which can be used to determine the existence of trapezoidal waveform effect on the flow. When both triangular sine and cosine components are dominate, the oscillating flow can result in significant trapezoidal waveform performance. The dimensionless oscillating frequency, amplitude, and fluid material parameters, are primary factors affecting the flow performance of an oscillating flow in a capillary tube. I. I NTRODUCTION Theoretical studies on the laminar oscillatory flow from a capillary tube have receive attention in the literature due to their applications in natural systems (respiratory system, circulatory system) as well as engineering systems (pulse com- bustors, reciprocating pumps). Several sinusoidal waveform profiles in a capillary tube have been studied to determine the oscillating pressure effect on the fluid flow. Richardson and Tyler [1] investigated Newtonian viscous flow in a capillary tube due to sinusoidal pressure waveform and revealed how the oscillating flow could result in a different velocity profile near the wall surface. Since then, diverse researches and studies have been devoted to oscillatory flow performance in capillary tubes. Rahaman and Ramkissoon [2] investigated viscoelastic upper convected Maxwell passed in the pipes and showed how the oscillating flow performance depend on the time-dependent pressure waveform of an arbitrary function. Hariharan et al. [3] reported a theoretical and experimental result of the peristaltic transport of non-Newtonian fluid, modeled as power law and Bingham fluid, in a diverging tube with different wall wave- forms: sinusoidal, multi-sinusoidal, triangular, trapezoidal and square waves. Results obtained suggest that square wave has the best pumping characteristics of all the waveforms, and the triangular wave has the worst characteristics. Yin and Ma [4] presented a novel analytical solution for velocity, temperature distributions and a Nusselt number for an oscillating laminar flow in a round pipe driven by a sinusoidal waveform. The result reveals that the dimensionless oscillating frequency, amplitude, and Prandtl number, have the direct effects on the heat transfer performance of an oscillating flow in a capillary tube. Zhao and Cheng [5] showed that in addition to the sinusoidal waveform, other waveforms exist in an oscillating pipes such as the triangular waveform. The existence of triangular waveforms, made it necessary to further investigate the existence of other waveforms effect of oscillating motion on the flow rate performance of oscillating flow in a capil- lary tube. Recently, Yin and Ma [6] derived an analytical solution of laminar Newtonian flow with a triangular pressure waveform. Results reveal that the heat transfer coefficient of the oscillating flow depends on the fluid characteristics and oscillating waveform. The triangular waveform of oscillating motion can lead to a greater heat transfer coefficient. Various rheological models have proposed and available to the literature to describe the rheological behavior of non- Newtonian fluids [7 − 9]. One of this type of rheological model is the rate type fluid model described by Dunn and Rajagopal [7]. Oldroyd-B fluid is one of the subclasses of these fluid models that exhibit strange features such as elastic and memory effects exhibited by most polymer and biological fluid. The objective of present paper is to establish the existence of trapezoidal waveform from triangular waveform of Oldroyd- B flud of rate type. The oscillating flow with a trapezoidal waveform has not been investigated before. The unanswered questions regarding trapezoidal waveforms, made it necessary to study the effect of oscillating motion on the fluid flow in a capillary tube. In the current investigation, the oscillating flow with a triangular waveform is modeled by using an infinite Fourier series. In order to demonstrate the existence of trapezoidal waveform effect, the triangular sine and cosine waveforms of the oscillating flow is presented. A triangular reduces to a trapezoidal when combined between sine and cosine waveform and hence, trapezoidal waveforms can be treated as a special case of triangular waveform. Using the analytical solution of velocity field, the effects of waveform frequency, waveform amplitude, and material parameter on the flow rate performance of oscillating flow are analyzed. 14