The peristaltic transport of a Couple stress fluid in the horizontal tapered asymmetric channel is examined, concealed by long wavelength and low Reynolds number assumptions. The tapered asymmetric channel is produced due to the peristaltic motion on the walls of the non-uniform channel having alternative amplitudes and phases. Excellent velocity, pressure gradient, and stream function results have been determined. Numerical evaluation has also been found for investigating the pumping characteristics with various flow values of parameters. It has been shown that the peristaltic transport pumping decreases with an increase in non-uniform rheological parameters. Further, our conclusion is created to be very relevant to  when the absence of the tapered asymmetric nature of the channel and couple stress fluid parameter.
Physiologists have found that peristaltic flow plays a key role in transport of fluid in many biological systems. Especially, a peristaltic mechanism may be regarded in chyme movement in the gastrointestinal tract, in transport of the ovum in female fallopian tube, in urine transport from kidney to bladder through urethra and also in industries for the transport of noxious fluid in nuclear industries, as well as in roller pumps, in blood pumps, in heart lung machine, vasomotion of small blood vessels, and many others. The problem of the mechanism of peristaltic motion has attracted the care of many detectives since the first investigation of Latham . The initial mathematical model of peristalsis viewed by train of sinusoidal waves in an infinitely long symmetric channel or tube has been initially studied by Fung and Yih  and Shapiro et al. . Later, researchers have extended a number of analytical, numerical and experimental studies of peristaltic transport of different fluids have been reported under different conditions with reference to physiological and mechanical situations (Refs. [4-11] and references therein). Undoubtedly, the most of physiological and industrial fluids including blood act as non-Newtonian fluids. Since, the observer of peristaltic motion of non-Newtonian fluids may assist to get better realizing of the biological systems. Even the latest researchers in the field are expecting for the analysis of peristaltic flows under different aspects [12-17]. It is a well known fact that, couple stress fluid is very useful in understanding several physical problems because it possesses the mechanism to illustrate rheological fluids such as blood, lubricants containing small amount of high polymer additive, liquid crystals, human blood, synovial joint, electro-rheological fluids and synthetic fluids. In this view the model formulated by Stokes  and represents to predict micro structural characteristics (particle size) of physiological suspensions with good precision. Mekheimer  studied the problem of the peristaltic transport of a couple stress fluid in a uniform and non-uniform channel. Effect of the induced magnetic field on peristaltic flow of a magneto-couple stress fluid was studied by Mekheimer . His study indicates that the current density at the center of the channel is higher for a couple stress fluid than a Newtonian fluid, and it also decreases as the transverse magnetic field increases. Pandey and Chaube  have investigated the peristaltic transport of a couple stress fluid in a channel with compliant walls. They have determined that the mean velocity at boundaries decreases with increasing couple-stress parameter and wall damping and increases with increase of wall tension and wall elasticity. The peristaltic flow of a couple stress fluid under the effect of induced magnetic field in an asymmetric channel have been observed by Nadeem and Akram , It is valuable to mention that the intra uterine fluid flow in a sagittal cross-section of non-pregnant uterus caused by myometrial contractions is a peristaltic–type fluid motion. In view of the above mentioned reasons, a new mathematical/theoretical analysis of peristaltic transport of a couple stress fluid in the most generalized form of the channel mentioned as the tapered asymmetric channel is investigated. To the best of our knowledge, no endeavor has been reported yet to discuss the peristaltic flow of a couple stress fluid through in the tapered asymmetric channel under the assumptions of long wavelength and low Reynolds number. The exact solutions of velocity distribution, pressure gradient and steam function are obtained. The influence of various pertinent parameters on the flow characteristics are discussed in details with the help of graphs.
II. MATHEMATICAL FORMULATION
The present investigation has discussed the peristaltic transport of a couple-stress fluid in the horizontal tapered asymmetric channel. The governing equations have been modeled and evaluated using long wavelength, and low Reynolds number approximations have been adopted. The correct expressions for horizontal pressure gradient, horizontal velocity and stream function are analytically determined. The main decisions of the present study are as follows:
1) The large values of Couple stress fluid parameter correspond to that of a Newtonian fluid.
2) The magnitude of the velocity of a Newtonian fluid is higher than a couple-stress fluid.
3) The rate of peristaltic pumping decreases with increasing non-uniform parameter and couple stress fluid parameter.
4) The capacity of the circulating bolus rises with a raises of a non-uniform parameter.
5) In figure 11, we conclude that our results are in accordance with  when the amplitudes of both walls are equal and in the absence of the phase difference.
 T.W. Latham, Fluid motion in a peristaltic pump, Cambridge: MS Thesis, MIT, 1966.
 Y.C. Fung, C.S. Yih, Peristaltic transport, J. App. Mech. 35 (1968) 669 - 675.
 A.H. Shapiro, M.Y. Jaffrin, S.L. Weinberg, Peristaltic pumping with long wavelengths at low Reynolds number, J. Fluid Mech. 37 (1969) 799 - 825.
 M.Y. Jaffrin, A.H. Shapiro, Peristaltic pumping, Annual Review of Fluid Mechanic, 3 (1971) 13 - 37.
 O. Eytan, D. Elad, Analysis of intra-uterine motion induced by uterine contractions, Bull. Math. Biol. 61 (1999) 221-238.
 M. Mishra, A. Ramachandra Rao, Peristaltic transport of a Newtonian fluid in an asymmetric channel, Z. Angew. Math. Phys. 54 (2003) 532-550.
 M. Kothandapani, S. Srinivas, Non-linear peristaltic transport of a Newtonian fluid in an asymmetric channel through a porous medium, Phys. Lett. A, 372 (2008) 1265-1276.
 D. Tripathi, A mathematical model for the peristaltic flow of chyme movement in small intestine, Math. Biosci. 233 (2) (2011) 90-97.
 C. Dhanapal, J. Kamalakkannan, J. Prakash, J. Kothandapani, Analysis of peristaltic motion of a nanofluid with wall shear stress, microrotation and thermal radiation effects, Appl. Bionics Biomech. 412374 (2016) 412374.
 J. Prakash, E.P. Siva, D. Tripathi, M. Kothandapani, Nanofluids flow driven by peristaltic pumping in occurrence of magnetohydrodynamics and thermal radiation, Mater. Sci. Semicond. Process. 100 (2019) 290-300.
 R. Vijayaragavan, P. Tamizharasi, A. Magesh, Brownian motion and thermoporesis effects of nanofluid flow through the peristaltic mechanism in a vertical channel, J. Porous Media, 25(6) (2022) 65-81.
 K.K. Raju, R. Devanathan, Peristaltic motion of a non-Newtonian fluid, Rheol. Acta. 11 (1972) 170-178.
 G. Radhakrishnamacharay, Long wave length approximation to peristaltic motion of a power law fluid, Rheol. Acta 21 (1982) 30-35.
 L.M. Srivastava, V.P. Srivastava, Peristaltic transport of blood: Casson model-II, J.Biomech. 17 (1984) 821-829.
 M. Kothandapani, J. Prakash, Effects of thermal radiation parameter and magnetic field on the peristaltic motion of Williamson nanofluids in a tapered asymmetric channel, Int. J. Heat Mass Transf. 81 (2015) 234-245.
 A. Magesh, M. Kothandapani, Heat and mass transfer analysis on non-Newtonian fluid motion driven by peristaltic pumping in an asymmetric curved channel, Eur. Phys. J. Spec. Top. 230 (2021) 1447-1464.
 A. Magesh, M. Kothandapani, V. Pushparaj, Electro-osmatic flow of Jeffery fluid in an asymmetric micro-channel under the effect of magnetic field, J. Phys. Conf. Ser. 1850 (2021) 012102.
 L.M. Srivastava, V.P. Srivastava, S.N. Sinha, Peristaltic Transport of a Physiological Fluid: Part I. Flow in Non-Uniform Geometry, Biorheol. 20 (1983) 153-166.
 L.M. Srivastava, V.P. Srivastava, Peristaltic Transport of a Power-Law Fluid: Application to the Ductus Efferentes of the Reproductive Tract, Rheol. Acta, 27 (1988) 428-433.
 V. K. Stokes, Couple stresses in fluids, Phys. Fluids, 9 (1966) 1709-1715.
 Kh. S. Mekheimer, Peristaltic transport of a couple stress fluid in a uniform and non-uniform channels, Biorehology, 39 (2002) 755765.
 Kh. S. Mekheimer, Effect of the induced magnetic field on peristaltic flow of a couple stress fluid, Phys. Lett. A, 372 (2008) 4271-4278.
 S. K. Pandey, M. K. Chaube, Study of wall properties on peristaltic transport of a couple stress fluid, Meccanica, 46 (2011) 1319-1330.
 S. Nadeem, S. Akram, Peristaltic flow of a couple stress fluid under the effect of induced magnetic field in an asymmetric channel, Arch. Appl. Mech. 81 (2011) 97-109.
 J. Kamalakkannan, C. Dhanapal, M. Kothandapani, non–Linear peristaltic flow of power-law fluid in the tapered asymmetric channel, International Journal for Research in Applied Science and Engineering Technology, 11 (01) (2023) 1545-1554.