**1. Introduction**

Synovial fluid is secreted to the cavity by its inner membrane called Synovial [1]. It is a biological fluid filling the Synovial joint-cavity's several-micrometers-thick layer between the interstitial cartilages [2]. The main component of Synovial fluid is ultrafiltration of the blood plasma devoid of high-molecular proteins, blood cells, and aggressors. Synovial fluid supports joints via high effective cartilage lubrication, while its essential component is an added lubricant called hyaluronan/hyaluronic acid [3]. Several studies showed that the viscoelastic features of Synovial fluid occur due to hyaluronic acid [4]. Hyaluronic acid is natively present in the Synovial fluid in relatively high concentrations [5]. It is experimentally [6] confirmed that the viscoelastic features of Synovial fluids strongly rely on a concentration of hyaluronic acid; therefore, the magnitude of polymerization is substantial, because the volume of hyaluronic random coils exhibits a momentous role in the viscoelastic attributes of Synovial fluid [7,8].

Furthermore, Synovial fluid contains mixtures that reveal a viscoelastic fashion. When a Synovial fluid is propagating with versatile conditions where there is no instantaneous input, then it performs as a Stokesian fluid. When it is only subject to immediate input, then its viscoelastic charactersitics manifests itself. Hron et al. [9] examined the flow analysis of three separate models that could be referred to as Synovial fluid models. These models fit into the type of generalized viscous fluids, whereas only one goes fits into the class of a shear-thinning model in which the power-law exponent relies upon the concentration.

Moreover, incompressible non-Newtonian liquids have attracted great interest in recent years. Perhaps this is due to academic curiosity and their several industrial applications including synthetic lubricants, colloidal fluids, and liquid crystals. It is found that various physiological fluids reveal non-Newtonian behavior. Non-Newtonian characteristics produce satisfactory results when analyzing the mechanism of peristalsis propagating in lymphatic vessels, blood vessels, ductus afferents, intestines, the motion of urine in the human body, food bolus moving through esophagus, the movement of spermatozoa in a vas deferens, the blending of food material, Chyme motion, cilia propagation, blood circulation, and the propagation of bile in a bile duct. A peristaltic movement is a fluid transport that happens because of the contraction and extension of smooth walls. Recently, many authors have determined the peristaltic mechanism in various boundary and initial conditions. Notably, Mekheimer et al. [10] calculated the peristaltic phenomenon of magnetized couple-stress fluid along with the effects of the induced magnetic field. He further achieved the exact analytics solutions for the velocity profile. Srinivas and Kothandapani [11] examined the mass and heat transfer impact on the peristaltic transportation of viscous liquid. They formulated the governing flow using the lubrication approach and obtained the exact solution. Further, they assumed that fluid is travelling in a porous medium having compliant walls. Riaz et al. [12] modeled the unsteady peristaltic flow of Carreau fluid propagating through a small intestine and presented analytic solutions using the perturbation method. Akram et al. [13] explored the behavior of lateral walls on the non-uniform, peristaltic-propelled three-dimensional flow of the couple stress fluid model. Ellahi et al. [14] also discussed the three-dimensional motion of Carreau fluid with an external uniform magnetic field. They used the Homotopy perturbation scheme to obtain the solutions of the obtained non-linear partial differential equations. They determined that the magnetic field is a significant factor in the preservation of the flow field. Bhatti et al. [15] examined the behavior of the oblique magnetic field with heat transfer on the uniform peristaltic motion containing small particles. They presented the exact solutions for the fluid and particulate phases, whereas numerical integration was used to determine the pumping characteristics. Sinha et al. [16] presented the peristaltic motion of viscous liquid containing a variable viscosity under the inclusion of heat exchange and the static magnetic field with asymmetric geometry. They obtained the perturbation solutions under the slip conditions and temperature jump. Shit et al. [17] examined the asymmetrical motion of a micropolar fluid with the induced magnetic field. They obtained exact results for micro-rotation components, magnetic force function, the velocity profile, and the current density profile. A mathematical analysis of a micropolar fluid in an artery having composite stenosis was measured by Ellahi et al. [18]. Bhatti et al. [19] evaluated the peristaltic propulsion of magnetized solid particles in Biorheological fluids. They considered the model of Casson fluid and obtained the exact results for liquid and particulate phase against velocity and temperature profile. Peristaltic motion through a porous channel was presented by Maiti and Misra [20]. They discussed the bile flow with in ducts in the pathological state. Bhatti et al. [21] considered the combined electric and magnetic field impact on the propulsion of the peristaltic third-grade fluid model containing small particles. They further considered the heat transfer effects and obtained the analytical results using Homotopy perturbation methods.

Furthermore, Kabov et al. [22] experimentally discussed the two-phase flow propagating through a microchannel. Mekheimer and Elmaboud [23] addressed the impression of heat exchange and magnetic field on the viscous-fluid model stimulated in peristaltic fashion. They explained the influence of endoscope and bioheat transfer. Elmaboud and Mekheimer [24] addressed the nonlinear peristaltic motion of second-grade fluid propagating through a porous geometry. They further applied the perturbation method to solve the velocity equations, whereas pumping features and friction forces were evaluated by numerical integration. Khan et al. [25] studied the behavior of changeable viscosity of the Jeffrey fluid model propagating through the asymmetric porous channel. Transient peristaltic flow through a permeable finite channel was determined by Tripathi [26]. Chaube et al. [27] discussed the peristaltic flow of the power-law model using the creeping flow regime. Shit et al. [28] also discussed the role of velocity slip on the wavy motion of the couple stress fluid model. They mainly focused on a peristaltic movement in the digestive system. Later, Shit et al. [29] governed the peristaltic biofluid flow through a microchannel. Moreover, they also considered the EMHD ("Electro-Magnetohydrodynamic") and velocity slip due to a hydrophobic/hydrophilic collision between negatively charged walls. Recently, Zeeshan et al. [30] addressed the behavior of the Sisko fluid model propagating across a non-uniform peristaltic channel. They obtained the second order solution using the Homotopy perturbation method. Some more useful studies related to the topic can be seen in [31,32].

According to literature surveyed, it is observed that no results have been presented yet to examine the behavior of Synovial fluid on peristaltic propulsion through an asymmetric channel. According to our knowledge, not a single mathematical model is given in the literature describing the behavior of Synovial fluid for peristaltic flow. The governing fluid holds the properties of incompressibility and irrotational and constant density. Furthermore, mass transport is also taken into account to discuss the present flow. Mass transportation is also an important phenomenon in the propagation of mass from one region to another region. Therefore, the primary theme of the current study is to present a theoretical and mathematical analysis of the said topic to fill this gap in the literature. The graphical results are presented for two different models of Synovial fluid.
