Flow analysis of Carreau fluid model induced by the ciliary cells, smooth muscle cells and pressure gradient at the ampullar region entrance

This theoretical analysis considers a biomechanical model in which the Carreau fluid model characterizes the viscoelastic nature of growing human embryo and secreted fluid. This model incorporates transport mechanisms that involve the swaying motions of ciliary cells, peristaltic contractions of smooth muscle cells and pressure gradient at the ampullar region entrance. Series form solutions of the resulting partial differential equations are obtained using the regular perturbation method. A theoretical estimate of effects of the condition of pressure gradient, geometric parameters and fluid model parameters on the flow variables that have relevance to the problem of growing embryo transport in the human fallopian tube is presented through the discussion of graphs. Furthermore, an analogy between the linearly viscous fluid, and the shear thinning and shear thickening characteristics of the Carreau fluid model is also presented. The pertinence of the obtained results with growing embryo transport in the human fallopian tube revealed that when shear thickening characteristics of the Carreau fluid model are considered then complete mitotic divisions take place properly with an estimated appropriate residue time about 3–4 days. Smaller size trapped boluses of the secreted fluid make the smooth forwarding of the growing embryo in the human fallopian tube when shear thinning characteristics of the Carreau fluid model are taken into account. Key modulators: progesterone ( $$P_{4})$$ and estradiol ( $$E_{2}$$ ), prostaglandin $$E_{2}$$ ( $$PGE_{2}$$ ) and prostaglandin $$F_{2\alpha }$$ ( $$PGF_{2\alpha }$$ ) constraint the growing embryo transport.