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Boundary layer studies suggest that the velocity and temperature boundary layer thicknesses are lower for higher Reynolds numbers.

It is also noticed from Fig. 5 that momentum boundary layer thickness reduces rapidly when K ≠ 0, whereas the case of thermal and concentration boundary layer thicknesses are quite opposite to this, i.e., both thicknesses decrease when K = 0.

The boundary layer thicknesses are of physical importance for this problem, defined by δ f / x = 2 η f δ / R e x 1 / 2, δ θ / x = 2 η θ δ / R e x 1 / 2, (47).

Homann (see White [45]) extended this problem to the axisymmetric stagnation flow and found that the solution differs a little from the plane flow, where the displacement and boundary layer thicknesses are slightly smaller and the wall shear stress is slightly larger.

Added to that, the effects of nonlinear growth of boundary layer thickness are rarely included in these models.

The temperature and thermal boundary layer thickness are decreased corresponding to an increase in the values of the Prandtl number.

As a general result, it should be noted that for decreasing values of the fractional coefficient α, the fluid velocity and boundary layer thickness are decreasing.

It is observed that the velocity profile (f' eta )) and momentum boundary layer thickness are enhanced when (lambda > 0) (assisting flow), while opposite behavior is noted for (lambda < 0) (opposing flow).

(b) In the situation of assisting flow ((lambda > 0)), both the velocity profile (f^{prime} (eta )) and momentum boundary layer thickness are increase, while opposite performance is perceived in the case of opposing flow ((lambda < 0)).

Temperature is recorded as 0.17726, 0.18163, 0.18604, 0.19050, and 0.19501 for the Eckert number Ec = 0.0, 5.0, 10.0, 15.0, and 20.0, respectively, at the same position of η = 3.20, and the temperature profiles increase by 10.01%, that is, velocity boundary layer thickness and thermal boundary layer thickness are unchanged.

Boundary layer thickness are increased by Clay swelling, and flow channel are blocked by clay off and migration, that the main reason of KW smaller than K∞.