Fundamental flow problems considering non-Newtonian hyperbolic tangent fluid with Navier slip: Homotopy analysis method

The presented research article deals with the classical fundamental flows (Poiseuille and Couette) of an incompressible hyperbolic tangent fluid while considering the Navier slip at the walls. The governing equations are solved using the homotopy analysis method. The velocity expressions are obtained for each problem and the effect of the flow parameters are discussed while being supplemented by graphical displays. Increasing the slip parameter reduces the fluid velocity for both problems, respectively. An increase in the Weissenberg number shows that skin friction at the lower wall reduces. This shows that a dominant elastic force is crucial in reducing the skin friction.


Introduction
The contact point between a fluid and a solid surface poses interesting questions. Many investigations have been undertaken and yet plenty of questions remain unanswered. As in the case of many engineering problem approximations, no-slip condition at the contact point of a moving fluid and a solid boundary has been found to be impractical for many cases [27]. Hence, the Navier slip at the contact point of fluid and solid body have been used prominently for theoretical studies as it has been found to be more realistic in view of real life applications. The Navier slip condition shows that the velocity of the moving fluid is directly proportional to the shear stress at the contact point of the fluid and solid boundary. The constant of proportionality is called the slip length. The non-dimensional slip parameter have been observed to be related to the direction of the fluid flow [24]. Usage of non-Newtonian fluids in industrial settings have slowly gained prominence in the last few decades. The hyperbolic tangent fluid is a subclass of non-Newtonian fluid which offers a shear thinning behavior and plays an important role in chemical engineering applications. Fluids such as paints, blood, ketchup or cream are examples of tangent hyperbolic fluid. Compared to other types of non-Newtonian fluids, the tangent hyperbolic fluid is rather simple, easily computable and is physically robust. Investigations in the present literature have reported few results of Navier slip on hyperbolic tangent fluid flows. However, the flow regimes have mostly been based on peristaltic flows [11][12][13], while a few others have been on flows over flat surfaces [4,7,21]. Skin friction has been reported to be diminished when the power law index was increased [15]. While the Weissenberg number has been reported to increase the hyperbolic tangent fluid flow in a channel [3]. Non-Newtonian fluid flow problems poses highly nonlinear differential equations and must be tackled intricately. The homotopy analysis method (HAM) [17,19] has been developed for such purposes. The HAM has been used successfully to solve numerous types of equations [1,2,9,18,20,23,28,29,31] and has been found to be extremely reliable [6,26].
In light of the presented literature review, the gap in the literature of Poiseuille and Couette flows of a tangent hyperbolic fluid is studied in the presented article. The Navier slip effect is assumed at the walls of the channel. For the Poiseuille flow, the flow is pressure driven and the walls are assumed to be stationary. For the case of the Couette flow, the flow is pressure driven while also being supplemented by the movement of the upper channel wall in the direction of the flow. The article is divided into several sections each focusing on a different aspect of the investigation. The second section focuses on the mathematical formulation of the tangent hyperbolic fluid in the Cartesian coordinates for the present problem in consideration. In the third section, the homotopy analysis method is introduced which is used to solve the nonlinear differential equation governing the flow. Section 4 is dedicated to the discussion of the results which are obtained in the previous section. The concluding remarks of the study are presented in the last section.

Governing equation formulation
Let us consider an incompressible non-Newtonian hyperbolic tangent fluid. A hyperbolic tangent fluid is a four constant fluid model describing the shear thinning effects of the fluid. The Cauchy stress tensor for the non-Newtonian hyperbolic tangent fluid can be written as [14,25] where τ is the extra stress tensor, κ ∞ is the shear rate viscosity at infinity, κ 0 is the initial shear rate viscosity, Γ is the time dependent material constant, n is the power law index and β is the shear rate. As the hyperbolic tangent fluid is a shear thinning fluid, we can conclude that κ ∞ = 0 and Γβ 1 such that tanh(Γβ) ≈ (Γβ). Thus, the Cauchy stress tensor (1) takes the form For a laminar, fully developed fluid flow, the velocity field is reduced to U = (u(y), 0, 0). In the absence of external body forces and a constant pressure gradient acting along the flow direction, the momentum equation of a hyperbolic tangent fluid can be written as [32] Introducing the non-dimensional terms as we have where s is the power law index, We is the Weissenberg number, and P is the dimensionless pressure gradient.
An important aspect of fluid flows is the skin friction at the fluid-solid interface. The skin friction is the resistance exerted by a solid body on a moving fluid. For the present analysis, the skin friction offered by the channel wall on the flowing hyperbolic tangent fluid is expressed in the non-dimensional form as

Poiseuille flow
For the Poiseuille fluid flow, consider the hyperbolic tangent fluid flowing between two infinitely long parallel horizontal plates which are at a distance 2L apart. The lower wall is taken to be at y = −L, while the upper wall is at y = L. The Navier slip at the walls are also considered. Thus, the conditions of the velocity at the walls of the channel can be written as [8] u Invoking Eq. (4), we have where γ = α/L is the dimensionless slip parameter.

Couette flow
For the Couette flow, again consider the hyperbolic tangent fluid flowing between two infinitely long horizontal plates kept at a distance 2L apart such that the upper wall is at y = L and the lower wall is at y = −L. For the general Couette flows, the fluid flow is aided by the movement of the upper plate in the direction of the flow. Hence, considering the existence of the Navier slip, the condition of the velocity at the lower wall and upper moving wall of the channel can be described as [8] u Invoking Eq. (4), we have where U w = u w /U 0 is the wall velocity parameter. The general expression of the fluid flow velocity can be obtained by taking s = 0 and γ = 0 and integrating Eq. (5) for both cases.

Analytical solution by HAM
We define a set of base functions as {Y n |n 0} to define the velocity expression in the form where C n is a coefficient. Using the boundary conditions for each problem, the initial guess approximations [18] are chosen as U 0 (Y ) = 0 for the Poiseuille flow, for the Couette flow.
The linear operator such that where K 1 and K 2 are unknown constants. Let h be the convergence controlling parameter and q ∈ [0, 1] be the embedding parameter. With reference to Eq. (5), the nonlinear operator is defined as The zero-order deformation equation is defined using L, h, q and N as When q = 0, the solution is the initial guess approximation and at q = 1, the final solution is obtained provided h = 0 Thus, it may be observed that as the value of q grows from 0 to 1, ψ(Y ; q) varies steadily from the initial guess U 0 to he final exact expression U(Y ). Using Taylor's series, ψ(Y ; q) is expanded in a power series of the embedding parameter q as shown below such that Using Eqs. (21) and (20) and assuming that the series converges for q = 1, we have The following vectors are defined Differentiating Eq. (17) with respect to q, m-times and then dividing it by m! and setting q = 0, the m th -order deformation equation is obtained as Accordingly, the boundary equations are reduced to It may be noted that the boundary conditions for both the Poiseuille and Couette flows are now equal as the constant terms are now eliminated. Also where The number of terms in the series solution should be such that the solution converges sufficiently. In order to achieve that the series is convergent sufficiently and quickly, the convergence control parameter has to be chosen carefully. As already defined in the literature [17], the q values have been narrowed down to the range [−2, 0]. To get the optimum value of q in the given range, we check the minimum residual error using the expression [16] where K is a positive integer and Δt = 1/K. In the present analysis, the value of K is taken as 20. Fig. 1 shows the best range of h values that may be considered for the present analysis. It is apparent that the optimal range for the h value coincide for both the Poiseuille and Couette flows. Consequently, the optimum value for h with the least amount of error coincide for both the flows. The h−values for different order of approximation is shown in Table 1. For the present analysis, the optimum value of h has been taken as −1.32.

Discussion
From Figs. 2 and 3, we can observe the effect of slip parameter on the velocity profile for both the Poiseuille and Couette flows, respectively. A high value of slip parameter γ indicates that there is a high interaction between the fluid particles and the solid wall. Hence, we can see that an increase in the slip parameter reduces the flow velocity significantly. This is a common phenomenon in flows with the Navier slip and has been discussed at length in the literature [5,10,22]. For the Poiseuille flow, we can see that the increase in slip parameter reduces the skin friction while the skin friction for the Couette flow is increased significantly. However, the reaction of skin friction to the increase in power law index by the Poiseuille and Couette flows is similar as shown in Figs. 10 and 11, respectively. We can see that there is a prominent increase in skin friction when the power law index is increased. The skin friction displayed in Figs. 8-11 are taken as a function of the Weissenberg number. It can be seen that the increase in the Weissenberg number reduces the friction at the wall. This shows that with the dominance of elastic forces over the viscous forces, friction at the walls of the channel may be reduced accordingly.

Conclusions
The investigation focused on the Poiseuille and Couette flows of a pressure driven non-Newtonian hyperbolic tangent fluid flow while considering the effect of the Navier slips at the walls. The momentum equation of the flow was reduced to a non-dimensional form after appropriate transformation. The homotopy analysis method was applied to solve the nonlinear second order governing differential equation. The velocity expression was obtained and the effect of pertinent flow parameters for both the Poiseuille and Couette flows was studied and discussed. The increase in value of the power law index significantly enhanced the velocity and skin friction of both flows. However, the effects of varying the value of the Weissenberg number on the velocity and skin friction were not too significant although they were noticeable. As noticed in the case of Newtonian fluids, flow reversal has also been observed in the present investigation. This is caused by the deceleration of flow due to the presence of the Navier slip at the walls [30].