1 Introduction

Non-Newtonian fluid mechanics interactions between structures (like surfaces, channels, tubes, and wedges) and different types of fluid have been investigated during the last four decades. In 1940, Rosenhead^[1] developed an analytical solution in terms of elliptic functions of the two-dimensional radial viscous fluid flow inside convergent or divergent wedge shape. About two and a half decades later, three important studies have shed to light. Tanner^[2] found a link between the apparent viscosity and the third rate of the invariant deformation in the case of inelastic non-Newtonian fluid inside conical walls. Moreover, he suggested a model at low Reynolds numbers that explains the viscometric experiments of inelastic fluid behavior in a cone. Simultaneously, Tadmor^[3] found analytically the velocity distribution for various Reynolds numbers of non-Newtonian fluid inside cylindrical annuli based on the power law model. About two decades later, Moffatt et al.^[4] found analytically that velocity discontinuity was rapidly growing with the increase in the Reynolds number in the case of two-dimensional flow of unmixed fluid between converged walls with the variable viscosity. During the next two decades, researchers, such as Hull and Pearson^[5], Durban^[6], Brewster et al.^[7], and Bird et al.^[8], used analytical methods to obtain approximate solutions for the converging incompressible slow viscoelastic fluid flow through cones and wedges. Brewster et al.^[7] found that the flow possessed the outer region in the presence of complex boundary layers with transitions at the walls using explicit solutions. Bird et al.^[8] found that, in the case of two-phase flow, the outlet flow area resistance was beyond the range of other regions where the fraction value of air bubbles was maximal. They also managed to control the air mass ratio at the end of the cone channel to reduce the flow resistance in the region. Shortly thereafter, Voropayev et al.^[9], Rahimi et al.^[10], and Nofal^[11] found similarity solutions for the two-dimensional steady state incompressible viscous flow problem of non-Newtonian fluid inside the convergent and divergent cones with and without source/sink based on the power-law model. Rahimi et al.^[10] also considered the wall friction and found that the pressure gradient was a key parameter in the Newtonian and the non-Newtonian fluid flow behaviors, and good agreement between the lubrication theory and asymptotic solution was found for small cone angles and small friction parameter values.

Additional two phenomena that may be relevant in the literature context are the magneto-hydro-dynamics (MHD) and the electroosmotic flow. A quick overview for these issues will be brought here.

In 1970, Girishwar^[12] solved analytically the steady incompressible axisymmetric flow of electrically conducting-viscous fluid between two concentric rotating cylinders composed of an insulating material under the influence of radial magnetic field. He found that the velocity field component values were decreased compared with those appearing in the classical hydrodynamic, and the tangential velocity became fully developed for the relatively small axial distance due to the presence of magnetic field. However, it required a greater axial distance for larger Reynolds numbers. About two decades later, Timol and Timol^[13] developed a three-dimensional model of an incompressible viscous flow over a semi-infinite plate under the influence of the magnetic field and a pressure gradient with or without suction/injection cases, using the appropriate similarity transformations. Recently, Makinde and Mhone^[14] claimed that a small divergence of walls might cause the non-stability behavior, while a small convergence had a stabilizing effect for the case of small disturbances in the Jeffery-Hamel (J-H) flow type including the MHD effect when magnetic number values were relatively small. Moreover, they found that an increase in the magnetic field intensity had a strong stabilizing effect on both diverging and converging channel geometries. Similarly, Sadeghy et al.^[15] found analytically that the parameters (the Reynolds number, the Weissenberg number, the half-angle, and the magnetic number) had crucial effects on the velocity profile. Moreover, the magnetic field played a main role in delaying flow dependency on fluid's elasticity when the magnetic force was found to suppress separation in diverging channels. Two years later, Makinde et al.^[16] found analytically using Bessel and Lommel functions that the axial velocity in the axisymmetric J-H flow through the porous medium, decreased significantly with an increase in the Hartmann number, and the tangential velocity was found to decrease due to the increase in the magnetic field. Other recent studies that were produced by Alam and Khan^[17], Shrama and Singh^[18], Dib et al.^[19], Hayat et al.^[20], and Usman et al.^[21] concentrated on external MHD field effects on the steady two-dimensional nonlinear flow through convergent/divergent channels of viscous incompressible electrically conducting fluid. Alam and Khan^[17] investigated the instability range limits of the solution using the power series that was based on Padé-Hermite approximations, and they found the limit margin for each MHD pattern. In addition, Shrama and Singh^[18] concentrated on the natural convection flow with variable electrical conductivity and heat generation along an isothermal vertical plate. Their main findings indicated that the fluid velocity was increased in the presence of heat generation volumetric rate or due to the increase in the electrical conductivity parameter. Additionally, the fluid temperature was found to increase in the presence of heat generation volumetric rate or due to the increase in the magnetic field intensity whereas it was decreased due to the increase in the electrical conductivity parameter (not much effective). Advanced analytical solutions based on the Adomian decomposition method, the numerical solution, and the homotopy method were obtained by Dib et al.^[19], Hayat et al.^[20], and Usman et al.^[21], respectively. Hayat et al.^[20] found that variable viscosity and thermal conductivity tended to increase the pressure gradient in the case of a mixed convection peristaltic flow of electrically conducting fluid in an inclined asymmetric channel when the fluid viscosity and the thermal conductivity were assumed to vary linearly as a function of temperature. Likewise, the viscosity parameter, the Grashoff number, and the inclination angle had similar effects on the velocity while the velocity wasn't affected by the change in the thermal conductivity parameter. Additionally, they found that the variable viscosity tended to decrease the fluid temperature and the viscosity parameter, and the inclination angle had similar effects on the heat transfer rate near the wall.

Electrostatics has a tremendous effect on many aspects of our life, such as mechanical, biological, and electrical engineering. In 1999, a research that considered the biological aspect was performed and presented by Fogolari et al.^[22]. The researchers examined electrostatics phenomena in the context of biomolecular systems, which are linearized forms of Poisson-Boltzmann equations (PBE) (LPBE). They compared PBE solutions with LPBE solutions and their application to biomolecular systems. Three years later, Chen and Santiago^[23] published their work on optimal electroosmotic micro-pumps of pressure capacity and flow rate alongside high thermodynamic efficiency between parallel plates based on the field-induced ion drag. The researchers found that their model behavior coinciding with experimental results of actual electroosmotic micro-pump test showed a linear ratio between the pressure and the flow rate. In addition, setting up high electric field strength values caused the flow rate to increase with the electrical Joule heating, which yields an upper limit for the operating voltage. Similarly, six years later, Berli and Olivares^[24] found that nonlinear effects induced by the shear-dependent viscosity of non-Newtonian fluid are limited to the pressure-driven component of the flow. Moreover, they found a mutual linkage between electroosmosis and streaming current when considering wall depletion effects of colloids in polymer solutions. In 2008, Zhao et al.^[25] made their own analytical contribution to the electroosmotic flow of power-law fluid through a slit channel. They found that the dynamic viscosity was decreased monotonically from the centerline to the wall in the case of pseudoelastic. However, in the case of dilatant fluid, fluid is in-viscid at the centerline, and then the viscosity increases gradually. Moreover, at the wall, the dynamic viscosity values are close to the unit one, and the velocity near the center region of the channel approaches the generalized Smoluchowski velocity regardless of the power law index values. However, in the case of pseudoplastic fluid, the effect of electrical double layer thickness parameter plays a main role in the Smoluchowski velocity compared with that in dilatant fluid. In the same year, Bohinc et al.^[26] published their analytical study on the electrolyte solution inside the cylindrical geometry with charged surfaces. They found that the absolute value of the electrostatic potential as well as the concentration decreases with the increasing distance from the charged surface. One year later, Afonso et al.^[27] and later Ghosal^[28] built an analytical model for the micro-and nano-fluidic viscoelastic flow between parallel plates, and analytical solutions were developed under the combined influence of electro-kinetic and pressure forces with the Debye-Hückel approximation.

In 2010, Vasu and De^[29] published their papers on the electroosmotic flow of power-law fluid in a rectangular microchannel at the high zeta potential. They developed the electrical double layer potential distribution without assuming the Debye-Hückel linear approximation. They found that average velocity values of pseudoplastic fluid were larger in the microchannel than that of dilatants for the same given operating conditions. One year later, Zhao and Yang^[30] developed an exact solution for the electroosmotic flow of non-Newtonian fluid inside the parallel plate microchannel in terms of hypergeometric functions alongside the new experimental scheme method for determining the rheological properties based on the generalized Smoluchowski velocity. They also found that either increasing the electro-kinetic parameter or decreasing the power law index leads to many plug-like velocity profiles where the local velocity distribution, the average velocity and the generalized Smoluchowski velocity coincide for the specific case. In 2014, Sherwood et al.^[31] gave the analysis on end in the context of the finite circular cylindrical pore that transverses a membrane with the specific given thickness including the electroosmosis phenomenon when the membrane outer surfaces are charged. They found that, if the membrane thickness is relatively larger than the cylindrical pore radius, then the end effects can be neglected, and both calculation results of infinite and finite cylindrical pores coincide.

Now, three combined models of both MHD and electroosmotic flow studies are presented. In 2011, Bhattacharyya and Layek^[32] conducted a research study on the MHD boundary layer flow of an electric conducting incompressible non-Newtonian dilatant fluid flow in a divergent channel with suction or blowing states when channel walls are designed in the porous shape, and the electric current was applied along the channel intersection line resulting with the magnetic field. They found that the boundary layer flow of the dilatant fluid may be generated without being separated if only the appropriate suction or blowing velocity is being used through its porous walls. Moreover, they showed that this kind of flow can even exist without the magnetic field. Moreover, it is found that the boundary layer flow may occur even in the presence of blowing. About three years later, Escandón et al.^[33-34] presented their study on temperature distributions in a parallel flat plate microchannel with composed steady electroosmotic and magneto-hydrodynamic driven forces. The analysis^[33] was based on the Phan-Thien-Tanner model, and their main conclusion was that the undesirable Joule heating in the microchannels can be reduced by a combination of both forces. In addition, the researchers^[34] found that the limits of small Hartmann numbers and low electrical conductivity in the buffer solution correspond to the range where the electric and magnetic effects can be used to move a charged solution in the flow control.

The current paper develops the J-H flow of non-Newtonian fluid with nonlinear viscosity and wall slip conditions based on the previous studies (see Refs. [35]-[37]) inside a convergent wedge. Unlike the usual power law model, this paper develops nonlinear viscosity which is dependent only on the tangential coordinate function.

2 Problem formulation model

Consider a steady two-dimensional flow of incompressible viscous fluid inside a convergent wedge (non-parallel walls) with friction on the walls. The cylindrical polar coordinates (r, θ, z) are selected to model the flow system as shown in Fig. 1, and the flow is intersected by the z-axis. It is assumed that the motion is purely radially dependent on r and θ only. Moreover, there are no changes with respect to the z-axis. The governing equations of motion are elaborated here as follows. The equation of continuity can be written as^[38]

(1)

where V, ρ, and u_r are the velocity vector, the constant fluid density, and the radial velocity component, respectively.

Now, according to the general equation of momentum, it can be written as

(2)

where F_B and τ are the body forces and stress vectors, respectively. Body forces are assumed to be neglected. The following viscosity terms are dependent on the tangential direction function:

(3)

where u_θ = u_z = 0, and (no velocities in the perpendicular and tangential axes). The general velocity gradient (∇V) can be represented by the Jacobian matrix as shown in Ref. [39]. Also, μ is the dynamic viscosity function, where μ₀ and n represent the material parameter and the flow consistency index, respectively. As introduced by Zhao et al.^[25], Zhao and Yang^[30] and Rahimi et al.^[10], the apparent (effective) viscosity is defined as a function of the shear rate. However, in this current study, the apparent viscosity will be assumed to be dependent on the tangential direction only. Usually, the apparent viscosity is defined as a function of the shear rate including the radial direction. However, the study will examine the nonlinear viscosity as a function of the tangential direction regardless the radial direction since the geometry effects that will be considered are in the tangential direction. Substituting the above relationships (3) into Eq. (2) results in the following radial and tangential displacement equations:

(4)

(5)

The equations of momentum are derived by using the fully cylindrical coordinates flow equations as developed by Membrado and Pacheco^[40]. Hence, Eq. (4) is obtained as

(6)

Differentiating Eq. (6) along the tangential direction leads to

(7)

Multiplying Eq. (5) by r together with differentiation performance along the radial direction yields

(8)

Therefore, by substituting Eq. (8) into Eq. (7), the following nonlinear differential equation is obtained:

(9)

with the slip boundary conditions as follows:

(10)

where m ≥ 0 is the friction coefficient factor. These conditions represent the relative slip friction between the fluid and the wall. A smooth boundary is described by m = 0, and the perfectly rough wall is obtained by assuming m = ∞. Similar studies on the wall friction condition (10) are available for a variety of non-Newtonian fluid such as elasto-viscoplastic and structural fluid^{[35, 41-43]}. The third condition is joined to the other two conditions (10), which provides^[35]

(11)

where A represents the cross sectional radial surface area of the flow field with the element area A = rdθ within the deformation zone. α represents the wedge semi-angle where the flow field domain is confined between rigid rough walls (θ = ± α) that enable the slip condition. Q is the steady state planar flow rate. In order to simplify the model, the following transformation will be used using Eq. (1) and the physical description as shown in Fig. 1:

(12)

Thus, substituting Eq. (12) into Eq. (9) yields

(13)

Then, multiplying both sides of Eq. (13) by r³ (r ≠ 0) and the density ρ and dividing Eq. (13) by Q (Q ≠ 0) yield

(14)

Assuming leads to the following nonlinear differential equation:

(15)

Substituting Eq. (3) into Eq. (15) yields

(16)

Now, in order to simplify solutions, two suggestions for g(θ) will be introduced,

(17)

Note that the case of the constant g function was solved and discussed by the author in the context of nano-fluid (see Ref. [44]). According to Eq. (17), substituting g(θ) into Eq. (16) yields the following two equilibrium states:

(18)

(19)

where , and the appropriate forms of conditions (10) and (11) are

(20)

(21)

Due to the flow field symmetry assumption, the condition (20) is simply written as

(22)

In the next section, an approximate analytical solution to the problem will take place for special cases.

3 Approximate solutions

In this section, an analysis of Eqs. (18) -(19) will be discussed here for an extreme parameter value. Moreover, Eqs. (18) -(19) will be solved numerically using MATLAB. The numerical procedure based on the initial value problem (IVP) will be introduced here. Firstly, Eq. (18) will be examined. Suppose that we have Newtonian fluid (n=1) where the inertia effect is neglected (Re → 0 or µ → ∞), and then the obtained differential equation is

(23)

The appropriate solution to Eq. (23) is

(24)

with the appropriate constants

(25)

Now, assume that the half cone angle is small enough such that 0° < θ ≤ 20°, and then by using the Taylor series, the exponential function can be expanded and approximated by the following relationship:

(26)

Note that the numerical error is smaller than 10^-2. Accordingly, the following nonlinear differential equation is obtained:

(27)

Equation (27) cannot be solved analytically without far reaching approximations but only by the complex series solution and will not be discussed here. We will turn to develop semi-analytical approximate solutions to Eq. (19) based on the series solution. Firstly, suppose that we have non-Newtonian fluid (n > 0, n ≠ 1) which fulfills d → ∞,

(28)

Now, Eq. (27) can be separated into two distinct equations by the following conditions:

(29a)

(29b)

The solution to Eq. (29a) under conditions (21) -(22) is

(30)

where

(31)

Note that the friction coefficient value is also determined and dependent on the wedge semi-angle.

In the case of Newtonian fluid (n=1) and d ≠ 0, Eq. (19) turns to be

(32)

Note that Eq. (32) has no dependency on the constant d. Integrating Eq. (32) along the θ-direction leads to the following equation:

(33)

where c₁ is a constant. Now, numerically and analytically approximate solutions to Eq. (33) will be presented. Suppose that f² ≈ ± f and 0° < θ ≤ 20°, then, the final solutions, including homogenous and particular parts, are

(34a)

(34b)

Now, the constants A₁, A₂, and c₁ will be determined by applying conditions (21) -(22) to Eqs. (34a) and (34b),

(35)

In the case that the Reynolds number becomes zero, Eqs. (34a) and (34b) turn to be the solution (24). Another method for solving Eq. (32) will be performed without using the assumption (f² ~ ± f, 0° < θ ≤ 20°). The following method for solving Eq. (32) is based on the following series:

(36)

where the previous assumption (f²≈ ± f, 0° < θ ≤ 20°) is not considered. Substituting Eq. (36) into Eq. (32) yields

(37)

Applying subscripts manipulations leads to

(38)

The coefficient development of Eq. (38) for the first eight members including conditions (21) -(22) applied to Eq. (35) yields

(39)

with five unknown constants a₀, a₂, a₄, a₆, and a₈. Note that the constant value c₁ is determined by Eq. (37), but does not participate in the solution process. The first eight coefficients are sufficient to use in order to obtain accurate solutions, especially for small wedge semi-angles as shown in the next section. The MATLAB program will be used in order to have a closed solution by using the FSOLVE function. In the next section, comparison will be performed between the approximate analytical solutions and the previous results in the literature.

In the case that n and s ≠ 1, the obtained equilibrium is derived by substituting the series solution substitution (35) into Eq. (29b),

(40)

The coefficient development of Eq. (40) for the first eight members including conditions (21) -(22) applied to (36) yields

(41)

The suggested parabolic approximations which fulfill boundary conditions (21) -(22) are brought as follows^[35]:

(42)

where α ≠ 0.

4 Results and discussion

In this section, parametric investigations based on solutions (24), (34), (39), and (41) -(42) through illustrative results will be presented and discussed. Solutions (24) and (41) -(42) will be discussed separately. Initially, suppose that the following parameters are considered: α =2.5°, and Re=3. According to the Newtonian case (n=1) where the variable viscosity behaves according to Eq. (17b) when d ≠ 0, two kinds of flow field solutions have been obtained, i.e., the analytical solutions (34a)-(34b) and the numerical solution (39), respectively. Seemingly, the normalized Newtonian velocity (f) is found to decrease gradually with the tangential direction progress as shown in Fig. 2. Moreover, the increase in the friction coefficient (m) causes the normalized Newtonian velocity profile values to decrease as shown in Fig. 2. Here, the case that the friction coefficient (m) value becomes large enough to represent "infinity" is for m=10⁶, and the condition (22) becomes the no-slip boundary condition f(α)=0. Moreover, the obtained solutions (34a), (34b), and (39) coincide for the specific data. The obtained solution has good agreement with Refs. [17] and [19]. The opposite extreme case that m=0 or f'(α) = 0 is illustrated in Fig. 2 with the horizontal line parallel to the θ-axis, and good agreement is found with Ref. [35].

However, the increase in the wedge semi-angle (α) value causes the normalized Newtonian velocity profile to decrease as shown in Fig. 3, good agreement is found between solutions for the small wedge semi-angle, and the most accurate solution is the numerical (semi-analytical) solution (39).

The initial observation on the Reynolds number (Re) in Fig. 4 shows that the normalized Newtonian velocity profile values increase as long as Re increases for the chosen parameters (m=100, and α =10°).

Additionally, for the specific extreme parameters choice (Re=250, α =20°), when the Reynolds number and the wedge semi-angle are relatively large, the normalized Newtonian velocity (34b) in Fig. 5 does not show physical behavior.

Now, we will turn to discuss on another comparison between the parabolic approximation (42) and the Newtonian flow solution (24). The observation shows that both solutions have no dependency on the Reynolds number and coincide for each value of the friction coefficient (m) as shown in Fig. 6. Similarly, both solutions profiles decrease as long as the friction coefficient increases as shown in Fig. 6. Moreover, like before, it is also obtained that the increase in the wedge semi-angle value (α) causes the velocity profile to decrease, as shown in Fig. 7.

Finally, the last case that will be discussed here deals with non-Newtonian fluid with nonlinear viscosity that behaves according to (29b) and fulfills d → ∞. It is found that the variable viscosity power constant (s) has no effect on the solution. In addition, relatively large values of the wedge semi-angle and friction coefficients have significant effects on the normalized velocity profile, as shown in Figs. 8 and 9.

In conclusion, it seems that both non-Newtonian and Newtonian flows have similar qualitative behaviors, and the nonlinear viscosity term has influential effects on the flow.

5 Concluding remarks

The J-H flow model of the non-Newtonian flow type inside a convergent wedge (inclined walls) with the wall friction is derived and expressed with a nonlinear ordinary differential equation with appropriate boundary conditions based on similarity relationships. Unlike the usual power law model, this paper develops the nonlinear viscosity which is based only on the tangential coordinate function. Two kinds of solutions are developed, i.e., analytical and semi-analytical (numerical) solutions. However, the analytical solution is developed for Newtonian and non-Newtonian cases, and the non-Newtonian analytical solution relies on the assumption that f² ~ ± f and 0 < θ ≤ 20°. On the other hand, the non-Newtonian semi-analytical solution is not developed by using the latter assumption. In addition, the parabolic approximation is used to compare between the Newtonian analytical solution and the certain approximation.

Parametric investigations including comparison between various solutions are performed. It is found that the normalized Newtonian velocity decreases gradually with the tangential direction progress. Also, the increase in the friction coefficient (m) causes the decrease in the normalized Newtonian velocity profile values. Additionally, for small values of wedge semi-angle, the present solutions are in good agreement with the previous results in the literature. The increase in the wedge semi-angle value causes the decrease in the normalized velocity profile, although the increase in the Reynolds number (Re) causes the normalized velocity function values to increase. Additionally, for the specific parameter choice, when the Reynolds number or the wedge semi-angle is relatively large, the normalized velocity behavior based on the analytical solution is not found to be suitable for the solution.

Moreover, the parabolic approximation shows excellent agreement with another Newtonian flow solution, and both solutions have no dependency on the Reynolds number. Similarly, both solutions profiles decrease as long as the friction coefficient (m) or the wedge semi-angle value (α) increases.

Finally, the specific case of the non-Newtonian fluid with nonlinear viscosity behavior is examined. It is found that the relatively large value of wedge semi-angle or friction coefficient has significant effects on the normalized velocity profile.

To sum it up, both non-Newtonian and Newtonian flows have similar qualitative behaviors, and the nonlinear viscosity term has influential effects on the flow.