Nomenclature
x, y,                                        dimensional spatial coordinates;
dp/dx,                                     pressure gradient;
U_c,                                         velocity of upper wall;
U,                                           velocity vector;
τ,                                            deviatoric stress tensor;
U_b,                                        dimensional streamwise velocity;
e_x,                                          unit vector of streamwise direction;
K,                                           consistency;
τ_xy,                                        component of stress tensor;
τ₀,                                         characteristic stress/pressure;
τxy,                                        dimensionless component of stress tensor;
C_o,                                        Couette number;
C_i,                                          integral constants;
C_oi,                                        critical Couette numbers;
C_ou, C_{o d},                          critical Couette numbers;
x, y,                                        dimensionless spatial coordinates;
h,                                           half channel width;
ρ,                                           density;
p,                                           pressure;
γ˙ ,                                         deviatoric strain rate tensor;
V_cf,                                       injection/suction velocity;
e_y,                                         unit vector of normal direction;
γ˙_ij                                        components of strain rate tensor;
n,                                           power-law index;
U₀,                                       characteristic velocity;
U_b,                                        dimensionless streamwise velocity;
R_cf,                                       crossflow Reynolds number;
τ_{xy, 0},                                   constant solution of stress;
y₀,                                        stagnation point of velocity;
U_{b, 0},                                    linear solution of velocity;
a, b,                                       combinations of R cf and Co. 1 Introduction

The Couette-Poiseuille flow,as a fundamental problem,has been widely studied both theoretically and experimentally for a long time. For a Newtonian fluid,a linear velocity profile for a Couette flow and a parabolic velocity profile for a Poiseuille flow can be derived with no difficulty. However ,significant variations of these flows emerge when non- Newtonian effects are taken into account . During the last several decades,there were plenty of studies on the Couette-Poiseuille flow with various non-Newtonian fluid models. Bittleston and Hassager ^[1] studied the circular Couette-Poiseuille flow for the Bingham fluid with the combination of wall rotation and imposed pressure gradient. Filip and David^[2] and Liu and Zhu^[3] studied the axial Couette-Poiseuille flow for the Robertson-Stiff fluid and the Bingham fluid,respectively. Kalyon and Malik^[4] focused on the axial Poiseuille flow of a Herschel-Bulkley fluid subjected to the wall slip condition. Ferr´as et al.^[5] presented a set of analytical solutions for both Newtonian and inelastic non-Newtonian fluids (including power-law,Bingham,Herschel-Bulkley, etc.) in the Couette-Poiseuille flow with planar geometry under various slip boundary conditions. Hayat and Javed^[6] derived the exact solution of peristaltic flows of a power-law fluid in an asymmetric channel with compliant walls. In these studies,non-Newtonian effects (including shear-thinning,shear-thickening,etc.) play an important role in the flow distribution.

It is well-known that injection/suction through a porous wall is able to provide an effective method for the control of wall bounded flows,which drives studies on the plane CouettePoiseuille flow combined with a uniform crossflow. A uniform crossflow will emerge while fluids are introduced into the flow field through one porous side wall and out of the flow field through another porous side wall with an identical velocity. For Newtonian fluids,analytical solutions were given by Guha and Frigaard^[7]. For Bingham fluids,Tsangaris et al.^[8] worked on the plane Couette-Poiseuille flow between two equally porous parallel plates,which provided a homogenous crossflow throughout the flow field. Chen and Zhu^[9] extended the work of Tsangaris et al.^[8] to a general situation where the slip effect at the porous walls was considered.

In this paper,the plane Couette-Poiseuille flow of a power-law fluid with a uniform crossflow is investigated,focusing on the effects of shear thinning/thickening features. It is an extension of the work of Guha and Frigaard^[7] for Newtonian fluids. For power-law fluid flows,the main difficulties are the nonlinearity of the constitutive equation and the existence of multiple possible cases of flows. Therefore,an important task in this paper is to assign the specific regions of the parameter plane to individual flow cases based on the exact solutions (see Refs. ^{[1, 3, 8, 9]}). In this way,the precise restriction about each flow case will be obtained in the phase diagram clearly. Moreover,it is worth noticing that these restrictions exhibit high dependence on the non-Newtonian effects.

The rest of this paper is organized as follows. In Section 2,the problem is established mathematically by proposing the geometry,governing equations,and boundary conditions. In Section 3,detailed solution procedures and exact solutions for each specific flow case are carefully derived. The results about distinguishing individual flow cases in the parameter plane are presented,verified,and discussed in Section 4. The conclusions of this paper are summarized in Section 5. 2 Problem establishment

As shown in Fig. 1,we consider an incompressible steady plane channel flow of a power-law fluid between two infinite parallel porous plates,where the equal injection/suction velocity V_cf is applied at the lower and upper walls and a uniform y-direction crossflow is imposed throughout the flow field. In the streamwise direction (x),the flow is driven by a constant negative pressure gradient (dp/dx < 0) and the wall movement. The upper wall y = h moves at a constant speed U_c while the lower one y = −h remains stationary. All dimensional variables are denoted by an over-bar.

The governing equations include the continuity and momentum equations as follows:

where ρ,U,P,and τ denote the density,the velocity vector,the pressure,and the deviatoric stress tensor of the fluid.

Let the y-component of the velocity be constant and the x-component of the velocity depend only on y . Then,the velocity vector U in the plane Cartesian coordinate system (x,y) can be expressed as

where e_x and e_y denote the unit vectors of the streamwise and normal directions,respectively.

The constitutive relation for power-law fluids is

where K and n are defined as the consistency and the power-law index of the power-law model, respectively,and τ and γ˙ represent the deviatoric stress tensor and the strain rate tensor, respectively. |γ˙| is the absolute value of the second invariant of the strain rate tensor defined by

The fluid exhibits shear-thinning or shear-thickening features when

The expression of γ˙ is

According to the velocity expression (2),the continuity equation (1) and the y-direction momentum are naturally satisfied,while the momentum equation in the streamwise direction is reduced to the following one:

where the stress term is

The following non- slip boundary conditions are applied at the wall:

Nondimensionalization is supposed to be necessary before the problem is well established. Based on this purpose,we use the following dimensionless variables for scaling:

where U₀ is the maximum velocity of the plane Poiseuille flow for the power-law fluid expressed by

and τ₀ = K(U₀/H₀)ⁿ. Then,we can obtain the dimensionless form of the momentum equation (6) as follows:

the dimensionless form of the constitutive equation (7) as follows:

and the dimensionless form of the boundary condition (8) as follows:

To establish the problem mathematically,three dimensionless parameters associated with the velocity distribution are introduced: the power-law index n,the crossflow Reynolds number (R_cf) defined by

and the Couette number (C_o) defined by

Here,C_o is set to be either a positive number or a negative one,implying that the upper wall may move along or against the streamwise direction. Therefore,we assume that R_cf > 0 during the solution procedure,in view of the fact that the velocity profile for R_cf < 0 can be achieved by an axisymmetric transformation combined with a translation. 3 Solution pro cedures and individual flow cases

In this section,we shall propose detailed solution procedures aiming at all possible n,which serves to be valid for both analytical and numerical solutions. Moreover,we shall deal with several choices of n as examples to show what we can obtain through the theoretical derivations. 3.1 Solution procedures for all n

The integration of (10) with respect to y reveals the relational expression between U_b and τ_xy as follows:

where C₁ is an integral constant. Hence,we turn to find the solutions for τ_xy instead of U_b . Taking the absolute value of both sides of (11),we get

Substituting this back into (11) shows

Substituting (14) into (10) y ields an equation only about τ_xy as follows:

Note that (15) has a constant solution

which implies from either (13) or (14) that the velocity profile is a straight line. The velocity gradient dU_b /dy may be obtained by two ways. One is the combination of (14) and (16),and the other is to utilize the boundary conditions in (12).

Therefore,a necessary condition for the linear velocity profile is

Moreover,we try to prove that (18) is also a sufficient condition. According to (12) and (13),we have

If τ_xy ( −1) > τ_xy,0 ,τ_xy is known to increase monotonically from (15),which indicates

Conversely,if τ_xy ( −1) < τ_xy,0 ,then

Hereto ,(18) is proved to be the necessary and sufficient condition for the linear velocity profile. Substituting (19) into (13) y ields the expression of τ_xy (y) as follows:

As shown in (13),U_b (y) can be obtained right after τ_xy (y). From here,the solution procedures are separated into two different directions. 3.1.1 R_cfC_o > 2( n + 1n )n

At this time,τ_xy (y) > τ_xy,0 is ensured. Then,from (15),we have

Integrating (21) with respect to y reveals

The values of C₁ and C₂ can be obtained with the combination of (19) and (22). Consequently, we obtain the expressions of τ_xy (y) and U_b (y). 3.1.2 R_cfC_o < 2( (n + 1)/n )ⁿ

At this time,τ_xy ( −1) > τ_xy,0 ,and τ_xy decreases monotonically. Thus,τ_xy (y) < τ_xy,0 is ensured,which means

If dU_b /dy |_{y = − 1} ≤ C_o/2,the decrease in dU_b /dy leads to U_b (1) < C_o,which brings out

Now,if y = y_o∈ ( −1,1) and

integrating (15) with respect to y reveals

If τ_xy (1) ≥ 0 and τ_xy (y) ≥ 0,then (22) is applied. If τ_xy ( −1) ≤ 0 and τ_xy (y) ≤ 0,then the lower half of (23) is applied. 3.2 Analytically solvable circumstances for certain n

In consideration of the analytically solvable circumstances of (22) and (23),we notice that the analytic integral can be achieved for certain n,such as n = 1/N or n = N,where N is a positive integer. Hence,several representative values of n including shear-thinning/thickening effects are chosen here to demonstrate individual flow cases accompanied with accurate restrictions concerning both C_o and R_cf so that the specific regions in the parameter plane C_o-R_cf,related to specific cases of the flow,can be recognized. 3.2.1 Newtonian circumstance (n = 1)

From the existing results,(15) can be reduced to

Then,we have τ_xy = dU_b/dy ,which consists of the results of Guha and Frigaard^[7]. Additionally, (18) reduces to R_cfC_o = 4,which accords with the result obtained by Guha and Frigaard^[7] again. 3.2.2 Shear-thinning circumstance (n = 0.5)

Substitute n = 0.5 into (18). Then,when R_cfC_o = 2 √3,there are

According to the solution procedures,we can rewrite (22) as the following form after integration when R_cfC_o > 2 √3:

Then,we can obtain

where

With the help of (20),U_b (y) emerges like

To find C₃ ,we substitute (26) back into (19) and organize the obtained result to get

From (25),we can find that C₂ < 0 and C₃ < 1,which leads to

where

Now,it turns to R_cfC_o < 2 √3,which makes (23) become the following form after integration:

Rearranging (30),we have

U_b (y) will emerge by utilizing (20). To find y 0,we substitute (31) back into (19) first. Then, we can get

Hereto,this flow case is closed. If τ_xy (y) ≥ 0 is valid throughout y ∈ [−1,1],then (22) becomes the following form after integration:

from which we have

where

With the help of (20),U_b (y) emerges like

To find C₅ ,we substitute (34) back into (19) first. Then,we have

From (33),we can find C₄ < 0. Then,we have C₅ < 1,which leads to

where

To satisfy τ_xy (y) ≥ 0,(34) gives

which is,based on (36),equivalent to

Hence,the restriction for this flow case is

where

On the other hand,if τ_xy (y) ≤ 0 is valid throughout y ∈ [−1,1],then the lower half of (23) becomes the following form after integration: 1

from which we have

With the help of (20),U_b (y) emerges like

To find C₆ ,we substitute (42) back into (19) and organize the obtained result to get

To s a tis fy τ_xy (y) ≤ 0,(41) gives

which is,based on (44),equivalent to

Hence,the restriction for this flow case is

where

3.2.3 Shear-thickening circumstance (n = 2)

Substitute n = 2 into (18). Then,we can obtain that when R_cfC_o = 9/2,there are

According to the solution procedures,we can rewrite (22) as the following form after integration when R_cfC_o > 9/2:

With the help of (19),equations about C₁ and C₇ emerge as

Eliminating C₇ gives

With k now n C₁ ,τ_xy ( −1) and τ_xy (1) can be obtained from (19),which induces the relational expression between y and τ_xy as follows:

where

Therefore,the solution of U_b (y) arises from the combination of (51) and (13).

Now,it turns to R_cfC_o < 9/2 ,which make (23) become the following form after integration:

With the help of (19),equations about C₁ and y₀ emerge as

Carrying a similar procedure as the former case,we can obtain U_b (y).

If τ_xy (y) ≥ 0 is valid throughout y ∈ [−1,1],then (22) becomes the following form after integration:

With the help of (19),equations about C₁ and C₈ emerge as

Eliminating C₈ gives

From C₁ ,τ_xy ( −1) and τ_xy (1) can be obtained from (19),which induces the relational expression between y and τ_xy as follows:

where

Therefore,the solution of U_b (y) arises from the combination of (57) and (13).

Different from the circumstance when R_cfC_o > 9/2,an additional restriction is necessary in this case,which is

where Co3(R_cf) satisfies τ_xy (1) = 0.

If τ_xy (y) ≤ 0 is valid throughout y ∈ [−1,1],then the lower half of (23) becomes the following form after integration:

With the help of (19),aligning about C₁ and C₉ emerges

Eliminating C₉ gives

From C₁ ,τ_xy (−1) and τ_xy (1) can be obtained from (19),which induces the relational expression between y and τ_xy as follows:

where τ_xy ∈ [τ_xy ( −1),τ_xy (1)]. Therefore,the solution of U_b (y) arises from the combination of (62) and (13).

Once again,an additional restriction is necessary in this case,which is

where C_o 4(R_cf) satisfies τ_xy ( −1)=0. 4 Result s and discussion

Since the solution procedures suitable for the numerical approach at all possible n and the analytical solutions at certain representative n are derived in detail,we shall present the velocity profiles through both the numerical and analytical approaches for comparison. Moreover,five cases are considered with different value ranges of C_o and R_cf,and the distributions of individual flow cases on the parameter plane C_o-R_cf are determined at different n. Eventually,some particular features existing on the phase diagram are proposed and discussed. 4.1 Shear-thi nni ng ci rcum stance (n = 0.5)

Based on Section 3.2.2,the following five flow cases are demonstrated:

Case I Convex velocity profile

Case II Linear velocity profile

Case III Concave velocity profile (the maximum speed is at the moving wall)

Case IV Concave velocity profile (the maximum speed is in the channel)

Case V Concave velocity profile (the maximum speed is at the stationary wall)

Here,

To compare the analytical results with the numerical ones,several combinations of C_o and R_cf are chosen. For C_o = 4,Cases I-IV can be revealed by certain R_cf (see Fig. 2). The values 0.05,0.3,0.6,0.866 03,and 1.1 of R_cf refer to Cases IV,IV,III,II,and I,respectively. The heavy line in Fig. 2 is supposed to exhibit a linear velocity profile. Particularly,C_o = 0.5 is set to check Case IV with various R_cf,which includes variations of τ_xy (y) from positive values to negative ones (see Fig. 3). In both figures,the analytical and numerical results are in good accordance. The distributions of all flow cases on the parameter plane C_o-R_cf are given in Fig. 4,where details of the phase diagram will be discussed in the last part of this section.

4.2 Shear-thi ckeni ng ci rcum s tance (n = 2)

Based on Section 3.2.3,the following five flow cases are demonstrated:

Case I Convex velocity profile

Case II Linear velocity profile

Case III Concave velocity profile (the maximum speed is at the moving wall)

Ca se IV C_o ncave velo c ity pr o file (the ma x imum sp eed is in the cha nnel)

Case V Concave velocity profile (the ma x imum speed is at the stationary wall)

Here ,

For C_o = 4,Cases I-III can be revealed by certain R_cf (see Fig. 5). Besides,for C_o = 2.5, Cases I-IV can be revealed by certain R_cf (see Fig. 6). The heavy lines in Figs. 5 and 6,again, are supposed to exhibit linear velocity profiles. Good accordance is found in both figures. The distributions of all flow cases on the parameter plane C_o-R_cf are given in Fig. 7.

4.3 Numerical results for more general cases (n =0.3 and n =1.3)

As show n in the theo r e tica l der iva tio ns,the demarcation line between Cases III and IV meets the requirement of τ_xy (1) = 0,while that between Cases IV and V meets the requirement ofτ_xy ( −1) = 0. Therefore,we can obtain the phase diagram for general n numerically with these requirements. Here,we choose n = 0.3 and n = 1.3 as representatives of general shear-thinning and shear-thickening circumstances,respectively. For C_o = 2.5,Cases I-IV can be revealed by certain R_cf,which are shown in Fig. 8 for n = 0.3 and in Fig. 9 for n = 1.3. Distributions of all flow cases on the parameter plane C_o-R_cf are given in Fig. 10 for n = 0.3 and in Fig. 11 for n = 1.3.

4.4 Verification of demarcation points at R_cf → 0

Substitute R_cf = 0 into (15). Then,Cases I and II disappear,and two critical Couette numbers C_ou and C_od corresponding to the demarcation points of Cases III,IV,and V are available for all n,i.e.,

These values denote the demarcation points for the plane Couette-Poiseuille flow of power-law fluids with no crossflow. For n = 0.5,investigating (64) as R_cf → 0 with the Taylor expansion reveals

which satisfies

For n = 2,investigating (65) as R_cf → 0 with the Taylor expansion reveals

Then,we can obtain

4.5 Details of phase diagram

Here,we discuss the details exhibited in the phase diagram for both n = 0.5 and n = 2. As shown in Figs. 4 and 7,all flow cases occupy certain regions in the parameter plane except Case II,which occurs merely on the solid line. Nevertheless,the dash lines do not correspond to particular cases and serve like demarcation lines only. Limit points which the demarcation lines approach as R_cf → 0 are already demonstrated in the above part. One distinction between the two figures is that the domain belonging to Case IV in Fig. 4 is larger than that in Fig. 7,while the domains for other cases are reduced. This indicates that a concave velocity profile with the maximum speed in the channel is more likely to emerge in the shear-thinning circumstance.

Now,we turn to focus on the dot-dash line appearing in Fig. 4 other than Fig. 7. The vertical dot-dash line,with R_cf = π²/16√3,emerges as an additional restriction of Case V and an asymptotic line for the lower demarcation dash line. In other words,Case V is not valid even for C_o → −∞ when R_cf exceeds a certain value for n = 0.5,while Case V is valid for all possible R_cf as long as C_o is smaller than the critical Couette number for n = 2.

In fact,it is worth noting that the distinctions presented above can also be found in addition to n = 0.5. We make extra efforts to perform more n for the exact solutions and divisions of the phase diagram,such as n = 1/3 and n = 3. The results suggest,which are omitted here for simplicity,that similar distributions can also be found. Particularly,the asymptotic feature emerges when n = 1/3 other than n = 3. Hereto,this asymptotic behavior is supposed to be remarkable distinctions for the shear-thinning circumstance distinguished from the shearthickening one. 5 Conclusions

In this paper ,the exact solution for the plane Couette-Poiseuille flow of a power - law fluid with a uniform crossflow is obtained,which is an extension of the work of Guha and Frigaard^[7] for Newtonian fluids. We present detailed solution procedures suitable for both analytical and numerical treatments with all n,and focus on the exact solutions of two representative choices of n corresponding to shear-thinning (n = 0.5) and shear-thickening (n = 2) circumstances, respectively. The results for Newtonian fluids from Guha an Frigaard^[7] can be reproduced as we set n = 1 in our solution procedures. Unlike a unified expression for the exact solution of the Newtonian circumstance as given by Guha and Frigaard^[7],multiple possible flow cases exist under the shear-thinning and shear-thickening circumstances. For both shear-thinning and shear-thickening circumstances,five different flow cases are studied separately. With precise restrictions about each flow case,the domains belonging to specific flow cases are identified clearly in the parameter plane C_o-R_cf. According to the phase diagrams,one of the five flow cases occupies a larger domain where n < 1 other than n > 1,implying that the flow case is more likely to happen under the shear-thinning circumstance than the shear-thickening circumstance. The demarcation points at R_cf → 0 in the phase diagram,which denote the plane Couette-Poiseuille flow of a power-law fluid with no crossflow,are verified. An additional restriction for the flow case,in which n < 1,is proved to be necessary. Therefore,the asymptotic line exhibited in the phase diagram under the shear-thinning circumstance is found to be an important feature,which is distinguished from that under the shear-thickening circumstance.