On determining Navier's slip parameter at a solid boundary in flows of a Navier–Stokes fluid

While the assumption of the “ no-slip ” condition at a solid boundary is unquestioningly applied to study the flow characteristics of a Navier – Stokes fluid, there was considerable debate among the early pioneers of fluid mechanics, Du Buat, Girard, Navier, Coulomb, Poisson, Prony, Stokes, and others, as to the proper condition that has to be met at a solid boundary due to a fluid, such as water flowing adjacent to the same. Contemporary usage of the no-slip boundary condition notwithstanding, in our previous study [M (cid:1) alek and Rajagopal, “ On a methodology to determine whether the fluid slips adjacent to a solid surface, ” Int. J. Non-Linear Mech. 157 , 104512 (2023)], we outlined a methodology to test the validity of the assumption. In this study, we continue the investigation further by providing a scheme for determining the slip parameter that characterizes the extent of slip, if one presumes that Navier ’ s slip boundary condition is satisfied. We find that depending on whether the volumetric flow rate is greater or less than the volumetric flow rate corresponding to the no-slip case, different scenarios present themselves regarding what transpires at the boundary.


I. INTRODUCTION
While the assumption that the fluid adjacent to an impervious boundary has zero normal component of the velocity with respect to the boundary is physically appropriate, the same cannot be said when it comes to the tangential component of the velocity being zero.The assumption of "no-slip" at a solid impermeable boundary was given the stamp of approval by Stokes 1 for sufficiently slow flows in channels and pipes but has become the mainstay with regard to boundary conditions at a solid impermeable boundary.Recently, in Ref. 2, we have evaluated the status of the no-slip boundary condition and provided an experimental procedure to determine the aptness of the assumption in internal flows of a classical Navier-Stokes fluid.
We have illustrated our methodology by investigating five unidirectional shear flows of a Navier-Stokes fluid: flows in pipes or between parallel plates (cylindrical and plane Poiseuille and Couette flows), and flows past an inclined plane.Interestingly, the criterion that characterizes the no-slip regime uses (easily) measurable experimental quantities such as the viscosity of the fluid, its density, the pressure gradient (pressure drop), the volumetric flow rate, and geometrical dimensions (such as the radius of the pipe, the distance between the planes, the angle of the inclined plane, etc.).
In this study, we consider the same set of special unidirectional flows as studied in Ref. 2, but we focus on the specification of possible flow regimes provided that the criterion for no-slip is not met.Assuming that the conditions (including geometry) of the regimes are unchanged with regard to the five flows studied previously, and that the fluid slips according to Navier's slip (see Ref. 3) that relates the tangential components of the velocity and the normal traction linearly, we are able to determine the exact value of Navier's slip parameter.The slip parameter at the solid boundary is however determined only if the volumetric flow rate is greater than the critical volumetric flow rate, which is the value of volumetric flow rate corresponding to the no-slip condition.If the volumetric flow rate is less than the critical volumetric flow rate associated with the no-slip boundary condition, one of several scenarios is possible (see comments in Sec.VIII).Here, we propose a situation similar to that envisaged by Girard, namely, a layer of the fluid near the wall sticks to it and does not deform, that is the layer behaves as though it were rigid layer.Of course, it is possible that the fluid under consideration is not a Navier-Stokes fluid in the first place.
Let us recall from Ref. 2 that Stokes 1 was far from convinced that the no-slip boundary condition for a fluid flowing past a solid boundary at the point of contact was felicitous in general flows.Several of Stokes' forerunners who devoted their research to mechanics, including Du Buat, Coulomb, Girard, Navier, Poisson, Prony, and others, provided competing assumptions concerning the boundary conditions based on physical considerations, see Goldstein. 4he choice of boundary conditions has significant impact on the character of the flows inside the flow domain, see, e.g., Ref. 5 for analytical solutions in a simple geometrical settings and Refs.6-9 for advantages that Navier's slip and other slipping boundary conditions have, in comparison to the no-slip boundary condition, on rigorous mathematical properties of long-time and large-data weak solutions of initial-and boundary-value problems concerning (internal) flows of incompressible fluids.
The structure of the paper is as follows.In Sec.II, we develop the governing equations, formulate Navier's slip boundary condition, and document that Navier's slip parameter has to be positive.In Sec.III, we provide a method to determine Navier's slip parameter for Poiseuille flow in a pipe.We also study the situation when the Navier's slip parameter cannot be determined.In Secs.IV-VI, we extend the approach to other unidirectional flow problems, namely, plane Poiseuille flow, plane Couette flow, and cylindrical Couette flow.These flows provide more possible scenarios due to potentially different behavior of the fluid at separate parts of the solid boundary, i.e., the lower vs upper plate or the inner vs outer cylinder.Finally, in Sec.VII, we apply the approach to flows down an inclined plane due to gravity.Section VIII contains concluding remarks.The system of governing equations formulated in cylindrical coordinates is given in the Appendix.

II. GOVERNING EQUATIONS AND NAVIER'S SLIP
Our discussion in this paper is concerned with flows of incompressible Navier-Stokes fluids that are characterized by a linear relation between the Cauchy stress tensor and the velocity gradient.Allowing the possibility that the fluid is non-Newtonian such as, for example, a power-law fluid, one could give a different flavor to this study.We intend to do this later as we think the understanding of the boundary conditions is somehow more challenging in comparison to the modeling of rheological properties of the fluid in the bulk as boundary conditions depend on the materials on either side of the boundary.
For incompressible fluids, the system of governing equations for the unknown velocity field v and the pressure (mean normal stress) Here, . is the constant (positive) density, b is the specific body force, l is the constant (positive) dynamic (shear) viscosity, and D stands for the symmetric part of the velocity gradient, i.e., At the solid impermeable part of the boundary of the flow domain, we will assume, besides the condition of impermeability, that the criterion for no-slip (specified later in the case of particular simple shear flows) is not met and that the fluid slips past the boundary according to Navier's slip condition.To formulate these conditions, we need to fix the notation to be used.Let n denote the unit outward normal vector at a given point of the boundary and z s be the projection of a vector z (defined at a point of interest on the boundary) to the tangent plane (constructed at that point of the boundary), i.e., z s :¼ z À ðz Á nÞn.Then, the boundary conditions that we would like to consider take the following form: (2.5) where the nonzero slip parameter j has to be positive as shown next.
If the flow is internal (i.e., it takes place in a fixed bounded container with no inflows or outflows) and we take the scalar product of v and (2.2) (where we set b ¼ 0 for simplicity), integrate the result over the flow domain X, use the relation divT Á v ¼ divðTvÞ À T Á D together with Gauss' theorem, we obtain, after integration of the result over the time interval ð0; tÞ, that Noticing that T is according to Eq. (2.3) symmetric and (2.5) holds, we observe that Tv Á n ¼ ðTnÞ s Á v s .Hence, (2.7) leads to ð (2.8) The second and third terms on the left-hand side, considered as a whole, represent the overall dissipation of the system and in accordance with the second-law of the thermodynamics it should be nonnegative.As the two terms are of different physical nature (the second term corresponds to frictional effects inside the fluid itself, while the third term is due to friction mechanisms on the boundary, i.e., it reflects the properties of both the fluid and the solid), each of them itself is usually assumed to be non-negative, see, for example, Refs.13, 14, and 6 for more details.As a consequence, we observe that the requirement that the second term in Eq. (2.8) is non-negative together with Eq. (2.3) implies that l > 0 and analogously the requirement that the third term is non-negative together with Eq. (2.6) gives that j > 0.
Note that letting (formally) j tend to infinity in Eq. (2.6), we obtain the no-slip condition v ¼ 0: (2.9) 'The fact that j has to be positive will play a key role in our considerations below.

III. POISEUILLE FLOW IN A PIPE
We study the flow of a Navier-Stokes fluid in a cylindrical pipe of infinite length of radius R. A constant flow rate Q is driven by a constant negative pressure gradient c.We assume that b ¼ 0 (no gravity) and v ¼ vðrÞe z .Then, the symmetric part of the velocity gradient D simplifies to [see Eq. (A2)] From the constitutive equation (3.1), we then obtain that The requirement that the velocity is bounded at r ¼ 0 implies that c 1 ¼ 0. Let Q be the volumetric flow rate, i.e., Q ¼ Ð R 0 2prvðrÞ dr.Using Eq. (3.3) with c 1 ¼ 0, we obtain Hence, Subtracting and adding the term cR 2 =ð4lÞ, we get , then v(R) ¼ 0, and there is no-slip.Note that the quantities l, R, c (the pressure drop), and Q can be (easily) measured.
, then it follows from Eq. (3.4) (if the assumptions that the flow is unidirectional, steady, and that the pressure is continuous (constant) across the tube etc. are appropriate) that vðRÞ 6 ¼ 0 and there is slip.Assuming that the fluid slips past the wall according to the Navier's slip condition (2.6), we will be able to show that such a situation is possible only if Indeed, in the studied geometrical setting, the impermeability condition (2.5) is met and (2.6) reduces to Referring to Eq. (3.2) (with c 1 ¼ 0) and (3.4) with r ¼ R, the condition (3.6) leads to As j and ÀcR have to be positive, we see that we can specify j by Eq.
(3.7) only if the denominator in the fraction of Eq. (3.7) is positive, i.e., if Eq. (3.5) holds.Said differently, if then Navier's slip condition (2.6) and the solution of the form (3.4) are not compatible.Below, following the ideas of Girard, 15,16 we provide one possible scenario of how the solution could look like in the case when the pressure gradient c and the volumetric flow rate Q satisfy (3.8).
Let us assume that (3.8) holds.This also means that Q and c (as well as l and R) are fixed.Let us consider R 2 ð0; RÞ that will be specified later.We look for flow that is at rest near the wall in the region characterized by the condition r 2 ð R; RÞ and that solves the unidirectional shear flow problem with no-slip condition for r satisfying 0 r < R.This means that for r satisfying 0 r < R, the solution is of the form [compare with Eq. (3.4)] We will now determine R by requiring that rvðrÞ dr: This leads to It is easy to check that the condition R < R coincides with Eq. (3.8).To conclude, assuming that an incompressible Navier-Stokes fluid with shear viscosity l flows in a pipe of radius R as a steady unidirectional simple shear flow with a given volumetric flow rate Q and the pressure gradient Àc we investigate the case when the no-slip condition is not fulfilled.We have found that 4 and if we assume that the fluid slips past the boundary according to Navier's slip boundary condition, we have been able to specify the exact value of Navier's slip parameter j, see formula (3.7).
, the fluid cannot meet Navier's slip boundary condition (as otherwise Navier's slip parameter j would be negative, which contradicts the second law of thermodynamics and the fluid would flow near the wall in the opposite direction to the direction in the inner part of the pipe).Such a solution does not seem to be in keeping with physics.
, the following scenario is possible: the fluid near the outer cylinder is "stuck" to the wall and the whole layer of the thickness R À R together with the wall behaves as a rigid body.In the inner region when r satisfies 0 r < R, the fluid flows according to formula (3.9).The precise value of R is determined from the given data l, R, Q, and Àc (and the natural assumption that the fluid exhibits no-slip when r ¼ R), see formula (3.10).
The behavior described above seems plausible, see also Fig. 1 for such a viewpoint.

IV. PLANE POISEUILLE FLOW
In this case, the flow of a Navier-Stokes fluid is supposed to take place between two parallel plates located at y ¼ 0 and y ¼ h.The effect due to gravity is neglected, i.e., b ¼ 0. The flow with a flow rate Q is driven by the constant pressure gradient c that is negative.Also, we assume that v ¼ uðyÞi which gives a very simple form for D and this together with the constitutive equation ( 2 where b and d are arbitrary constants, while c is the negative constant representing the pressure gradient.The constitutive equation du dy ¼ s l then leads to In what follows, we assume that both the plates are made of the same material and the fluid at both the upper and the lower plate interacts in a like manner.It is possible that the interaction at the top and bottom plates could be different, and an example of the same is the consequence of the roughness of the two plates being different or one of the plates being hydrophobic and the other hydrophilic and the fluid in question being water.We discuss such a possibility in Sec.V when analyzing plane Couette flow.
In the given situation when the plate-fluid responses are identical at the upper and lower plates, it is reasonable to assume that the velocity profile will be symmetric with respect to the line y ¼ h=2.Then, it is natural to assume uðhÞ ¼ uð0Þ: (4.4) We could alternatively impose the condition u 0 ðh=2Þ ¼ 0. Applying the condition (4.4) to the formula given in Eq. ( 4.3), we observe that Substituting this in Eq. ( 4.3) and requiring further that we obtain Hence, This implies that , then uð0Þ ¼ 0, u(h) ¼ 0 and the solution of the form exhibits no-slip on the lower and the upper plates.If Q 6 ¼ À ch 3 12l , then it follows from Eq. (4.7) that uð0Þ 6 ¼ 0 and uðhÞ 6 ¼ 0 as well.If Q > À ch 3 12l , then we will be able to determine the (positive) Navier's slip parameter j appearing in Eq. (2.6).Indeed, assuming that the fluid slips via Navier's slip condition (2.6), which in the considered geometrical setting takes the form uð0Þ ¼ sð0Þ j and uðhÞ ¼ À sðhÞ j ; then, using these conditions together with Eqs.(4.2), (4.5), and (4.7), we obtain which stays positive only if 12l , then the form (4.7) seems to be nonphysical as u is negative in the vicinity of the plates, which means that the fluid should flow against the pressure gradient.This is why we provide a different scenario, in the spirit of the one presented in Sec.III.
We assume that the fluid near the plates, more specifically, in the layers characterized by y 2 ð0; h=2 À hÞ and by y 2 ðh=2 þ h; hÞ, is stuck to the plates, and the layer and the wall behave as a rigid body at rest.The precise value of h 2 ð0; h=2Þ will be specified later.Then, for the channel characterized as the set where y 2 ðh=2 À h; h=2 þ hÞ, the solution is the same as the solution corresponding to no-slip boundary conditions on the plates y ¼ h=2 6 h.Comparing it with Eq. (4.8), we observe that the solution is of the form This leads to the following formula for the flow rate: where we have used the substitution y ¼ h=2 À z.The above formula then gives Note that the requirement h < h=2 is satisfied due to our assumptions concerning Q, c, h, and l.
To summarize our findings, setting Q crit :¼ À ch 3 12l , we have identified three types of solutions are possible assuming that the fluid is a Navier-Stokes fluid and the same type of boundary conditions hold on the lower and the upper plates; this choice of the boundary conditions implies that the solution we have been seeking is symmetric with respect to the channel axis.If Q ¼ Q crit , then, the fluid satisfies the noslip boundary condition on the plates.If Q > Q crit , then, assuming Navier's slip boundary condition on the plates, we determine the value of the slip parameter j, see formula (4.9).Finally, if Q < Q crit , then, assuming that fluid near the plates adheres to them, the layers together with the plates behave as rigid bodies and in the inner cylinder the fluid flows as a Navier-Stokes fluid subject to no-slip boundary condition at the interface, and we can determine the precise thickness of the layers, see Eq. (4.11).Visualization is omitted as it would be very similar to Fig. 1.

V. PLANE COUETTE FLOW
Similarly as in Sec.IV, the flow of a Navier-Stokes fluid takes place between two parallel plates located at y ¼ 0 and y ¼ h, the flow being engendered due to the application of a shear stress on the upper plate.Again, b ¼ 0 and v ¼ uðyÞi, but the pressure p is supposed to be constant.It then follows from Eq. (4.1) that divT ¼ ðs 0 ; 0; 0Þ where 0 denotes d dy through the whole section.We assume that a shear stress s app is applied on the fluid by moving the upper plate in contact with the fluid (this being equal and opposite to the shear stress exerted by the fluid on the plate), i.e., sðhÞ ¼ s app : (5.1) Without loss of generality, we can assume that s app > 0. Due to s app , the upper plate moves with velocity V that can be (easily) measured.As in Sec.IV, we assume that we know the volumetric flow rate Q and it is a fixed positive constant.Then, Hence, In the rest of this section, we will discuss in detail the following three situations: Then, it follows from Eq. (5.4) that uð0Þ ¼ 0, and there is no-slip on the lower plate.Looking then at the upper plate moving with velocity V, we observe that if u(h) ¼ V, then there is no-slip also on the upper plate.This [i.e., u(h Using Eqs.(5.3) and (5.4), we conclude that Note that in this case, the fluid exhibits the no-slip on the lower plate and Navier's slip with j h given in Eq. (5.7) on the upper plate.
we assume that the fluid sticks to the upper plate, and the whole structure moves as a rigid body with speed V of the upper plate in the layer characterized by y 2 ð h; hÞ, where h 2 ð0; hÞ will be specified later.This means that u(y) ¼ V for y 2 ð h; hÞ.On ð0; hÞ, the solution is linear [see Eq. (5. 2)] and satisfies the no-slip boundary conditions uð0Þ ¼ 0 and uð hÞ ¼ V. Hence, uðyÞ ¼ Vy= h for ð0; hÞ.The precise value of h is determined from the knowledge of the flow rate Q.Indeed, which implies that h ¼ 2ðVh À QÞ=V: while the requirement that h > 0 follows from Vh > Q, which is a natural condition on the data (Vh is the maximal flow rate capacity of the channel if the upper plate moves with velocity V, the lower plate is at the rest and the motion is generated by s app ).All three cases connected with the no-slip boundary condition on the lower plate are depicted in Fig. 2.
(ii) Let Q À sapp 2l h 2 > 0. Assuming that the fluid slips according to Navier's slip boundary condition (2.6), which takes the form uð0Þ ¼ sð0Þ j ; we can fix the positive j.Indeed, (5.8) Next, we look at the upper plate.As the constructed solution satisfies we observe that if speed V of the upper plate associated with the applied shear stress s app is such that then there is no-slip at the upper plate.Note that in this case, we have again different slipping mechanisms at the lower and the upper plate.
then the Navier's slip condition (5.6) leads to j h ¼ 2s app lh 2lhV À 2lQ À s app h 2 ; (5.9) which is due to the condition on V, Q, h, l, and s app positive.Thus, the fluid slips according to Navier's slip boundary condition on the upper and lower plates but the specific value of the slip parameter j can be different.In fact, we consider a layer near the upper plate, characterized by y 2 ð h; hÞ, moving as a rigid body with constant velocity V, while on ð0; hÞ, the velocity will be linear satisfying uð hÞ ¼ V and uð0Þ ¼ sð0Þ=j 0 , where j 0 is given in Eq. (5.8).It then follows from Eq. (5.2) that The precise value of h can be determined from the requirement that which leads to a quadratic equation for h.We depict all three cases associated with Navier's slip boundary condition on the lower plate in Fig. 3.
(iii) Let ½Q À sapp 2l h 2 < 0. In this case, we require that a layer adjacent to the lower plate, i.e., for y 2 ð0; hÞ, is at the rest, and the fluid flowing in the region y 2 ð h; hÞ satisfies the condition uð hÞ ¼ 0. Here, h 2 ð0; hÞ.Clearly, u(y) ¼ 0 if 0 y h and uðyÞ ¼ sapp l ðy À hÞ.We determine h from the condition This gives Considering the behavior of the solution at the upper plate, we observe that if then there in no-slip on the upper plate.If V > sapp l ðh À hÞ, the assumed Navier's slip boundary condition of the form (5.6) leads to (5.10) In this situation, there is a layer near the bottom plate where the fluid is at rest and the fluid slips according to Navier's slip boundary condition, with j h given in the above formula, on the upper plate.If V < sapp l ðh À hÞ, we assume the existence of a layer of thickness h À h near the upper plate moving as a rigid body with constant speed V, which is the velocity of the upper plate as measured.The conditions uð hÞ ¼ 0 and uð h Þ ¼ V lead to The value of h will be again determined from the condition on the flow rate: All three cases identified in the case (iii) are depicted in Fig. 4.

FIG. 2. Different flow regimes for plane Couette flow if
. This condition implies the no-slip on the lower plate.Depending on the relation among V, s app , h, l, and Q, we identify three different regimes at the upper plate.These regimes are identified by means of (easily) measurable quantities l, h, s app , Q, and V.

FIG. 3. Different flow regimes for plane Couette flow if
There is Navier's slip on the lower plate with the slip parameter identified by Eq. (5.8).There are three possible regimes identified at the upper plate.These regimes are identified by means of experimentally measurable quantities l, h, s app , Q, and V.

FIG. 4.
Different flow regimes for plane Couette flow if Q < s app h 2 =ð2lÞ.The layer adjacent to the lower plate behaves as a rigid body.There are three scenarios identified at the upper plate.These situations are identified by means of the (easily) measurable quantities l, h, s app , Q, and V.

VI. CYLINDRICAL COUETTE FLOW
Next, we consider the flow of a Navier-Stokes fluid between concentric cylinders with radii R i (inner cylinder) and R o (outer cylinder), 0 < R i < R o .Couette flow is characterized by the following conditions: Then, referring to Eq. (A2), we have Here and in the rest of this section, 0 denotes the derivative with respect to r.We assume that the outer cylinder is rotating due to the applied torque resulting in the shear stress s o , i.e., Equation ( 6.4) and the boundary condition (6.6) imply that Next, using Eq. ( 6.3), we observe from Eq. (6.7) that which leads to To fix the constant D, we use the equation for p [see Eq. (6.5)] and assume that where p i and p o are given pressures that could be measured by pressure transducers at the outer surface of the inner cylinder and the inner surface of the outer cylinder, respectively.Integrating Eq. (6.5) between R i and r and using the first condition in Eqs.(6.9) and (6.8), we obtain Using the second condition in Eq. (6.9), we get the following quadratic equation for D: , this quadratic equation has (one or two) real solutions if and only if b 2 À 4ac !0. This imposes a restriction on the admissible set of problem parameters l, ., s o ; R o ; R i ; p o and p i .In what follows, we assume that D is fixed (We also do not discuss further the possibility that there are two such D.).
It follows from Eq. (6.8) that the angular velocity xðrÞ :¼ v u ðrÞ=r is given by As a consequence of the applied torque on the outer cylinder [see Eq. ( ], the outer cylinder moves with the constant (experimentally measurable) velocity where X o stands for the angular speed of the outer cylinder.The inner cylinder is supposed to be at rest, i.e., fixed, referring to Eq. (6.8), we observe from Eq. (6.11) . We distinguish the following three situations: , and (iii . Then, xðR i Þ ¼ 0 and there is no-slip on the inner cylinder.Three possibilities are considered at the outer cylinder.
First, if the experimentally measurable parameters s o , D, l, and X o satisfy which means that there is no-slip at the outer cylinder.
Second, if 2lX o À 2D þ s o > 0, then, assuming Navier's slip boundary condition in the form we can determine j o : Third, if 2lX o À 2D þ s o < 0, then we assume that the fluid near the outer cylinder is stuck to the outer cylinder, and the whole layer moves with the outer cylinder as a rigid body with angular velocity X 0 .The fluid responds as a Navier-Stokes fluid in the domain characterized by r 2 ðR i ; R o À hÞ, where h > 0 will be specified later.In this situation, The precise values of D and h are determined from the requirement that and from the knowledge of p i ; p o and the requirement that p is continuous at R o À h.Using Eq. (6.10), the latter leads to the condition while Eq.(6.15) gives By solving the system of two equations (6.16) and (6.17), one obtains, under certain additional conditions associated with the quadratic equation (6.16), the values D and h that determine the solution in the form (6.14).
. In this case, assuming that the Navier-Stokes fluid slips according to Navier's slip boundary condition, we can determine the value of the slip parameter j i .Indeed, from we obtain, using Eqs.(6.7) and (6.8), that Proceeding in the same way as in the case (i) above, we can distinguish three situations at the outer cylinder.As the procedure is identical, we skip the details and refer the reader to the material discussed above concerning the procedure.
. In this case, we need to require that the fluid is stuck to the outer cylinder for r 2 ðR i ; R i þ hÞ, while the fluid flows as a Navier-Stokes fluid for r Again, proceeding as in the case (i) above, we can distinguish three situations at the outer cylinder.We skip all further details.Visualization of all these nine cases is also omitted as it would be similar to Figs. 2-4 concerning plane Couette flows.

VII. FLOW DOWN AN INCLINED PLANE DUE TO GRAVITY
Let us consider the flow of a Navier-Stokes fluid down an inclined plane.In the coordinate system associated with the inclined plane, the gravitational force in the chosen coordinate system takes the form .b ¼ ð.g sin h; À.g cos h; 0Þ, where g is the acceleration due to gravity and h is the angle of inclination.
Let us now assume that Q 6 ¼ .gðsinhÞh 3 3l and the fluid slips according to Eq. (2.6) along the inclined plane.In the considered geometrical setting, (2.6)For given h, h, ., l, and Q, we determine h from the condition Q ¼ Ð h h uðyÞ dy with u given in Eq. (7.12).Thus, Note that h 2 ð0; hÞ which follows from Eq. (7.10).To conclude, setting Q crit :¼ .gðsinhÞh 3 =ð3lÞ, we distinguish three different scenarios regarding the behavior of the Navier-Stokes fluid in the vicinity of the inclined plane.If Q ¼ Q crit , then fluid adheres to the boundary and no-slip boundary condition is satisfied.If Q > Q crit , then, assuming Navier's slip boundary condition on the boundary (i.e., on the inclined plane), we can determine the value of the slip parameter j, see formula (7.9).Finally, if Q < Q crit , then, assuming that fluid near the inclined plane adheres to the wall, the layer together with the wall behaves as a rigid body and the fluid flows above this layer as a Navier-Stokes fluid subject to no-slip boundary condition at the interface, we can determine the precise thickness of the layer, see Eq. (7.13).All these three cases are depicted in Fig. 5.

VIII. CONCLUSION
For unidirectional flows of an incompressible Navier-Stokes fluid with constant viscosity, it is possible to determine the validity of the no-slip boundary condition on the walls from the knowledge of the viscosity l, the volumetric flow rate Q, the pressure gradient Àc, and the geometrical dimensions (the radii of cylinders or the distance between the boundary plates).This approach, developed in Ref. 2 for five different types of flow, thus guarantees the validity of the no-slip boundary condition if the volumetric flow rate is equal to a certain critical value Q crit (depending on l, Àc, and some other relevant macroscopic values).In this study, we investigated the situation when the volumetric flow rate does not fulfill the condition guaranteeing noslip.
We have found that • If Q > Q crit and if we assume that the fluid slips past the boundary according to Navier's slip boundary condition, we have been able to specify the exact value of Navier's slip parameter j. • If Q < Q crit , the fluid cannot satisfy Navier's slip boundary condition (as otherwise Navier's slip parameter j would be negative, which contradicts the second law of thermodynamics and the fluid would flow near the wall in the opposite direction than in the inner part of the pipe).Such a solution does not seem to be in keeping with physics.
5. Different flow regimes for flows down an inclined plane.These regimes depend on the relationship among (easily) measurable l, ., h, Q, and h.V C Author(s) 2024

Physics of
• If Q < Q crit , the following scenario is possible: the fluid near the outer cylinder is stuck to the wall and the layer together with the wall behaves as a rigid body.In the inner region, the Navier-Stokes fluid flows subject to no-slip at the interface.The precise thickness of the layer is determined from the given macroscopic data.By interpreting the layer in the vicinity of the wall as a rigid solid, we relax the assumption of the continuity of the stress at the interface between the layer and the fluid flowing in the smaller domain.The scenario described above resembles the situation described in a recent study. 17Other variants are systematically treated in the case of Poiseuille pipe flow and for plane Couette flow in Ref. 18.
5][26][27][28][29][30][31][32][33][34] More recent studies concerning the nature of the boundary condition at a solid surface above which a fluid flows can be found in Refs.17 and 35-42.However, these investigations are not relevant to the kind of flows considered by Du Buat on the basis of which Stokes initially advocated the no-slip boundary condition.That being said, the procedure outlined in Ref. 2 and in this paper will allow one to determine whether the fluids adhere or slip at the boundary, and if it slips, it allows us to determine the slip parameter, even in the case of boundaries with different roughnesses and those that are coated with hydrophilic or hydrophobic material.

APPENDIX: GOVERNING EQUATIONS WRITTEN IN THE CYLINDRICAL COORDINATES
Balance of linear momentum (2.2) The symmetric part of the velocity gradient D ¼ DðvÞ ¼ 1 2 ½rv þðrvÞ T : The constitutive equation (2.3): Here, p :¼ À 1 3 ðT rr þ T uu þ T zz Þ.This implies that the left-hand side of the equation is traceless, and hence, reading the same on the right-had side, one has then the fluid slips at the upper plate.If one assumes Navier's slip boundary condition(2.6),then in the case of a 10e standard way of expressing the Navier-Stokes constitutive relation is at odds with causality as the cause (stress) is expressed in terms of the effect (velocity gradient) (see Rajagopal12and M alek et al.10for a discussion of the same).Note also that the incompressibility constraint (2.1) is included automatically in the constitutive equation (2.3), see Refs.10-12.
4)Note that Eq. (2.3) is equivalent to the more standard constitutive form for a Navier-Stokes fluid, namely, T ¼ ÀpI þ 2lD.