A Hypergeometric Integral Framework for Pressure-Driven Carreau--Yasuda Flow in Circular Tubes and Plane Channels

1. Introduction

Steady pressure-driven flow in a straight tube or between parallel plates is one of the basic boundary-value problems of transport theory and rheology [1,2,3]. The momentum balance is particularly simple: the shear stress is an explicitly known linear function of the transverse coordinate. For a Newtonian fluid, the linear constitutive relation then produces the parabolic Hagen–Poiseuille profile. For a generalized Newtonian fluid, however, the shear stress is a nonlinear function of the shear rate, and constitutive inversion becomes the main analytical difficulty.

Several analytical and semi-analytical strategies have been developed for non-Newtonian tube and slit flow. These include direct constitutive inversion, variational formulations, and the use of stress, shear rate, velocity, or flow rate as an independent variable [4,5,6,7]. Such methods are useful in capillary rheometry and provide analytical benchmarks for numerical solvers. The Carreau–Yasuda model, originating in the constitutive relation proposed by Carreau [8] and its subsequent generalization by Yasuda et al. [9], is particularly useful because, in its usual shear-thinning regime, its parameters describe two Newtonian plateaus connected by a broad power-law transition.

For plane Poiseuille flow, Griffiths [10] obtained exact Carreau-fluid solutions and identified conditions under which the analytical construction is valid. Kutev and Tabakova [11] analyzed Carreau–Yasuda Poiseuille flow under a variable pressure gradient, including the steady case, and distinguished parameter regimes admitting classical and generalized solutions. More recently, Santesarti et al. [12] developed a quasi-analytical description of Carreau–Yasuda-type flow in slowly tapered pipes using a truncated power-law approximation. These studies illustrate the continuing interest in analytical structures that separate constitutive nonlinearity from geometric weighting.

Wang [13] derived a shear-rate-parametrized solution for steady Carreau–Yasuda flow in a circular tube. The main analysis provided parametric expressions for the radial coordinate and velocity, together with a hypergeometric representation of the cross-sectional average velocity. The appendix of the same work applied the construction to plane Poiseuille flow and obtained the corresponding velocity and average-velocity formulas. Consequently, the basic parametric solutions for both circular tubes and plane channels, including their geometry-specific average velocities, have already been reported.

In that earlier work, however, the non-elementary integrals were evaluated as individual problem-specific expressions with the aid of a computer algebra system, and only a brief indication of their connection with Euler’s integral representation was provided. The global invertibility of the shear-rate parametrization was not established explicitly, and the tube and plane-channel averages were not recognized as particular members of a common hierarchy of radial moments.

The purpose of the present paper is therefore not to claim new basic velocity or average-velocity formulas for these two geometries. Instead, it provides a proof-oriented reconstruction and generalization of the mathematical structure underlying the previously reported results. Three principal contributions are established.

First, the normalized stress map

$$s⟼s\tilde{\eta }\left(s\right)$$

(1)

is proved to be strictly increasing over the Carreau–Yasuda parameter range specified below. This result establishes the global well-posedness of the shear-rate parametrization and shows that the parametric velocity curve covers the complete cross-section exactly once.

Second, a master integral is derived directly from Euler’s integral representation of the Gauss hypergeometric function through two explicit changes of variables and a complete matching of the hypergeometric parameters. Derivative, recurrence, and power-series identities are then obtained from the same integral family. This construction replaces case-by-case symbolic integration with a reproducible analytical procedure.

Third, the master integral is used to derive a finite hypergeometric representation for every nonnegative integer radial moment. The previously reported plane-channel and circular-tube average velocities are recovered as the second- and third-order members of this hierarchy, respectively. Their different formulas are thereby shown to arise solely from the different geometric weights in the cross-sectional averages. The standard Newtonian and normalized power-law limits are recorded as direct consequences of the moment formulation.

This organization separates three mathematical ingredients:

1.

the monotone constitutive map, which determines the transverse parametrization;

2.

the special-function integral, which determines the velocity and moment integrals;

3.

the geometric weight, which selects the radial moment entering the cross-sectional average.

The resulting framework requires no pointwise inversion of the nonlinear Carreau–Yasuda flow curve.

The paper is organized as follows. Section 2 introduces the dimensionless formulation and proves the strict monotonicity of the normalized stress map. Section 3 derives the master hypergeometric integral and its derivative, recurrence, and series representations. Section 4 applies the master integral to the parametric velocity profile. Section 5 establishes the finite moment hierarchy and the unified average-velocity formula. Section 6 treats the Newtonian and normalized power-law limits. Section 7 discusses the mathematical scope and limitations of the framework.

2. Mathematical Formulation

The shear-rate parametrization and velocity quadrature for circular-tube flow were derived in Ref. [13]. Because the normalized shear stress is also linear across one half of a plane channel, the same local formulation applies there. We therefore recall only the notation and identities required for the analysis below. The new argument in this section is the proof that the stress map is globally monotone.

2.1. Constitutive Notation

The Carreau–Yasuda viscosity [8,9] is

$$\eta \left(\dot{\gamma }\right)={\eta }_{\infty }+({\eta }_0-{\eta }_{\infty }){\left[1+{\left(\lambda \dot{\gamma }\right)}^a\right]}^{(n-1)/a}.$$

(2)

We assume

$$a>0,n>0,\lambda \ge 0,{\eta }_0>{\eta }_{\infty }\ge 0.$$

(3)

The common shear-thinning range is $0<n<1$, but the monotonicity result below only requires $n>0$.

Let R denote the tube radius, or the half-gap in a plane channel. Let ${\dot{\gamma }}_{\mathrm{w}}$ and ${\tau }_{\mathrm{w}}$ be the wall shear rate and wall shear stress, respectively, with ${\eta }_{\mathrm{w}}=\eta \left({\dot{\gamma }}_{\mathrm{w}}\right)={\tau }_{\mathrm{w}}/{\dot{\gamma }}_{\mathrm{w}}$. Following Ref. [13], set

$$\begin{array}{cccccccc}\hfill \tilde{r}& ={\displaystyle \frac{r}{R}},\hfill & \hfill s& ={\displaystyle \frac{\dot{\gamma }}{{\dot{\gamma }}_{\mathrm{w}}}},\hfill & \hfill \tilde{u}& ={\displaystyle \frac{u}{R{\dot{\gamma }}_{\mathrm{w}}}},\hfill & \hfill \tilde{\lambda }& =\lambda {\dot{\gamma }}_{\mathrm{w}},\hfill \\ \hfill \tilde{\eta }& ={\displaystyle \frac{\eta }{{\eta }_{\mathrm{w}}}},\hfill & \hfill {\tilde{\eta }}_0& ={\displaystyle \frac{{\eta }_0}{{\eta }_{\mathrm{w}}}},\hfill & \hfill {\tilde{\eta }}_{\infty }& ={\displaystyle \frac{{\eta }_{\infty }}{{\eta }_{\mathrm{w}}}},\hfill & \hfill \Delta \tilde{\eta }& ={\tilde{\eta }}_0-{\tilde{\eta }}_{\infty },q={\displaystyle \frac{n-1}{a}}.\hfill \end{array}$$

(4)

The normalized viscosity is therefore

$$\tilde{\eta }\left(s\right)={\tilde{\eta }}_{\infty }+\Delta \tilde{\eta }{\left[1+{\left(\tilde{\lambda }s\right)}^a\right]}^q.$$

(5)

The choice of wall viscosity as the scale gives $\tilde{\eta }\left(1\right)=1$.

2.2. The Normalized Stress Map

The fully developed momentum balance and the constitutive law give

$${\displaystyle \frac{\tau \left(r\right)}{{\tau }_{\mathrm{w}}}}=\tilde{r}=s\tilde{\eta }\left(s\right).$$

(6)

Hence the radial coordinate is represented parametrically by

$$\tilde{r}=\rho \left(s\right),\rho \left(s\right)=s\tilde{\eta }\left(s\right),0\le s\le 1.$$

(7)

The endpoint values are $\rho \left(0\right)=0$ and $\rho \left(1\right)=1$. This map was used in Ref. [13]; here we make its global invertibility explicit.

Proposition 1

(Strict monotonicity). Under the assumptions in Equation (3), the map $\rho :[0,1]\to [0,1]$ defined by Equation (7) is strictly increasing. It therefore has a continuous inverse on $[0,1]$.

Proof. 

Set $X={\left(\tilde{\lambda }s\right)}^a$. Using Eqs. (5) and (7),

$$\rho \left(s\right)=s\left[{\tilde{\eta }}_{\infty }+\Delta \tilde{\eta }{(1+X)}^q\right].$$

(8)

Since

$${\displaystyle \frac{\mathrm{d}X}{\mathrm{d}s}}=a{\tilde{\lambda }}^a{s}^{a-1},s{\displaystyle \frac{\mathrm{d}X}{\mathrm{d}s}}=aX,$$

(9)

differentiation gives

$$\begin{array}{cc}\hfill {\rho }^{\prime }\left(s\right)& ={\tilde{\eta }}_{\infty }+\Delta \tilde{\eta }{(1+X)}^q+\Delta \tilde{\eta }sq{(1+X)}^{q-1}{\displaystyle \frac{\mathrm{d}X}{\mathrm{d}s}}\hfill \\ \hfill & ={\tilde{\eta }}_{\infty }+\Delta \tilde{\eta }{(1+X)}^{q-1}\left[(1+X)+aqX\right].\hfill \end{array}$$

(10)

Because $aq=n-1$, the bracket simplifies to $1+nX$. Consequently,

$${\rho }^{\prime }\left(s\right)={\tilde{\eta }}_{\infty }+\Delta \tilde{\eta }{(1+X)}^{q-1}(1+nX).$$

(11)

Every factor in the second term is nonnegative, and at least one of ${\tilde{\eta }}_{\infty }$ and $\Delta \tilde{\eta }$ is positive. Since $n>0$ and $X\ge 0$, ${\rho }^{\prime }\left(s\right)>0$ on $[0,1]$. The endpoint values then show that $\rho$ maps $[0,1]$ bijectively onto $[0,1]$. □

Remark 1.

Proposition 1 gives a direct mathematical justification for the shear-rate parametrization. An explicit elementary formula for ${\rho }^{-1}$ is not required; strict monotonicity is enough to ensure that each radial position corresponds to exactly one shear rate.

2.3. Velocity in the Shear-Rate Parameter

The velocity quadrature was also obtained in Ref. [13]. We record only the form needed later. Since $\dot{\gamma }=-\mathrm{d}u/\mathrm{d}r$ and $\tilde{r}=\rho \left(s\right)$, the chain rule gives

$${\displaystyle \frac{\mathrm{d}\tilde{u}}{\mathrm{d}s}}=-s{\rho }^{\prime }\left(s\right).$$

(12)

With $\tilde{u}\left(1\right)=0$, direct integration followed by integration by parts gives

$$\begin{array}{cc}\hfill \tilde{u}\left(s\right)& ={\int }_s^1\xi {\rho }^{\prime }\left(\xi \right)\mathrm{d}\xi \hfill \\ & =1-s\rho \left(s\right)-W\left(s\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill W\left(s\right)& ={\int }_s^1\rho \left(\xi \right)\mathrm{d}\xi ={\int }_s^1\xi \tilde{\eta }\left(\xi \right)\mathrm{d}\xi .\hfill \end{array}$$

(14)

The new special-function analysis begins with the exact evaluation of this integral.

3. The Master Hypergeometric Integral

3.1. Closed Form

For $a>0$, $\mathrm{\Lambda }\ge 0$, $m>-1$, $p\in \mathbb{R}$, and $x\ge 0$, define

$${\mathcal{J}}_{m,p}(x;\mathrm{\Lambda })={\int }_0^x{\xi }^m{\left[1+{\left(\mathrm{\Lambda }\xi \right)}^a\right]}^p\mathrm{d}\xi .$$

(15)

Set

$$\beta ={\displaystyle \frac{m+1}{a}}.$$

(16)

Theorem 1

(Master integral). The integral in Equation (15) is

$$\begin{array}{cc}\hfill {\mathcal{J}}_{m,p}(x;\mathrm{\Lambda })& ={\displaystyle \frac{x^{m+1}}{m+1}}{}_{2}F_1\left(\beta ,-p;\beta +1;-{\left(\mathrm{\Lambda }x\right)}^a\right).\hfill \end{array}$$

(17)

Proof. 

First set $\xi =xt$, so that $\mathrm{d}\xi =x\mathrm{d}t$. The lower and upper limits become $t=0$ and $t=1$, respectively. Hence

$$\begin{array}{cc}\hfill {\mathcal{J}}_{m,p}(x;\mathrm{\Lambda })& =x^{m+1}{\int }_0^1{t}^m{\left[1+{\left(\mathrm{\Lambda }x\right)}^a{t}^a\right]}^p\mathrm{d}t.\hfill \end{array}$$

(18)

Next set

$$y=t^a,t=y^{1/a},\mathrm{d}t={\displaystyle \frac{1}{a}}{y}^{1/a-1}\mathrm{d}y.$$

(19)

It follows that

$$t^m\mathrm{d}t={\displaystyle \frac{1}{a}}{y}^{(m+1)/a-1}\mathrm{d}y.$$

(20)

With $\beta$ as in Equation (16), set

$$Z={\left(\mathrm{\Lambda }x\right)}^a.$$

(21)

Equation (18) becomes

$${\mathcal{J}}_{m,p}(x;\mathrm{\Lambda })={\displaystyle \frac{x^{m+1}}{a}}{\int }_0^1{y}^{\beta -1}{(1+Zy)}^p\mathrm{d}y.$$

(22)

Euler’s integral representation can be written as

$$\begin{array}{cc}\hfill {}_{2}F_1\left(A,B;C;z\right)& ={\displaystyle \frac{\mathrm{\Gamma }\left(C\right)}{\mathrm{\Gamma }\left(B\right)\mathrm{\Gamma }(C-B)}}\mathcal{E}(A,B,C;z),\hfill \end{array}$$

(23)

where

$$\begin{array}{cc}\hfill \mathcal{E}(A,B,C;z)& ={\int }_0^1{y}^{B-1}{(1-y)}^{C-B-1}{(1-zy)}^{-A}\mathrm{d}y,\hfill \end{array}$$

(24)

initially for $ReC>ReB>0$ [14,15]. Choose

$$A=-p,B=\beta ,C=\beta +1,z=-Z.$$

(25)

Then

$${(1-y)}^{C-B-1}=1,{(1-zy)}^{-A}={(1+Zy)}^p.$$

(26)

Moreover,

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{\Gamma }\left(C\right)}{\mathrm{\Gamma }\left(B\right)\mathrm{\Gamma }(C-B)}}& ={\displaystyle \frac{\mathrm{\Gamma }(\beta +1)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left(1\right)}}=\beta .\hfill \end{array}$$

(27)

Therefore,

$${\int }_0^1{y}^{\beta -1}{(1+Zy)}^p\mathrm{d}y={\displaystyle \frac{1}{\beta }}{}_{2}F_1\left(-p,\beta ;\beta +1;-Z\right).$$

(28)

Substitution into Equation (22), followed by $a\beta =m+1$, proves Equation (17). The symmetry of the two numerator parameters of ${}_{2}F_1$ has been used to write the result in the displayed order. □

Remark 2.

For the stated parameter range, $\beta >0$, $C-B=1>0$, and $z=-Z\le 0$. Thus the Euler integral used in the proof is regular on $0\le y\le 1$. No analytic continuation across a branch cut is needed for the physical Carreau–Yasuda problem.

3.2. Derivative and Recurrence Relations

The fundamental theorem of calculus immediately gives

$${\displaystyle \frac{\partial }{\partial x}}{\mathcal{J}}_{m,p}(x;\mathrm{\Lambda })=x^m{\left[1+{\left(\mathrm{\Lambda }x\right)}^a\right]}^p.$$

(29)

This identity is also an efficient verification of any hypergeometric implementation of Equation (17).

A recurrence follows from differentiating

$$F\left(x\right)=x^{m+1}{\left[1+{\left(\mathrm{\Lambda }x\right)}^a\right]}^p.$$

(30)

Indeed,

$$\begin{array}{cc}\hfill F^{\prime }\left(x\right)=& (m+1)x^m{\left[1+{\left(\mathrm{\Lambda }x\right)}^a\right]}^p\hfill \\ \hfill & +ap{\mathrm{\Lambda }}^a{x}^{m+a}{\left[1+{\left(\mathrm{\Lambda }x\right)}^a\right]}^{p-1}.\hfill \end{array}$$

(31)

Integrating from 0 to x gives the following result.

Proposition 2

(Parameter recurrence). For $m>-1$,

$$\begin{array}{cc}\hfill (m+1){\mathcal{J}}_{m,p}(x;\mathrm{\Lambda })=& x^{m+1}{\left[1+{\left(\mathrm{\Lambda }x\right)}^a\right]}^p\hfill \\ \hfill & -ap{\mathrm{\Lambda }}^a{\mathcal{J}}_{m+a,p-1}(x;\mathrm{\Lambda }).\hfill \end{array}$$

(32)

3.3. Power-Series Representation

For $\left|\mathrm{\Lambda }x\right|<1$, the generalized binomial series gives

$${\left[1+{\left(\mathrm{\Lambda }\xi \right)}^a\right]}^p={\displaystyle \sum _{\ell =0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{p}{\ell }}\right){\mathrm{\Lambda }}^{a\ell }{\xi }^{a\ell }.$$

(33)

Term-by-term integration yields

$${\mathcal{J}}_{m,p}(x;\mathrm{\Lambda })={\displaystyle \sum _{\ell =0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{p}{\ell }}\right){\displaystyle \frac{{\mathrm{\Lambda }}^{a\ell }{x}^{m+a\ell +1}}{m+a\ell +1}}.$$

(34)

In particular,

$$\begin{array}{cc}\hfill {\mathcal{J}}_{m,p}(x;\mathrm{\Lambda })=& {\displaystyle \frac{x^{m+1}}{m+1}}+{\displaystyle \frac{p{\mathrm{\Lambda }}^a{x}^{m+a+1}}{m+a+1}}\hfill \\ \hfill & +{\displaystyle \frac{p(p-1){\mathrm{\Lambda }}^{2a}{x}^{m+2a+1}}{2(m+2a+1)}}+O\left({\mathrm{\Lambda }}^{3a}\right).\hfill \end{array}$$

(35)

The hypergeometric expression in Equation (17) remains valid beyond the disk of convergence of this particular power series. An independent term-by-term verification is given in Appendix A.

4. Velocity Profile

Substitution of Equation (5) into Equation (14) gives

$$\begin{array}{cc}\hfill W\left(s\right)=& {\tilde{\eta }}_{\infty }{\int }_s^1\xi \mathrm{d}\xi \hfill \\ \hfill & +\Delta \tilde{\eta }{\int }_s^1\xi {\left[1+{\left(\tilde{\lambda }\xi \right)}^a\right]}^q\mathrm{d}\xi .\hfill \end{array}$$

(36)

The first integral is $(1-s^2)/2$. For the second, use Theorem 1 with $m=1$, $p=q=(n-1)/a$, and $\mathrm{\Lambda }=\tilde{\lambda }$. Define

$$\mathrm{\Phi }\left(s\right)=s^2{}_{2}F_1\left({\displaystyle \frac{2}{a}},{\displaystyle \frac{1-n}{a}};{\displaystyle \frac{a+2}{a}};-{\left(\tilde{\lambda }s\right)}^a\right).$$

(37)

Then

$${\mathcal{J}}_{1,q}(s;\tilde{\lambda })={\displaystyle \frac{\mathrm{\Phi }\left(s\right)}{2}}.$$

(38)

Since an integral from s to 1 is the difference of two lower-limit integrals,

$${\int }_s^1\xi {\left[1+{\left(\tilde{\lambda }\xi \right)}^a\right]}^q\mathrm{d}\xi ={\displaystyle \frac{\mathrm{\Phi }\left(1\right)-\mathrm{\Phi }\left(s\right)}{2}}.$$

(39)

Therefore,

$$W\left(s\right)={\displaystyle \frac{{\tilde{\eta }}_{\infty }}{2}}(1-s^2)+{\displaystyle \frac{\Delta \tilde{\eta }}{2}}\left[\mathrm{\Phi }\left(1\right)-\mathrm{\Phi }\left(s\right)\right].$$

(40)

Theorem 2

(Parametric velocity profile). Under Equation (3), the dimensionless velocity profile is the regular parametric curve

$$\begin{array}{cc}\hfill \tilde{r}\left(s\right)& =s\left[{\tilde{\eta }}_{\infty }+\Delta \tilde{\eta }{\left(1+{\left(\tilde{\lambda }s\right)}^a\right)}^q\right],\hfill \\ \hfill \tilde{u}\left(s\right)& =1-s\tilde{r}\left(s\right)-{\displaystyle \frac{{\tilde{\eta }}_{\infty }}{2}}(1-s^2)\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{\Delta \tilde{\eta }}{2}}\left[\mathrm{\Phi }\left(1\right)-\mathrm{\Phi }\left(s\right)\right].\hfill \end{array}$$

(42)

The radial component $\tilde{r}\left(s\right)$ is strictly increasing.

Proof. 

Equation (41) is the stress balance (7). Strict monotonicity follows from Proposition 1. Equation (42) follows by substituting Equation (40) into Equation (13). □

The expression can be checked without special-function differentiation. From the defining integral,

$$W^{\prime }\left(s\right)=-s\tilde{\eta }\left(s\right)=-\rho \left(s\right),W\left(1\right)=0.$$

(43)

Consequently, differentiating Equation (13) gives

$${\tilde{u}}^{\prime }\left(s\right)=-\rho \left(s\right)-s{\rho }^{\prime }\left(s\right)-W^{\prime }\left(s\right)=-s{\rho }^{\prime }\left(s\right),$$

(44)

which is Equation (12). At $s=1$, $\rho \left(1\right)=1$ and $W\left(1\right)=0$, so $\tilde{u}\left(1\right)=0$.

At the centerline, $s=0$ and $\rho \left(0\right)=0$. Since $\mathrm{\Phi }\left(0\right)=0$, the maximum velocity is

$${\tilde{u}}_{max}=1-{\displaystyle \frac{{\tilde{\eta }}_{\infty }}{2}}-{\displaystyle \frac{\Delta \tilde{\eta }}{2}}\mathrm{\Phi }\left(1\right).$$

(45)

5. Moment Hierarchy and Average Velocities

5.1. All Integer Radial Moments

Define the kth moment of the normalized radial map by

$${\mathcal{M}}_k={\int }_0^1{\rho }^k\left(s\right)\mathrm{d}s,k=0,1,2,\dots .$$

(46)

For integers $k\ge 0$ and $0\le j\le k$, define

$$H_{k,j}={}_{2}F_1\left({\displaystyle \frac{k+1}{a}},{\displaystyle \frac{j(1-n)}{a}};{\displaystyle \frac{a+k+1}{a}};-{\tilde{\lambda }}^a\right).$$

(47)

Notice that $H_{k,0}=1$.

Theorem 3

(Finite moment hierarchy). Every integer moment in Equation (46) has the finite hypergeometric representation

$${\mathcal{M}}_k={\displaystyle \frac{1}{k+1}}{\displaystyle \sum _{j=0}^k}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right){\tilde{\eta }}_{\infty }^{k-j}{(\Delta \tilde{\eta })}^j{H}_{k,j}.$$

(48)

Proof. 

From Equation (7),

$${\rho }^k\left(s\right)=s^k{\left[{\tilde{\eta }}_{\infty }+\Delta \tilde{\eta }{\left(1+{\left(\tilde{\lambda }s\right)}^a\right)}^q\right]}^k.$$

(49)

The binomial theorem gives

$$\begin{array}{c}\hfill {\rho }^k\left(s\right)={\displaystyle \sum _{j=0}^k}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right){\tilde{\eta }}_{\infty }^{k-j}{(\Delta \tilde{\eta })}^j{s}^k{\left[1+{\left(\tilde{\lambda }s\right)}^a\right]}^{jq}.\end{array}$$

(50)

Integrating term by term,

$$\begin{array}{c}\hfill {\mathcal{M}}_k={\displaystyle \sum _{j=0}^k}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right){\tilde{\eta }}_{\infty }^{k-j}{(\Delta \tilde{\eta })}^j{\mathcal{J}}_{k,jq}(1;\tilde{\lambda }).\end{array}$$

(51)

Theorem 1 gives

$$\begin{array}{cc}\hfill {\mathcal{J}}_{k,jq}(1;\tilde{\lambda })& ={\displaystyle \frac{1}{k+1}}{}_{2}F_1\left({\displaystyle \frac{k+1}{a}},-jq;1+{\displaystyle \frac{k+1}{a}};-{\tilde{\lambda }}^a\right)\hfill \\ \hfill & ={\displaystyle \frac{H_{k,j}}{k+1}},\hfill \end{array}$$

(52)

because $-jq=j(1-n)/a$. Substitution into Equation (51) proves Equation (48). □

Corollary 1

(Moment bounds). For $k\ge 1$,

$$0<{\mathcal{M}}_k<1,{\mathcal{M}}_{k+1}\le {\mathcal{M}}_k.$$

(53)

Proof. 

Proposition 1 and the endpoint values imply $0\le \rho \left(s\right)\le 1$, with strict inequalities for $0<s<1$. Thus $0<{\rho }^k\left(s\right)<1$ in the interior and ${\rho }^{k+1}\left(s\right)\le {\rho }^k\left(s\right)$. Integration gives the result. □

5.2. A Unified Geometric Formula

Let d denote the cross-sectional weight:

$$d=\left\{\begin{array}{cc}1,\hfill & \mathrm{plane}\mathrm{channel},\hfill \\ 2,\hfill & \mathrm{circular}\mathrm{tube}.\hfill \end{array}\right.$$

(54)

The normalized cross-sectional average is

$${\tilde{u}}_{\mathrm{avg}}^{\left(d\right)}=d{\int }_0^1\tilde{u}\left(\tilde{r}\right){\tilde{r}}^{d-1}\mathrm{d}\tilde{r}={\int }_0^1\tilde{u}\left(\tilde{r}\right)\mathrm{d}\left({\tilde{r}}^d\right).$$

(55)

Integration by parts gives

$$\begin{array}{cc}\hfill {\tilde{u}}_{\mathrm{avg}}^{\left(d\right)}& ={\left.\tilde{u}{\tilde{r}}^d\right|}_0^1-{\int }_0^1{\tilde{r}}^d\mathrm{d}\tilde{u}.\hfill \end{array}$$

(56)

The boundary term vanishes. Using $\mathrm{d}\tilde{u}=-s\mathrm{d}\tilde{r}$,

$$\begin{array}{cc}\hfill {\tilde{u}}_{\mathrm{avg}}^{\left(d\right)}& ={\int }_0^{1}s{\tilde{r}}^d\mathrm{d}\tilde{r}={\displaystyle \frac{1}{d+1}}{\int }_0^{1}s\mathrm{d}\left({\tilde{r}}^{d+1}\right).\hfill \end{array}$$

(57)

Changing to the parameter s and integrating by parts once more,

$$\begin{array}{cc}\hfill {\tilde{u}}_{\mathrm{avg}}^{\left(d\right)}& ={\displaystyle \frac{1}{d+1}}\left[{\left.s{\tilde{r}}^{d+1}\right|}_0^1-{\int }_0^1{\tilde{r}}^{d+1}\left(s\right)\mathrm{d}s\right].\hfill \end{array}$$

(58)

Since $\rho \left(0\right)=0$ and $\rho \left(1\right)=1$, this proves the following result.

Corollary 2

(Average velocity). For $d=1$ or $d=2$,

$${\tilde{u}}_{\mathrm{avg}}^{\left(d\right)}={\displaystyle \frac{1}{d+1}}\left(1-{\mathcal{M}}_{d+1}\right).$$

(59)

Combining with Theorem 3,

$${\tilde{u}}_{\mathrm{avg}}^{\left(d\right)}={\displaystyle \frac{1}{d+1}}\left[1-{\displaystyle \frac{{\mathrm{\Sigma }}_d}{d+2}}\right],$$

(60)

where

$${\mathrm{\Sigma }}_d={\displaystyle \sum _{j=0}^{d+1}}\left({\displaystyle \genfrac{}{}{0pt}{}{d+1}{j}}\right){\tilde{\eta }}_{\infty }^{d+1-j}{(\Delta \tilde{\eta })}^j{H}_{d+1,j}.$$

(61)

5.3. Circular Tube

For $d=2$, the required moment is

$$\begin{array}{cc}\hfill {\mathcal{M}}_3={\displaystyle \frac{1}{4}}[& {\tilde{\eta }}_{\infty }^3+3{\tilde{\eta }}_{\infty }^2\Delta \tilde{\eta }{H}_{3,1}\hfill \\ \hfill & +3{\tilde{\eta }}_{\infty }{(\Delta \tilde{\eta })}^2{H}_{3,2}+{(\Delta \tilde{\eta })}^3{H}_{3,3}].\hfill \end{array}$$

(62)

Thus

$${\tilde{u}}_{\mathrm{avg}}^{\mathrm{tube}}={\displaystyle \frac{1}{3}}(1-{\mathcal{M}}_3).$$

(63)

The three nontrivial hypergeometric factors are

$$H_{3,j}={}_{2}F_1\left({\displaystyle \frac{4}{a}},{\displaystyle \frac{j(1-n)}{a}};{\displaystyle \frac{a+4}{a}};-{\tilde{\lambda }}^a\right),j=1,2,3.$$

(64)

The quantities denoted $I_1,I_2,I_3$ in Ref. [13] correspond to

$$\begin{array}{cc}\hfill I_1& ={(\Delta \tilde{\eta })}^3{H}_{3,3},\hfill \\ \hfill I_2& ={\tilde{\eta }}_{\infty }{(\Delta \tilde{\eta })}^2{H}_{3,2},\hfill \\ \hfill I_3& ={\tilde{\eta }}_{\infty }^2\Delta \tilde{\eta }{H}_{3,1}.\hfill \end{array}$$

(65)

Equation (62) therefore explains both the hypergeometric parameters and the coefficients $1,3,3,1$: they arise directly from one application of the binomial theorem.

5.4. Plane Channel

For $d=1$,

$${\mathcal{M}}_2={\displaystyle \frac{1}{3}}\left[{\tilde{\eta }}_{\infty }^2+2{\tilde{\eta }}_{\infty }\Delta \tilde{\eta }{H}_{2,1}+{(\Delta \tilde{\eta })}^2{H}_{2,2}\right],$$

(66)

where

$$H_{2,j}={}_{2}F_1\left({\displaystyle \frac{3}{a}},{\displaystyle \frac{j(1-n)}{a}};{\displaystyle \frac{a+3}{a}};-{\tilde{\lambda }}^a\right),j=1,2.$$

(67)

The corresponding dimensionless average velocity is

$${\tilde{u}}_{\mathrm{avg}}^{\mathrm{plane}}={\displaystyle \frac{1}{2}}(1-{\mathcal{M}}_2).$$

(68)

Equations (66) and (68) recover the plane-channel average-velocity result reported in the Appendix of Ref. [13]. In the moment hierarchy, the plane-channel and circular-tube averages correspond to $k=2$ and $k=3$, respectively. The plane-channel and tube formulas are thus not separate integrations; they are the $k=2$ and $k=3$ members of the same moment hierarchy, respectively.

6. Limiting Regimes

6.1. Newtonian Limit

If $n=1$, then $q=0$ and the viscosity is independent of s. The wall normalization forces

$$\tilde{\eta }\left(s\right)=1,\rho \left(s\right)=s.$$

(69)

Consequently,

$${\mathcal{M}}_k={\int }_0^1{s}^k\mathrm{d}s={\displaystyle \frac{1}{k+1}}.$$

(70)

Equation (59) gives

$${\tilde{u}}_{\mathrm{avg}}^{\left(d\right)}={\displaystyle \frac{1}{d+1}}\left(1-{\displaystyle \frac{1}{d+2}}\right)={\displaystyle \frac{1}{d+2}}.$$

(71)

Hence

$${\tilde{u}}_{\mathrm{avg}}^{\mathrm{plane}}={\displaystyle \frac{1}{3}},{\tilde{u}}_{\mathrm{avg}}^{\mathrm{tube}}={\displaystyle \frac{1}{4}}.$$

(72)

6.2. Normalized Power-Law Limit

Assume $0<n<1$ and ${\tilde{\eta }}_{\infty }=0$. The wall normalization gives

$$\Delta \tilde{\eta }={(1+{\tilde{\lambda }}^a)}^{-q}.$$

(73)

Therefore,

$$\tilde{\eta }\left(s\right)={\left[{\displaystyle \frac{1+{\left(\tilde{\lambda }s\right)}^a}{1+{\tilde{\lambda }}^a}}\right]}^q.$$

(74)

For every fixed $s>0$,

$$\underset{\tilde{\lambda }\to \infty }{lim}\tilde{\eta }\left(s\right)=s^{aq}=s^{n-1}.$$

(75)

Thus

$$\rho \left(s\right)⟶s^n.$$

(76)

Because $0\le \rho \left(s\right)\le 1$, dominated convergence applies to every integer moment:

$$\underset{\tilde{\lambda }\to \infty }{lim}{\mathcal{M}}_k={\int }_0^1{s}^{kn}\mathrm{d}s={\displaystyle \frac{1}{kn+1}}.$$

(77)

It follows that

$$\underset{\tilde{\lambda }\to \infty }{lim}{\tilde{u}}_{\mathrm{avg}}^{\left(d\right)}={\displaystyle \frac{n}{1+n(d+1)}}.$$

(78)

In particular,

$${\tilde{u}}_{\mathrm{avg}}^{\mathrm{plane}}⟶{\displaystyle \frac{n}{2n+1}},{\tilde{u}}_{\mathrm{avg}}^{\mathrm{tube}}⟶{\displaystyle \frac{n}{3n+1}}.$$

(79)

The limiting velocity profile can also be obtained explicitly. Since $\rho \left(s\right)=s^n$,

$${\displaystyle \frac{\mathrm{d}\tilde{u}}{\mathrm{d}s}}=-s{\displaystyle \frac{\mathrm{d}}{\mathrm{d}s}}\left(s^n\right)=-n{s}^n.$$

(80)

Using $\tilde{u}\left(1\right)=0$,

$$\tilde{u}\left(s\right)={\displaystyle \frac{n}{n+1}}\left(1-s^{n+1}\right).$$

(81)

Since $s={\tilde{r}}^{1/n}$,

$$\tilde{u}\left(\tilde{r}\right)={\displaystyle \frac{n}{n+1}}\left(1-{\tilde{r}}^{(n+1)/n}\right).$$

(82)

6.3. Small- $\tilde{\lambda }$ Regime

Equation (35) gives systematic corrections to the low-shear Newtonian plateau. For the velocity integral, $m=1$ and $p=q$:

$$\begin{array}{cc}\hfill {\mathcal{J}}_{1,q}(x;\tilde{\lambda })=& {\displaystyle \frac{x^2}{2}}+{\displaystyle \frac{q{\tilde{\lambda }}^a{x}^{a+2}}{a+2}}\hfill \\ \hfill & +{\displaystyle \frac{q(q-1){\tilde{\lambda }}^{2a}{x}^{2a+2}}{2(2a+2)}}+O\left({\tilde{\lambda }}^{3a}\right).\hfill \end{array}$$

(83)

Analogous expansions for all moments follow by setting $m=k$ and $p=jq$ in Equation (35).

7. Discussion

The master-integral viewpoint changes the interpretation of pressure-driven Carreau–Yasuda flow in tubes and plane channels. The hypergeometric functions in the velocity and flow-rate formulas are not isolated outputs of symbolic integration. They are members of a single two-parameter integral family. Once Theorem 1 is established, every subsequent non-elementary integration reduces to parameter substitution and a finite binomial expansion.

The proof also clarifies why Euler’s representation is particularly well adapted to this problem. The substitution $y=t^a$ produces the exponent $(m+1)/a-1$. Choosing $C=B+1$ in Euler’s integral removes the $(1-y)$ factor entirely, and the Gamma-function prefactor collapses to B. The three hypergeometric parameters in the final expression are therefore fixed by the power m, the Carreau–Yasuda exponent p, and the transition parameter a; none of them is ad hoc.

The moment hierarchy in Theorem 3 is a further consequence that is not visible when the circular-tube and plane-channel integrals are treated separately. The hierarchy shows that an arbitrary integer moment of the radial stress map requires only $k+1$ hypergeometric evaluations. The binomial coefficients record the mixing of the infinite-shear plateau and the shear-dependent contribution. Physical flow rates require only the members $k=2$ and $k=3$, but higher moments may be useful in weighted transport calculations or in error estimates for reduced-order approximations.

Strict monotonicity of the stress map is equally important. The absence of an elementary expression for ${\rho }^{-1}$ does not make the solution ill-defined. Proposition 1 ensures that the parametric curve is single valued and covers the entire cross section exactly once. This is the precise condition needed for a parametric analytical solution.

The present framework has clear limitations. It assumes a monotone generalized Newtonian flow curve, steady fully developed motion, and no slip at the wall. Nonmonotone constitutive curves may produce multiple shear rates at the same stress and require a different selection principle. Wall slip would add a stress-dependent velocity offset but would not change the bulk integrals. Temperature dependence, thixotropy, viscoelastic memory, and developing flow lie outside the scalar constitutive setting considered here.

The same integral method applies whenever the normalized viscosity can be expanded into finitely many powers of $[1+{\left(\mathrm{\Lambda }s\right)}^a]$, or when an infinite expansion can be justified term by term. The Carreau–Yasuda law is especially convenient because its affine structure produces the finite hierarchy, as shown in Equation (48).

8. Conclusion

A unified hypergeometric framework has been established for steady, fully developed, pressure-driven Carreau–Yasuda flow in circular tubes and plane channels. The principal results are:

1.

the normalized stress map is strictly increasing under the stated Carreau–Yasuda parameter assumptions, establishing that the shear-rate parametrization is globally one-to-one;

2.

the non-elementary integrations required for the velocity profile and radial moments are generated by the master identity in Equation (17);

3.

every nonnegative integer radial moment admits the finite hypergeometric representation in Equation (48);

4.

the plane-channel and circular-tube average velocities are the $k=2$ and $k=3$ cases, respectively, of the unified formula in Equation (59);

5.

the Newtonian and normalized power-law limits follow directly from the same moment formulation.

The analysis uses only changes of variables, Euler’s integral representation, the binomial theorem, and integration by parts. It provides a transparent and reproducible derivation of the special-function formulas and avoids pointwise inversion of the nonlinear constitutive relation. In particular, it supplies the detailed mathematical derivation underlying the plane-channel and circular-tube formulas reported in Ref. [13] and recasts their average velocities as the $k=2$ and $k=3$ cases, respectively, of a general hierarchy of radial moments. The framework thereby separates the constitutive structure of the Carreau–Yasuda law from the geometric weighting that distinguishes the two flow configurations.