Axial laminar flow of viscoplastic fluids in a concentric annulus subject to wall slip

Abstract

The flow of non-Newtonian fluids in annular geometries is an important problem, especially for the extrusion of polymeric melts and suspensions and for oil and gas exploration. Here, an analytical solution of the equation of motion for the axial flow of an incompressible viscoplastic fluid (represented by the Hershel–Bulkley equation) in a long concentric annulus under isothermal, fully developed, and creeping conditions and subject to true or apparent wall slip is provided. The simplifications of the analytical model for Hershel–Bulkley fluid subject to wall slip also provide the analytical solutions for the axial annular flows of Bingham plastic, power-law, and Newtonian fluids with and without wall slip at one or both surfaces of the annulus.

Appendix

Axial laminar flow of power-law, Bingham plastic, and Newtonian fluids in a concentric annulus with wall slip at the inner and outer walls

For the annular flow of power-law fluids, there will be two flow zones for true slip (II, IV) and four zones for apparent slip (zones I, II, IV, and V), i.e., the plug flow zone (III) sandwiched in between the two deforming zones II and IV would be missing. Furthermore, $dV_z^{\rm II} /dr=dV_z^{{\rm IV}} /dr=0\;\mbox{at}\;r=\lambda R$ and, therefore, the equations for the power-law fluid case can be obtained from those of the foregoing section by setting λ ₁ = λ ₂ = λ and $\tau _0^\ast =0$. Thus, from Eqs. 11 and 12, the velocity distributions in zones II and IV are

$$ \label{eq28} \!\begin{aligned}[b] V_z^{\rm II} ={}& U_{{\rm s},1} +\left( {\dfrac{\Delta PR}{2mL}} \right)^{\frac{1}{n}}\displaystyle\int\nolimits_{\kappa R}^r \left( {\dfrac{\lambda^2R}{r}- \dfrac{r}{R}} \right)^{\frac{1}{n}}dr, \\ & \;\;\; {\kappa R\le r\le \lambda R} \end{aligned} $$

(26)

and

$$ \label{eq29} \!\begin{aligned}[b] V_z^{{\rm IV}} ={}& U_{{\rm s},2} + \left( {\dfrac{\Delta PR}{2mL}} \right)^{\frac{1}{n}}\displaystyle\int\nolimits_r^R \left( {\dfrac{r}{R}-\dfrac{\lambda^2R}{r}} \right)^{\frac{1}{n}}dr,\\ &\;\;\; {\lambda R\le r\le R,} \end{aligned} $$

(27)

respectively, in which the slip velocities U _s,1 and U _s,2 are given by Eq. 25. Noting that $V_z^{\rm II} \left( {\lambda R} \right)= V_z^{{\rm IV}} \left( {\lambda R} \right)$, the equation for determining λ is obtained as

$$ \label{eq30} f\left( {\lambda R} \right)=V_z^{\rm II} \left( {\lambda R} \right)-V_z^{{\rm IV}} \left( {\lambda R} \right)=0. $$

(28)

The flow rate of the power-law fluid through the annulus with apparent slip is obtained (after multiple manipulations of the integral terms following Hanks and Larsen (1979)) as

$$ \begin{array}{rll} \label{eq31} Q&=&\pi \left( {\frac{\Delta PR}{2mL}} \right)^{\frac{1}{n}} \frac{nR^3}{\left( {1+3n} \right)}\\ && \times \left[ {\left( {1-\lambda^2} \right)^{\frac{1}{n}+1}-\frac{1}{\kappa^{\frac{1}{n}-1}}\left( {\lambda ^2-\kappa^2} \right)^{\frac{1}{n}+1}} \right]\\ &&+\, \pi R^2\left( {U_{{\rm s},2} -\kappa^2U_{{\rm s},1} } \right)\\ &&-\,\frac{\pi \lambda^2R^2\left( {n-1} \right)}{\left( {1+3n} \right)}\left( {U_{{\rm s},2} -U_{{\rm s},1} } \right) \end{array} $$

(29)

For U _s,1 = U _s,2 = 0, flow rate equation for the no-slip case given by Hanks and Larsen (1979) is obtained.

Another case of interest of concentric annular flow problem with wall slip which follows from the general Herschel–Bulkley fluid analysis presented earlier is that of the flow of a Bingham plastic fluid. By setting m = μ and n = 1, for a Bingham plastic fluid, Eqs. 11 and 12 may be simplified as follows:

$$ \begin{array}{rll} \label{eq32} V_z^{\rm II} &=&U_{{\rm s},1} +\left( {\dfrac{\Delta PR^2}{4\mu L}} \right)\\ &&\times\left[ {\kappa^2-\left( {\dfrac{r}{R}} \right)^2-2\tau_0^\ast \left( {\dfrac{r}{R}-\kappa } \right)+2\lambda^2\ln \dfrac{r}{\kappa R}} \right],\\ &&\;\;\;{\kappa R\le r\le\lambda_1 R} \end{array} $$

(30)

$$ \begin{array}{rll} \label{eq33} V_z^{\rm IV} &=&U_{{\rm s},2} +\left( {\frac{\Delta PR^2}{4\mu L}} \right) \\ &&\times\left[ {1-\left( {\frac{r}{R}} \right)^2-2\tau_0^\ast \left( {1-\frac{r}{R}} \right)-2\lambda^2\ln \frac{R}{r}} \right], \\ & & \;\;\;{\lambda_2 R\le r\le R} \end{array} $$

(31)

From the above equations (or from Eqs. 13 and 14), the plug velocity $U_{plug} =V_z^{\rm II} \left( {\lambda_1 R} \right)=V_z^{\rm IV} \left( {\lambda _2 R} \right)$ may be obtained as

$$ \begin{array}{rll} \label{eq34} U_{plug} &=&U_{{\rm s},1} +\left( {\frac{\Delta PR^2}{4\mu L}} \right)\\ &&\times\left[ {\kappa ^2-\lambda_1^2 -2\tau_0^\ast \left( {\lambda_1 -\kappa } \right)+2\lambda ^2\ln \frac{\lambda_1 }{\kappa }} \right] \end{array} $$

(32a)

$$ \begin{array}{rll} \label{eq34b} U_{plug} &=&U_{{\rm s},2} +\left( {\frac{\Delta PR^2}{4\mu L}} \right)\\ &&\times\left[ {1-\lambda_2^2 -2\tau_0^\ast \left( {1-\lambda_2 } \right)+2\lambda^2\ln \lambda_2 } \right]. \end{array} $$

(32b)

Equating the plug velocity expressions from the above equations and simplifying, the equation for determining λ is obtained as

$$ \label{eq36} {\kern83pt}\lambda^2=\frac{\left( {\frac{4\mu L}{\Delta PR^2}} \right)\left( {U_{{\rm s},1} -U_{{\rm s},2} } \right)-\left( {1-\kappa^2+\lambda_2^2 -\lambda_1^2 } \right)+2\tau_0^\ast\left({1+\kappa -\lambda_2 -\lambda_1 } \right)}{2\ln \frac{\kappa \lambda_2 }{\lambda_1 }}. $$

(33)

Finally, the flow rate may be obtained, from Eq. 16, as

$$ \begin{array}{rll} \label{eq37} Q&=&\left( {\frac{\pi \Delta PR^4}{24\mu L}} \right)\left[ 3\left( {1-\lambda _2^4 +\lambda_1^4 -\kappa^4} \right)\right.\\ &&\;\quad\qquad\qquad-\,4\tau_0^\ast \left( 1-\lambda_2^3 -\lambda_1^3 +\kappa^3 \right)\\ &&\;\quad\qquad\qquad\left.-\,6\lambda^2\left( 1-\lambda_2^2 +\lambda _1^2 -\kappa^2 \right) \right]\\ &&+\,\pi R^2\left( {U_{{\rm s},2} -\kappa^2U_{{\rm s},1} } \right). \end{array} $$

(34)

It may be verified that for U _s,1 = U _s,2 = 0, the above equations simplify to the corresponding equations of Fredrickson and Bird (1958) for the annular flow of Bingham plastic without slip.

Considering the problem of a Newtonian fluid with shear viscosity, μ, flowing with slip velocities of U _s,1 and U _s,2 at r = κR and r = R, respectively, the velocity distribution may be obtained from any one of the Eqs. 26 and 27 by setting m = μ and, n = 1, i.e.,

$$ \begin{array}{rll} \label{eq38} V_z &=&\left( {\dfrac{\Delta P\,R^2}{4\mu L}} \right)\left[ {2\lambda^2\ln \frac{r}{\kappa R}-\left( {\left( {\frac{r}{R}} \right)^2-\kappa^2} \right)} \right]\\ &&+\,U_{{\rm s},1} ,\;\;\;\kappa R\,\le \,r\,\le \,R \end{array} $$

(35)

or

$$ \begin{array}{rll} \label{eq39} V_z &=&\left( {\dfrac{\Delta P\,R^2}{4\mu L}} \right)\left[ {2\lambda^2\ln \dfrac{R}{r}+\left( {1-\left( {\dfrac{r}{R}} \right)^2} \right)} \right]\\ &&+\,U_{{\rm s},2} ,\;\;\; \kappa R \le r \le R \end{array} $$

(36)

and

$$ \label{eq40} \lambda^2=\frac{\left( {1-\kappa^2} \right)-\frac{\left( {U_{{\rm s},2} -U_{{\rm s},1} } \right)4\mu L}{\Delta P\,R^2}}{2 \ln \left( {\frac{1}{\kappa }} \right)}. $$

(37)

The flow rate of the Newtonian fluid through the concentric annulus, i.e., $Q=2\pi \mathop \int\nolimits_{\kappa R}^R V_z rdr$:

$$ \begin{array}{rll} \label{eq41} Q&=&\left( {\frac{\pi \Delta PR^4}{8\mu L}} \right)\left( {1-\kappa^2} \right)\left( {1-2\lambda^2+\kappa^2} \right) \\ &&+\,\pi \left( {U_{{\rm s},2} -\kappa^2U_{{\rm s},1} } \right)R^2 \end{array} $$

(38)

Equations 32 and 33 may be reduced to the no-slip case for the axial annular flow of Newtonian fluids by setting U _s,2 = U _s,1 = 0 to generate the following well-known solution for the no-slip of Newtonian fluids in an axial annular flow (Bird et al. 2002):

$$ \begin{array}{rll} \label{eq42} \lambda^2&=&\frac{\left( {1-\kappa^2} \right)}{2\ln \left( {\frac{1}{\kappa }} \right)}\;\;\;\mbox{and}\\ Q&=& \left( {\frac{\pi \Delta PR^4}{8\mu L}} \right)\left( {1-\kappa^4- \frac{\left( {1-\kappa^2} \right)^2}{\mbox{ln}\left( {\frac{1}{\kappa }} \right)}} \right) \end{array} $$

(39)

For the apparent slip case, a Newtonian fluid with shear viscosity, μ, subject to apparent slip layers comprised of another Newtonian fluid with viscosity μ _b (where μ _b < μ), can be considered. Noting that β ₁ = δ ₁ /μ _b and β ₂ = δ ₂ /μ _b , the slip velocities at the two cylinder surfaces are not equal to each other, i.e., U _s,1 ≠ U _s,2 even if β ₁ = β ₂ or δ ₁ = δ ₂ since

$$ \label{eq43} \frac{U_{{\rm s},1} }{U_{{\rm s},2} }=\frac{\beta_1 }{\beta_2 }\frac{\left( {\lambda^2-\kappa^2} \right)}{\kappa \left( {1-\lambda^2} \right)}=\frac{\delta_1 }{\delta_2 }\frac{\left( {\lambda^2-\kappa^2} \right)}{\kappa \left( {1-\lambda^2} \right)}. $$

(40)