Theoretical Study of Oldroyd-B Visco-Elastic Fluid Flow Through Curved Pipes with Slip Effects in Polymer Flow Processing

Abstract

The characteristics of the flow field of both viscous and viscoelastic fluids passing through a curved pipe with a Navier slip boundary condition have been investigated analytically in the present study. The Oldroyd-B constitutive equation is employed to simulate realistic transport of dilute polymeric solutions in curved channels. In order to linearize the momentum and constitutive equations, a perturbation method is used in which the ratio of radius of cross section to the radius of channel curvature is employed as the perturbation parameter. The intensity of secondary and main flows is mainly affected by the hoop stress and it is demonstrated in the present study that both the Weissenberg number (the ratio of elastic force to viscous force) and slip coefficient play major roles in determining the strengths of both flows. It is also shown that, as a result of an increment in slip coefficient, the position of maximum velocity markedly migrates away from the pipe center towards the outer side of curvature. Furthermore, results corresponding to Navier slip scenarios exhibit non-uniform distributions in both the main and lateral components of velocity near the wall which can notably vary from the inner side of curvature to the outer side. The present solution is also important in polymeric flow processing systems because of experimental evidence indicating that the no-slip condition can fail for these flows, which is of relevance to chemical engineers.

Appendix

Due to the quasi-linear, coupled nature of the momentum conservation equations, a perturbation method is used. The perturbation parameter in the momentum equations is considered to be the curvature ratio ($\delta = \frac{r}{R}$). Considering the rectilinear form of the flow distribution in a straight pipe and consequently the absence of secondary flow in this situation ($\psi^{(0)} = 0$), stream functions start from the first order onwards. The appropriate series forms for the stress tensor, stream function and main velocity are:

$$ w = \sum\limits_{n = 0}^{\infty } {\delta^{n} } w^{(n)} (r,\phi ),\quad \psi = \sum\limits_{n = 1}^{\infty } {\delta^{n} } \psi^{(n)} (r,\phi ),\quad \tau = \sum\limits_{n = 0}^{\infty } {\delta^{n} } \tau^{(n)} (r,\phi ) $$

(A-1)

Introducing (A-1) into the momentum equation and arranging coefficients of $\delta^{0}$, the first characteristic equation of the primary (main) velocity is obtained as:

$$ \frac{{r^{4} \partial^{2} w^{(0)} }}{{\partial r^{2} }} + 4r^{4} + r^{2} \frac{{\partial^{2} w^{(0)} }}{{\partial^{2} \theta }} + r^{3} \frac{{\partial w^{(0)} }}{\partial r} = 0 $$

(A-2)

The zero-order solution of the main velocity $w^{(0)}$ with respect to the slip condition around the wall is:

$$ w^{(0)} = 1 - r^{2} + 2\beta_{v} $$

(A-3)

Upon substituting Eq. (A-3) into the Oldroyd-B Constitutive Equation the zeroth order solution os the stress components can be calculated as:

$$ \tau_{rs}^{(0)} = \frac{{\partial w^{(0)} }}{\partial r},\quad \tau_{\phi s}^{(0)} = \frac{1}{r}\frac{{\partial w^{(0)} }}{\partial \phi },\quad \tau_{ss}^{(0)} = 2We\beta \left( {\left( {\frac{{\partial w^{(0)} }}{\partial r}} \right)^{2} + \left( {\frac{1}{r}\frac{{\partial w^{(0)} }}{\partial \phi }} \right)^{2} } \right) $$

(A-4)

After collecting the coefficient of $\delta$ in Eqs. 17 and 16, the following equation can be obtained, respectively:

$$ \begin{aligned} \left( {2\text{Re} w^{(0)} \frac{{\partial w^{(0)} }}{\partial r} - \frac{{\partial \tau_{ss}^{(0)} }}{\partial r}} \right)\sin \phi & = - \frac{1}{r}\frac{{\partial^{2} \tau_{rr}^{(1)} }}{\partial r\partial \phi } - \frac{1}{{r^{2} }}\frac{{\partial^{2} \tau_{rr}^{(1)} }}{\partial \phi } + \frac{1}{r}\frac{{\partial^{2} \tau_{\phi \phi }^{(1)} }}{\partial r\partial \phi } + \frac{1}{{r^{2} }}\frac{{\partial \tau_{\phi \phi }^{(1)} }}{\partial \phi } \\ & \quad + \frac{{\partial^{2} \tau_{r\phi }^{(1)} }}{{\partial r^{2} }} + \frac{3}{r}\frac{{\partial \tau_{r\phi }^{(1)} }}{\partial r} - \frac{1}{{r^{2} }}\frac{{\partial^{2} \tau_{r\phi }^{(1)} }}{{\partial \phi^{2} }} \\ \end{aligned} $$

(A-5)

$$ \begin{aligned} \nabla^{2} w^{(1)} & = 4\cos \phi - Re\left( {\frac{{\partial w^{(0)} }}{\partial r}\frac{{\partial \psi^{(1)} }}{\partial \phi }} \right) - We\beta \frac{1}{r}\left( {\frac{{\partial^{3} w^{(0)} }}{{\partial r^{3} }}\frac{{\partial \psi^{(1)} }}{\partial \phi } - \frac{{\partial^{2} w^{(0)} }}{{\partial r^{2} }}\frac{{\partial^{2} \psi^{(1)} }}{\partial r\partial \phi } + 3\frac{1}{r}\frac{{\partial w^{(0)} }}{{\partial r^{2} }}\frac{{\partial \psi^{(1)} }}{\partial \phi }} \right. \\ & \quad \left. - \frac{{\partial w^{(0)} }}{\partial r}\frac{{\partial^{3} \psi^{(1)} }}{{\partial \phi \partial r^{2} }} + \frac{1}{r}\frac{{\partial w^{(0)} }}{\partial r}\frac{{\partial^{2} \psi^{(1)} }}{\partial \phi \partial r} - 3\frac{1}{{r^{2} }}\frac{{\partial w^{(0)} }}{\partial r}\frac{{\partial \psi^{(1)} }}{\partial \phi } - \frac{{\partial w^{(0)} }}{\partial r}\frac{{\partial^{3} \psi^{(1)} }}{{\partial \phi^{3} }}\right) \\ \end{aligned} $$

(A-6)

Using the perturbation series presented in Eq. (A-1) in constitutive equation, expressions for first order of stress tensor components can be obtained as:

$$ \begin{aligned} & \tau_{rr}^{(1)} = - 2\frac{\partial }{\partial r}\left( {\frac{1}{r}\frac{{\partial \psi^{(1)} }}{\partial \phi }} \right),\quad \tau_{\phi \phi }^{(1)} = 2\frac{\partial }{\partial r}\left( {\frac{1}{r}\frac{{\partial \psi^{(1)} }}{\partial \phi }} \right), \\ & \tau_{r\phi }^{(1)} = \frac{{\partial^{2} \psi^{(1)} }}{{\partial r^{2} }} - \frac{1}{r}\frac{{\partial \psi^{(1)} }}{\partial \phi } - \frac{1}{{r^{2} }}\frac{{\partial^{2} \psi^{(1)} }}{{\partial \phi^{2} }} \\ \end{aligned} $$

(A-7)

Solution to the equation (A-5 and A-6), considering Eq. (A-7) will be in the form of $\psi^{(1)} = g_{1} (r)\sin \phi$ and $w^{(1)} = f_{1} (r)\cos \phi$ which the function $g_{1} (r)$ and $f_{1} (r)$ are calculated using the slip boundary condition.

$$ \begin{aligned} g_{1} (r) & = - (1/48)r^{5} \text{Re} - (1/12)r^{5} We\beta - (1/24)r^{5} \beta_{v} \text{Re} + (1/288)\text{Re} r^{7} - (1/72)r\text{Re} /(1 + 2\beta_{v} ) \\ & \quad - (1/9)r\beta_{v} \text{Re} /(1 + 2\beta_{v} ) + (2/3)r^{3} \beta_{v} We\beta /(1 + 2\beta_{v} ) + (1/3)r^{3} \beta_{v}^{2} \text{Re} /(1 + 2\beta_{v} ) \\ & \quad - (1/12)rWe\beta /(1 + 2\beta_{v} ) - (1/2)r\beta_{v} We \cdot \beta /(1 + 2\beta_{v} ) - (1/4)r\beta_{v}^{2} \text{Re} /(1 + 2\beta_{v} ) \\ & \quad + (1/32)r^{3} \text{Re} /(1 + 2\beta_{v} ) + (1/6)r^{3} We\beta /(1 + 2\beta_{v} ) + (3/16)r^{3} \beta_{v} \text{Re} /(1 + 2\beta_{v} ) \\ \end{aligned} $$

(A-8)

$$ \begin{aligned} f_{1} (r) & = 1920r^{5} We^{2} \beta^{2} /(23040\beta_{v} + 11520) - 40r^{7} \text{Re}^{2} \beta_{v} /(23040\beta_{v} + 11520) \\ & \quad + 320r^{5} \text{Re}^{2} \beta_{v}^{2} /(23040\beta_{v} + 11520) + (11/32)r\beta We\text{Re} \beta_{v} /(2\beta_{v}^{2} + 3\beta_{v} + 1) \\ & \quad + 640r^{5} \beta We\text{Re} /(23040\beta_{v} + 11520) + 3840r^{5} \beta_{v} We^{2} \beta^{2} /(23040\beta_{v} + 11520) \\ & \quad + 2560r^{5} \beta We\text{Re} \beta_{v} /(23040\beta_{v} + 11520) - 5760r^{3} \beta We\text{Re} \beta_{v} /(23040\beta_{v} + 11520) \\ & \quad - 120r^{7} \beta We\text{Re} /(23040\beta_{v} + 11520) - (5/3)rWe^{2} \beta_{v}^{2} \beta^{2} /(2\beta_{v}^{2} + 3\beta_{v} + 1) \\ & \quad - 960r^{3} \beta We\text{Re} /(23040\beta_{v} + 11520) + 30r^{5} \text{Re}^{2} /(23040\beta_{v} + 11520) \\ & \quad - (5/6)rWe\beta \beta_{v}^{3} \text{Re} /(2\beta_{v}^{2} + 3\beta_{v} + 1) - 10r^{7} \text{Re}^{2} /(23040\beta_{v} + 11520) \\ & \quad - (3/4)r/(2\beta_{v}^{2} + 3\beta_{v} + 1) + 4r\beta_{v}^{3} /(2\beta_{v}^{2} + 3\beta_{v} + 1) - (5/2)r\beta_{v}^{2} \\ & \quad /(2\beta_{v}^{2} + 3\beta_{v} + 1) - (15/4)r\beta_{v} /(2\beta_{v}^{2} + 3\beta_{v} + 1) + (19/11520)r\text{Re}^{2} \\ & \quad /(2\beta_{v}^{2} + 3\beta_{v} + 1) + 8640r^{3} /(23040\beta_{v} + 11520) + r^{9} \text{Re}^{2} /(23040\beta_{v} + 11520) \\ & \quad - 40r^{3} \text{Re}^{2} /(23040\beta_{v} + 11520) + 17280r^{3} \beta_{v} /(23040\beta_{v} + 11520) - 320r^{3} \text{Re}^{2} \beta_{v} \\ & \quad /(23040\beta_{v} + 11520) + (1/6)rWe^{2} \beta^{2} /(2\beta_{v}^{2} + 3\beta_{v} + 1) + (127/1920)r\text{Re}^{2} \beta_{v}^{2} \\ & \quad /(2\beta_{v}^{2} + 3\beta_{v} + 1) + (209/11520)r\text{Re}^{2} \beta_{v} /(2\beta_{v}^{2} + 3\beta_{v} + 1) + (7/96)r\text{Re}^{2} \beta_{v}^{3} \\ & \quad /(2\beta_{v}^{2} + 3\beta_{v} + 1) - 7680r^{3} \beta We\text{Re} \beta_{v}^{2} /(23040\beta_{v} + 11520) - 3840r^{3} We^{2} E^{2} \\ & \quad /(23040\beta_{v} + 11520) + 2r^{9} \text{Re}^{2} \beta_{v} /(23040\beta_{v} + 11520) + 180r^{5} \text{Re}^{2} \beta_{v} /(23040\beta_{v} + 11520) \\ & \quad - 15360r^{3} \beta_{v} We^{2} \beta^{2} /(23040\beta_{v} + 11520) - 40r^{7} \text{Re}^{2} \beta_{v}^{2} /(23040\beta_{v} + 11520) \\ & \quad + (11/288)r\beta We\text{Re} /(2\beta_{v}^{2} + 3\beta_{v} + 1) + (77/144)r\beta We\text{Re} \beta_{v}^{2} /(2\beta_{v}^{2} + 3\beta_{v} + 1) \\ & \quad - 720r^{3} \text{Re}^{2} \beta_{v}^{2} /(23040\beta_{v} + 11520) + (7/6)r\beta_{v} We^{2} \beta^{2} /(2\beta_{v}^{2} + 3\beta_{v} + 1) \\ & \quad + 1920r^{5} \beta We\text{Re} \beta_{v}^{2} /(23040\beta_{v} + 11520) - 240r^{7} \beta We\text{Re} \beta_{v} /(23040\beta_{v} + 11520) \\ \end{aligned} $$

(A-9)

The same method can be used to drive the characteristic equation of order 2 which due to large size of equations are not presented here but are included once results are reported. The solution to the characteristic equation of order two are in the forms of [23, 24]:

$$ \psi^{(2)} = g_{2} (r)\sin (2\phi ) $$

(A-10)

$$ w^{(2)} = f_{20} (r) + f_{22} (r)\cos (2\phi ) $$

(A-11)

Solution of Flow Rate

The dimensionless flow rate through the pipe can be simply presented as:

$$ Q = \int\limits_{0}^{2\pi } {\int\limits_{0}^{1} {wrdrd\phi } } $$

(A-12)

where w (axial velocity) is considered to be in the form of a perturbation expansion as:

$$ w = w^{(0)} (r) + \delta f_{1} (r)\cos \phi + \delta^{2} \left( {f_{20} (r) + f_{22} (r)\cos 2\phi } \right) $$

(A-13)

Substituting the solutions of velocity components into Eqs. (A-12, A-13), the following equation for the flow rate is readily arrived at:

$$ \tilde{Q}_{s} = \frac{{\pi r_{o}^{2} }}{8\eta }G $$

(A-14)

where $Q_{s}$ is the dimensionless flow rate in a straight stationary pipe with the same pressure gradient and this value is equal to $\pi /2$. The magnitude of this parameter is:

$$ \frac{{Q_{c} }}{{Q_{s} }} = 1 + \delta^{2} \left( {\begin{array}{*{20}l} {(1/4180377600)\left( {87091200\delta^{2} + 3948134400\delta^{2} \beta_{v} We^{2} \beta^{2} + 1393459200\delta^{2} \beta_{v}^{2} We^{2} \beta^{2} } \right.} \hfill \\ { - 1281982464000\delta^{2} \beta_{v}^{5} + 232243200\delta^{2} We^{2} \beta^{2} - 51495\text{Re}^{4} \delta^{2} \beta_{v} - 57625152\text{Re}^{4} \delta^{2} \beta_{v}^{5} } \hfill \\ { - 5719092\text{Re}^{4} \delta^{2} \beta_{v}^{3} + 328043520\delta^{2} We\beta \text{Re} \beta_{v} - 62815680\text{Re}^{4} \delta^{2} \beta_{v}^{6} - 24730232\text{Re}^{4} \delta^{2} \beta_{v}^{4} } \hfill \\ { + 199938170880\delta^{2} We\beta \text{Re} \beta_{v}^{5} + 2905943040\delta^{2} We\beta \text{Re} \beta_{v}^{2} - 3570739200\delta^{2} \beta_{v} } \hfill \\ { - 740738\text{Re}^{4} \delta^{2} \beta_{v}^{2} - 20620800\text{Re}^{4} \delta^{2} \beta_{v}^{7} - 1541\text{Re}^{4} \delta^{2} - 2661120\text{Re}^{2} \delta^{2} + 535088332800\delta^{2} \beta_{v}^{7} } \hfill \\ { + 17418240\delta^{2} We\beta \text{Re} - 45635788800\delta^{2} \beta_{v}^{2} + 484923801600\beta_{v}^{2} + 191368396800\delta^{2} \beta_{v}^{4} We^{2} \beta^{2} } \hfill \\ { - 17805312000\text{Re}^{2} \delta^{2} \beta_{v}^{5} + 1688872550400\beta_{v}^{3} - 12831436800\text{Re}^{2} \delta^{2} \beta_{v}^{6} - 60117120\text{Re}^{2} \delta^{2} \beta_{v} } \hfill \\ { + 30965760\delta^{2} We^{4} \beta^{4} + 20437401600\delta^{2} \beta_{v}^{3} We^{2} \beta^{2} - 256744857600.\delta^{2} \beta_{v}^{3} + 9289728000\delta^{2} \beta_{v}^{2} We^{4} \beta^{4} } \hfill \\ { + 44714557440\delta^{2} \beta_{v}^{3} We^{4} \beta^{4} - 170311680\text{Re}^{3} \delta^{2} We\beta \beta_{v}^{7} + 24319733760\text{Re} \delta^{2} We^{3} \beta^{3} \beta_{v}^{3} } \hfill \\ { + 1708462080\text{Re}^{3} \delta^{2} We\beta \beta_{v}^{6} - 267544166400\delta^{2} \beta We\text{Re} \beta_{v}^{7} + 3068280\text{Re}^{3} \delta^{2} We\beta \beta_{v}^{2} } \hfill \\ { + 3893944320\text{Re} \delta^{2} We^{3} \beta^{3} \beta_{v}^{2} - 575648640\text{Re}^{2} \delta^{2} \beta_{v}^{2} - 10013068800\text{Re}^{2} \delta^{2} \beta_{v}^{4} } \hfill \\ { + 596175840\text{Re}^{3} \delta^{2} We\beta \beta_{v}^{4} - 535088332800\delta^{2} \beta_{v}^{6} We^{2} \beta^{2} - 41760\text{Re}^{3} \delta^{2} We\beta_{v} \beta } \hfill \\ { + 1765077120\text{Re}^{3} \delta^{2} We\beta \beta_{v}^{5} + 79246080\text{Re}^{3} \delta^{2} We\beta \beta_{v}^{3} - 11147673600\delta^{2} \beta_{v}^{5} We^{2} \beta^{2} } \hfill \\ { - 3091253760\text{Re}^{2} \delta^{2} \beta_{v}^{3} + 15809932800\text{Re}^{2} \delta^{2} \beta_{v}^{4} We^{2} \beta^{2} + 17644677120\delta^{2} We\beta \text{Re} \beta_{v}^{3} } \hfill \\ { + 91143843840\delta^{2} We\beta \text{Re} \beta_{v}^{4} + 42294528000\text{Re}^{2} \delta^{2} \beta_{v}^{5} We^{2} \beta^{2} + 321052999680\delta^{2} \beta_{v}^{5} We^{4} \beta^{4} } \hfill \\ { + 245744271360\delta^{2} \beta_{v}^{4} We^{4} \beta^{4} + 3240\text{Re}^{3} \delta^{2} We\beta + 774144000\delta^{2} \beta_{v} We^{4} \beta^{4} + 27136800\text{Re}^{2} \delta^{2} \beta_{v} We^{2} \beta^{2} } \hfill \\ { + 444528000\text{Re}^{2} \delta^{2} \beta_{v}^{2} We^{2} \beta^{2} + 309162147840\text{Re} \delta^{2} We^{3} \beta^{3} \beta_{v}^{6} + 74317824000\text{Re}^{2} \delta^{2} \beta_{v}^{7} We^{2} \beta^{2} } \hfill \\ { + 82585681920\text{Re}^{2} \delta^{2} \beta_{v}^{6} We^{2} \beta^{2} + 2942985830400\beta_{v}^{5} + 10644480\text{Re} \delta^{2} We^{3} \beta^{3} + 1070176665600\beta_{v}^{6} } \hfill \\ { + 995040\text{Re}^{2} \delta^{2} We^{2} \beta^{2} - 401316249600\delta^{2} \beta_{v}^{6} + 97557626880\text{Re} \delta^{2} We^{3} \beta^{3} \beta_{v}^{4} + 4180377600} \hfill \\ { + 281594880\text{Re} \delta^{2} We^{3} \beta^{3} \beta_{v} + 291325870080\text{Re} \delta^{2} We^{3} \beta^{3} \beta_{v}^{5} + 3697511040\text{Re}^{2} \delta^{2} \beta_{v}^{3} We^{2} \beta^{2} } \hfill \\ { + 3143643955200\beta_{v}^{4} + 5573836800\delta^{2} \beta We\text{Re} \beta_{v}^{6} + 71066419200\beta_{v} } \hfill \\ {\left. { - 834682060800\delta^{2} \beta_{v}^{4} } \right)/\left( {160\beta_{v}^{4} + 64\beta_{v}^{5} + 148\beta_{v}^{3} + 64\beta_{v}^{2} + 13\beta_{v} + 1} \right)} \hfill \\ \end{array} } \right) $$

(A-15)