Slow motion of couple stress fluid past a solid sphere in a virtual cell: slip effect

Abstract

The steady axisymmetric flow of an incompressible couple stress fluid past a solid sphere located at the center of a hypothetical spherical cavity is analytically investigated using the cell model technique. Here we assume that the inner sphere is solid and the outer one is fictitious. On the surface of the inner sphere, boundary conditions are the vanishing of normal velocity, slip boundary condition, and nil couple stress condition have been used. On the fictitious surface of the outer spherical cell, boundary conditions are Happel’s model, Kuwabara’s model, Kvashnin’s model, and Cunningham’s model applied. The couple stress fluid flow is governed in the cavity. The drag force and the wall correction factor acting on a slip sphere are evaluated. In special cases, we have discussed the drag force acting on the slip sphere in an unbounded case and the wall correction factor experienced by viscous fluid for all four models in a bounded case. Variations of the wall correction factor versus separation parameter for different values of the couple stress and slip parameters with a fixed value of the couple stress viscosity ratio parameter are presented graphically. Here we have also discussed the results of consistent couple stress theory. The tabulated results show that the wall correction factor increases monotonically as the couple stress viscosity ratio, slip, and separation parameters increases. It has the largest values when an interaction between the spherical cavity and inner sphere is very close and the smallest values when they are far from each other.

Appendix A

The system of linear equations for the steady flow of an incompressible couple stress fluid past a slip sphere of radius $r=a$ in a concentric spherical cavity of radius $r=b$ are given as

The boundary conditions are:

• on the surface of inner sphere $(r = a)$ The velocity component vanishes

  $$\begin{aligned} A a^{2} +\frac{B}{a} + C a^{4} + D a+ E\sqrt{a}K_{3 / 2}(\lambda a)\nonumber \\ + F\sqrt{a}I_{3 / 2}(\lambda a) = 0, \end{aligned}$$

  (8.1)

  The slip boundary condition

  $$\begin{aligned} Aa \beta _{1} - B\left( \frac{\beta _{1}}{2a^{2}} +\frac{3}{a^{3}}\right) + C \left( 2\beta _{1}a^{3} - 3 a^{2}\right) + D\frac{\beta _{1}}{2} \nonumber \\ - E a^{-\frac{1}{2}}\left( \left( \frac{\beta _{1}}{2} +\frac{3}{a}\right) K_{3 / 2}(\lambda a) +\left( \frac{a\beta _{1}}{2}+1\right) \lambda \right. \nonumber \\ \left. K_{1 / 2}(\lambda a) \right) - F a^{-\frac{1}{2}}\left( \left( \frac{\beta _{1}}{2} +\frac{3}{a}\right) I_{3 / 2}(\lambda a) \right. \nonumber \\ \left. -\left( \frac{a\beta _{1}}{2}+1\right) \lambda I_{1 / 2}(\lambda a) \right) = 0, \end{aligned}$$

  (8.2)

  The couple stress component vanishes

  $$\begin{aligned}{} & {} 10C(1-\tau ) + D(2+\tau ) \frac{2}{a^{3}} \nonumber \\{} & {} - E \lambda ^{2}a^{-\frac{3}{2}}\left( (2+\tau ) K_{3 / 2}(\lambda a) + \lambda a K_{1 / 2}(\lambda a)\right) \nonumber \\{} & {} - F\lambda ^{2}a^{-\frac{3}{2}} \left( (2+\tau ) I_{3 / 2}(\lambda a)-\lambda a I_{1 / 2}(\lambda a)\right) = 0, \end{aligned}$$

  (8.3)

• On the surface of the cell model $(r = b)$ Radial velocity is continuous

  $$\begin{aligned}{} & {} A b^{2} +\frac{B}{b} + C b^{4} + D b+ E\sqrt{b}K_{3 / 2}(\lambda b) \nonumber \\{} & {} + F\sqrt{b}I_{3 / 2}(\lambda b) = b^{2}, \end{aligned}$$

  (8.4)

  The couple stress component vanishes on the cell surface

  $$\begin{aligned}{} & {} 10C(1-\tau ) + D(2+\tau ) \frac{2}{b^{3}}\nonumber \\{} & {} -E \lambda ^{2}b^{-\frac{3}{2}}\left( (2+\tau ) K_{3 / 2}(\lambda b)+ \lambda b K_{1 / 2}(\lambda b)\right) \nonumber \\{} & {} -F\lambda ^{2} b^{-\frac{3}{2}}\left( (2+\tau ) I_{3 / 2}(\lambda b) -\lambda b I_{1 / 2}(\lambda b)\right) = 0, \end{aligned}$$

  (8.5)

  • Happel’s model:

    $$\begin{aligned}{} & {} B\frac{3}{b^{4}}+3bC +Eb^{-\frac{5}{2}}\left( 3K_{3 / 2}(\lambda b)+\lambda b K_{1 / 2}(\lambda b)\right) \nonumber \\{} & {} + Fb^{-\frac{5}{2}} \left( 3I_{3 / 2}(\lambda b)-\lambda bI_{1 / 2}(\lambda b)\right) = 0, \end{aligned}$$

    (8.6)

  • Kuwabara’s model:

    $$\begin{aligned}{} & {} 10Cb^{2}-\frac{2}{b}D+ E\lambda ^{2}b^{\frac{1}{2}}K_{3 / 2}(\lambda b)\nonumber \\{} & {} +F\lambda ^{2}b^{\frac{1}{2}} I_{3 / 2}(\lambda b) = 0, \end{aligned}$$

    (8.7)

  • Kvashmin’s model:

    $$\begin{aligned}{} & {} \frac{3}{b^{4}}B+8b C-\frac{1}{b^{2}}D\nonumber \\{} & {} +Eb^{-\frac{5}{2}}\left( (3+\lambda ^{2}b^{2})K_{3 / 2}(\lambda b) +\lambda bK_{1 / 2}(\lambda b)\right) \nonumber \\{} & {} + Fb^{-\frac{5}{2}} \left( (3+\lambda ^{2}b^{2})I_{3 / 2}(\lambda b)-\lambda bI_{1 / 2}(\lambda b)\right) = 0, \end{aligned}$$

    (8.8)

  • Cunningham’s model:

    $$\begin{aligned}{} & {} A-\frac{1}{2b^{3}}B+ 2b^{2} C + \frac{1}{2b}D\nonumber \\{} & {} -E\frac{1}{2}b^{-\frac{3}{2}}\left( K_{3 / 2}(\lambda b)+ \lambda b K_{1 / 2}(\lambda b)\right) \nonumber \\{} & {} -F\frac{1}{2}b^{-\frac{3}{2}} \left( I_{3 / 2}(\lambda b)-\lambda b I_{1 / 2}(\lambda b)\right) = 1. \end{aligned}$$

    (8.9)

The above equations represent the four systems of algebraic equations (Happel’s model: Eq.(8.1)-Eq.(8.6), Kuwabara’s model: Eq.(8.1)-Eq.(8.5) and Eq.(8.7) Kvashnin’s model: Eq.(8.1)-Eq.(8.5) and Eq.(8.8), Cunningham’s model: Eq.(8.1)-Eq.(8.5) and Eq.(8.9)) for the determining the unknown constants A, B, C, D, E and F. The solutions in each case are cumbersome and lengthy, so they are not presented here.