^{1}

^{*}

^{1}

^{1}

^{1}

^{1}

Based on the governing equations of the inner cyinder of the unsteady flow of the power law fluid in eccentric annuli with the inner cylinder reciprocating axially in bipolar coordinate system, the calculation formulae of tangential force were established, and the relevant numerical calculation method was given. Taking the aqueous solution of partially hydrolyzed polyacrylamides (HPAM) for examples, the tangential forces were calculated by using the formulae and numerical calculation method mentioned above; the curves of the tangential force on the wall of the inner cylinder of HPAM aqueous solution were plotted; and the effects on the tangential force of the flow behavior index of the power law fluid, the stroke and the stroke frequency of the inner cylinder were analyzed.

At present in our country, there is few theoretical research on the tangential force on the wall of the inner cylinder of the power law fluid flowing in eccentric annuli with the inner cylinder reciprocating axially. This force for sucker rod eccentric wear research has an important influence. Therefore, research on the tangential force has a practical significance. Based on the governing equations of the inner cyinder of the unsteady flow of the power law fluid in eccentric annuli with the inner cylinder reciprocating axially in the bipolar coordinate system [

Power law fluid with the inner cylinder reciprocating axially, _{o} is the radius of the inner cylinder of annuli, R_{i} is the radius of the inner cylinder, e is the eccentricity of the eccentric annulus.

The governing equations of the unsteady flow of the power law fluid in eccentric annuli with the inner cylinder reciprocating axially in the bipolar coordinate system by Haiqing Cui, Tao Gao [

By assumption flow conditions can be obtained the boundary conditions of dynamic flow equation.

where

From the formulae of shear stress on the wall of the inner cylinder

the calculation formulae of tangential force are given as follows

which can be expressed as

where

From the density of the power law fluid ρ, the flow behavior index of the power law fluid n, the consistency coefficient of the power law fluid k, the radius of the inner cylinder of annuli R_{o}, the radius of the inner cylinder R_{i}, the eccentricity of the eccentric annulus e, the stroke S, the stroke frequency of the inner cylinder f and the pressure gradient P, the distribution of the tangential force F_{z} can be calculated: firstly, through the equations ( 1a ), (1b), ( 1c ) and (1d), the velocity distribution of the power law fluid at anytime by using the finite difference method is calculated; then, through the formula (6), the tangential force by using the numerical integration method is calculated.

Take HPAM aqueous solution as an example, the radius of the inner cylinder of annulus R_{o} = 2.960 × 10^{−2} m, the radius of the inner cylinder R_{i} = 0.885 × 10^{−2} m, the pressure gradient of the aqueous solution P = 61.061 Pa/m and by using the calculation method mentioned above, the tangential forces on the wall of the inner cylinder of the HPAM aqueous solution are calculated and the relevant distribution curves are plotted as

Under different flow behavior index n, the curves of the tangential force to the cycle number N are shown as _{z} change a lot.

Under different eccentricities e, the curves of the tangential force to the cycle number are shown as _{z} change a lot.

Under different strokes S, the curves of the tangential force to the cycle number are shown as _{z} change a lot.

Under different frequencies f, the curves of the tangential force distribution to the cycle number are shown as _{z} change a lot.

1) Based on the governing equations of the inner cyinder of the unsteady flow of the power law fluid in eccentric annuli with the inner cylinder reciprocating axially in the bipolar coordinate system, the calculation formulae of tangential force on the wall of the inner cylinder of the power law fluid flowing in eccentric annuli with the inner cylinder reciprocating axially in the bipolar coordinate system were established, and the relevant numerical calculation method was given.

2) Taking the HPAM aqueous solution for examples, by using the calculation formulae and method mentioned above, the tangential forces of the HPAM aqueous solution on the wall of the inner cylinder were calculated and the tangential force curves were plotted.

3) The effects on the tangential force of the flow behavior index of the power law fluid, the eccentricity of the eccentric annulus, the stroke and the stroke frequency of the inner cylinder were analyzed; the influence of the eccentricity was more obvious.