Stress diffusion and high order viscoelastic effects in the 3D flow past a sedimenting sphere subject to orthogonal shear

Abstract

The effect of polymer stress diffusion in the unbounded flow past a sedimenting, freely rotating, rigid sphere subject to shear in a plane perpendicular to the direction of sedimentation is investigated analytically. Steady state, creeping, incompressible, and isothermal flow is assumed. For viscoelastic fluids following the Oldroyd-B constitutive model, three-dimensional results for the velocity vector, pressure, and viscoelastic extra-stress tensor are derived by including an artificial diffusion term in the constitutive equation and using regular perturbation theory with the small parameter being the Deborah number. The analytical solution reveals that the influence of the stress diffusion term on the results may be significant (and sometimes unexpected) and strongly depends on the magnitude of the dimensionless diffusion coefficient. For instance, it is shown that the critical Deborah number, below which a physical solution arises, decreases with the increase in the diffusion coefficient. Also, comparison against simulation results from the literature shows excellent agreement up to shear Weissenberg number (defined as the product of the imposed shear rate with the single relaxation time of the fluid) approximately equal to unity.

Appendix

The individuals contributions to the total drag force, up to fourth order in Deborah number, are:

Form drag: D _P ≈D _P0+De D _P1+De² D _P2+De³ D _P3+De⁴ D _P4, where

$$D_{P0} =2\pi $$

$$D_{P1} =0 $$

$$\begin{array}{@{}rcl@{}} D_{P2} &=&\pi \eta_{p} \left\{ -\frac{3951}{25025}+\frac{33535a^{2}}{14014}\right.\\ &&\left.\qquad\quad+\left( {\frac{3}{175}-\frac{127285a^{2}}{68796}} \right)\eta_{p} \right\}\\ D_{P3} &=&\pi \,\mu \,\eta_{p} \left\{ \frac{127(-10962+11155\,a^{2})}{45045}\right.\\ &&\qquad\qquad+\left( {\frac{1511802}{120120}-\frac{12542935\,a^{2}}{120120}} \right)\eta_{p}\\ &&\qquad\qquad\left.+\left( {-\frac{215514}{180180}+\frac{5604785\,a^{2}}{180180}} \right){\eta_{p}^{2}} \right\}\\ D_{P4} &=&\pi \,\eta_{p} \left\{ m_{P,0} +m_{P,1} \eta_{p} +m_{P,2} \eta _{p}^{2} +m_{P,3} {\eta_{p}^{3}}\right.\\ &&\left.\qquad\quad +\mu^{2}\left( {l_{P,0} +l_{P,1} \eta_{p} +l_{P,2} {\eta_{p}^{2}} +l_{P,3} {\eta_{p}^{3}} } \right) \right\} \end{array} $$

Friction drag: D _V ≈D _V0+De D _V1+D e ² D _V2+D e ³ D _V3+D e ⁴ D _V4, where

$$ D_{V0} =4\pi \notag $$

$$ D_{V1} =0 \notag $$

$$ D_{V2} \,=\,\pi \eta_{p} \left\{ {\frac{2403}{25025}\,+\,\frac{23735a^{2}}{7007}-\left( {\frac{9}{175}\,+\,\frac{538735a^{2}}{189189}} \right)\eta_{p} } \right\} \notag $$

$$\begin{array}{@{}rcl@{}} D_{V3} &=&\pi \,\mu \,\eta_{p} \left\{ \frac{6846}{143}+\frac{\text{2570236}{\kern 1pt}a^{2}}{\text{9009}}\right.\\ &&\qquad\qquad-\left( {\frac{\text{28398\thinspace \thinspace }}{\text{5005}}+\frac{\text{640883}a^{2}}{\text{12012}}} \right)\eta_{p}\\ &&\left.\qquad\qquad+\left( {\frac{60}{7}+\frac{\text{377413\thinspace }a^{2}}{\text{18018}}} \right){\eta_{p}^{2}} \right\} \notag\end{array} $$

$$\begin{array}{@{}rcl@{}} D_{V4} &=&\pi \,\eta_{p} \left\{ m_{V,0} +m_{V,1} \eta_{p} +m_{V,2} \eta _{p}^{2} +m_{V,3} {\eta_{p}^{3}}\right.\\ &&\left.\qquad\quad+\mu^{2}\left( {l_{V,0} +l_{V,1} \eta_{p} +l_{V,2} {\eta_{p}^{2}} +l_{V,3} {\eta_{p}^{3}} } \right) \right\} \notag\end{array} $$

Elastic drag: D _E ≈D _E0+De D _E1+De² D _E2+De³ D _E3+De⁴ D _E4, where:

$$ D_{E0} =D_{E1} =D_{E2} =0 \notag $$

$$\begin{array}{@{}rcl@{}} D_{E3} &=&-\frac{4\pi \,\mu \,\eta_{p} }{35}\left\{ 3(63+925a^{2})+\eta_{p} (63-900a^{2}\right.\\ &&\qquad\qquad\quad\left.+7(9+25a^{2})\eta_{p} ) \right\} \notag\end{array} $$

$$ D_{E4} =0 \notag $$

Note that m _j =(m _P,j +m _V,j )/6 and l _j =(l _P,j +l _V,j )/6,j=0,1,2,3 where m _j ,l _j also appear in the expression for the total drag, Eq. (33). The exact form of all these constants is the following:

$$\begin{array}{@{}rcl@{}} m_{P,0} &=&\!\!\frac{1191018}{11316305}\!\,+\,\!\frac{83050099793}{149918408640}a^{2}\,-\,\frac{2843447503601}{1484192245536}a^{4}\\ m_{P,1} &=&-\frac{203229914661}{6472926460000}+\frac{461167076437383887}{1071487855127289600}a^{2}\\ &&+\frac{18378215376689321853241}{2620109252142761258880}a^{4}\\ m_{P,2} &=&\frac{155605234737}{8238270040000}-\frac{49931383324670597}{214297571025457920}a^{2}\\ &&-\frac{1605323717329068271147}{227835587142848805120}a^{4}\\ m_{P,3} &=&-\frac{16987380103}{2384762380000}+\frac{135028136834209}{4110567219158400}a^{2}\\ &&+\frac{173007849818839694311}{63646378594561002240}a^{4}\\ \end{array} $$

$$\begin{array}{@{}rcl@{}} l_{P,0} &=&\frac{192893328}{12155}+\frac{4086250606}{323323}a^{2} \\ l_{P,1} &=&-\frac{58982301}{5720}+\frac{2993536913}{68068}a^{2} \\ l_{P,2} &=&\frac{1558368}{715}-\frac{575537264}{51051}a^{2} \\ l_{P,3} &=&-\frac{244917}{1430}+\frac{5015944}{17017}a^{2} \\ \end{array} $$

$$\begin{array}{@{}rcl@{}} m_{0} &=&\frac{2\,524\,971}{181\,060\,880}+\frac{38\,233\,944\,659}{899\,510\,451\,840}a^{2}\\ &&-\frac{2\,937\,202\,878\,593}{2\,968\,384\,491\,072}a^{4} \\ m_{1} &=&\frac{117\,018\,471}{681\,360\,680\,000}+\frac{1810219875810923}{10253472297868800}a^{2}\\ &&+\frac{13763975185670117186047}{5240218504285522517760}a^{4} \\ m_{2} &=&-\frac{447910935363}{181241940880000}-\frac{66062207387331247}{428595142050915840}a^{2}\\ &&-\frac{2340625218055859031859}{952767000779185912320}a^{4} \\ m_{3} &=&\frac{2426923127}{4769524760000}+\frac{369517655537579}{8221134438316800}a^{2}\\ &&+\frac{4528896505546350733}{4714546562560074240}a^{4} \end{array} $$

$$\begin{array}{@{}rcl@{}} l_{0} &=&-\frac{9196032}{12155}-\frac{33149131711}{969969}a^{2} \\ l_{1} &=&-\frac{699173433}{816816}+\frac{17463838019}{408408}a^{2} \\ l_{2} &=&\frac{209964}{715}-\frac{515925959}{17017}a^{2} \\ l_{3} &=&\frac{1276107}{2860}+\frac{1908703028}{153153}a^{2} \end{array} $$