Peristaltic flow of Phan-Thien-Tanner fluid: effects of peripheral layer and electro-osmotic force

Abstract

The two-layered electro-osmotic peristaltic flow of Phan-Thien-Tanner (PTT) fluid in a flexible cylindrical tube is analyzed. The core (inner) layer fluid satisfies the constitutive equation of PTT fluid model and the peripheral (outer) layer is characterized as a Newtonian fluid. For each region, the two-dimensional conservation equations for mass and momentum with electro-osmotic body forces are transformed from the fixed frame to the moving frame of reference. These equations are further simplified by invoking the constraints of long wavelength and low Reynolds number. Closed-form expressions for velocity and stream function are derived and then employed to investigate the pressure variations, trapping, interface region, and reflux for a variety of the involved parameters. The analysis reveals that the reflux and trapping can be restrained by appropriately tuning the electro-kinetic slip parameter and Deborah number. Further, the pumping efficacy can also be improved by adjusting the rheological and the electro-kinetic effects. These results may be helpful for improving the performance of the microfluidic peristaltic pump.

Appendix

Here, we provide the values of coefficients appearing in the interface polynomial Eq. (67):

$$ {A}_{14}=\left({U}_E-1\right)\left({\mu}_r-1\right), $$

$$ {A}_{12}=\frac{1}{24{\mu}_r}\left(3{\left(-1+{k}^2\right)}^2{P}_0\left(-1+{\mu}_r\right)+8{\mu}_r\left(3\left(-1+{k}^2+q+{U}_E-{k}^2{U}_E\right)+3\left(-1+{k}^2\right)\left({U}_E-1\right){\mu}_r-64{b}^2{\left({U}_E-1\right)}^3{\mu}_r^3+3{R}_0^2\left({U}_E-1\right)\left(-6+5{\mu}_r\right)\right)\right), $$

$$ {A}_{10}=\frac{1}{2{\mu}_r}\left({\left(-1+{k}^2\right)}^2{P}_0\left(-{R}_0^2\left(-1+{\mu}_r\right)+16{b}^2{\left({U}_E-1\right)}^2{\mu}_r^3\right)+2{\mu}_r\left({R}_0^2\left(4-4{k}^2-6q+15{R}_0^2\left({U}_E-1\right)+4\left(-1+{k}^2\right){U}_E\right)-2{R}_0^2\left(-2+2{k}^2+5{R}_0^2\right)\left({U}_E-1\right){\mu}_r+64{b}^2\left(-1+{k}^2+{R}_0^2\right){\left({U}_E-1\right)}^3{\mu}_r^3\right)\right), $$

$$ {A}_8=\frac{1}{8{\mu}_r}\left(-8{b}^2{\left(-1+{k}^2\right)}^4{P}_0^2\left({U}_E-1\right){\mu}_r^2+{\left(-1+{k}^2\right)}^2{P}_0\left(-5{R}_0^4+6{R}_0^4{\mu}_r-128{b}^2\left(-1+{k}^2+{R}_0^2\right){\left({U}_E-1\right)}^2{\mu}_r^3\right)+8{\mu}_r\left(5{R}_0^4\left(-1+{k}^2+3q-4{R}_0^2\left({U}_E-1\right)+{U}_{\mathrm{E}}-{k}^2{U}_{\mathrm{E}}\right)+2{R}_0^4\left(-3+3{k}^2+5{R}_0^2\right)\left({U}_E-1\right){\mu}_r-64{b}^2{\left(-1+{k}^2+{R}_0^2\right)}^2{\left({U}_E-1\right)}^3{\mu}_r^3\right)\right), $$

$$ {A}_6=\frac{1}{24}\left({b}^2{\left(-1+{k}^2\right)}^6{P}_0^3-480q{R}_0^6+24{b}^2{\left(-1+{k}^2\right)}^4{P}_0^2\left(-1+{k}^2+{R}_0^2\right)\left({U}_E-1\right){\mu}_r+12{\left(-1+{k}^2\right)}^2{P}_0\left(-{R}_0^6+16{b}^2{\left(-1+{k}^2+{R}_0^2\right)}^2{\left({U}_E-1\right)}^2{\mu}_r^2\right)+8\left({U}_E-1\right)\left(45{R}_0^8-3{R}_0^6\left(-4+4{k}^2+5{R}_0^2\right){\mu}_r+64{b}^2{\left(-1+{k}^2+{R}_0^2\right)}^3{\left({U}_E-1\right)}^2{\mu}_r^3\right)\right), $$

$$ {A}_4=\frac{1}{8{\mu}_r}{R}_0^8\left({\left(-1+{k}^2\right)}^2{P}_0\left(5+{\mu}_r\right)+8{\mu}_r\left(5-5{k}^2+15q+{R}_0^2\left({U}_E-1\right)\left(-6+{\mu}_r\right)+{\mu}_r-{k}^2{\mu}_r+\left(-1+{k}^2\right){U}_E\left(5+{\mu}_r\right)\right)\right), $$

$$ {A}_2=\frac{1}{2{\mu}_r}{R}_0^{10}\left(-{\left(-1+{k}^2\right)}^2{P}_0-2\left(4-4{k}^2+6q-{R}_0^2\left({U}_E-1\right)+4\left(-1+{k}^2\right){U}_E\right){\mu}_r\right), $$

$$ {A}_0=\frac{1}{8{\mu}_r}{R}_0^{12}\left({\left(-1+{k}^2\right)}^2{P}_0+8\left(1-{k}^2+q+\left(-1+{k}^2\right){U}_E\right){\mu}_r\right). $$

From Eq. (65), the real solution of P₀ can be obtained by employing Cardan-Tartaglia formula of algebraic cubic equation as follows:

$$ {P}_0=-\frac{B}{S}+\frac{S}{A}, $$

where $S={\left(-d+\sqrt{A^3{B}^3+{d}^2}\right)}^{\frac{1}{3}}$,

with d = 12(q − (U_E − 1))A²,  B = 1 + (μ_r − 1)k⁴ and A = b²μ_rk⁶.