MULTIPHASE PERISTALTIC FLOW

WHAT IS PERISTALSIS?

Peristalsis is a radially symmetrical contraction of muscles which propagates in a wave down the muscular tube. In humans, peristalsis is found in the contraction of smooth muscles to propel contents through the digestive tract.

PERISTALSIS

APPLICATIONS

The heart lung machine used during bypass surgery uses peristaltic flow as a mechanism to pump blood and avoid contact between blood and the components of the pump to prevent damage to RBC.

Shapiro, Jaffrin, & Weinberg. (1969). Peristaltic pumping with long wavelengths at low reynolds number. Journal of fluid mechanics , 799-825.

The reflux phenomenon in the peristaltic flow can be used to describe the travelling of bacteria against the flow from urinary bladder to kidney.

FROM MULTIPHASE POINT OF VIEW

Solid at the center – dispersed phase Fluid around the solid – continuous phase In actuality dispersed phase is much more

dispersed than depicted in the figure . The contents of the tube are normally a

particulate – fluid suspension.

MODELING PERISTALTIC FLOW

The situation can be suitable reconstructed for modeling purposes using a sine wave in the boundary of a flexible tube containing particulate fluid suspension

Final shape of tube

Initial shape of tube

Direction of wave propagation

Direction of wave propagation

Fluid / Particulate fluid suspension

WHY FLOW TAKES PLACE?

There is no external pressure gradient still the flow takes place ??The pressure falls from right to left in the contracted section ,owing to the viscous losses. Thus ,peristaltic wave produces a rising pressure in direction of wave.=>If the fluid is inviscid , there will be no flow.

PERISTALTIC FLOW FOR PARTICULATE FLUID SUSPENSION FLUID

Assumption:-Dispersed Phase continuumContinuous Phase continuumDispersed Phase particle-particle interactions can be neglected.

FOR FLUID PHASE

Momentum conservation :-

pressure drop viscous dissipation fluid-solid

interaction

Continuity :-

FOR PARTICULATE PHASE

Momentum conservation :-

pressure drop fluid solid

interaction

Continuity:-

TYPES OF INTERACTIONS IN A MULTIPHASE PARTICULATE – FLUID GRANULAR FLOW

Fluid-Fluid interaction Already accounted for in momentum conservation

Fluid-solid interaction The drag exerted by fluid on solid particles.

Solid-solid interaction Most complicated in nature as it depends on the volume fraction of particulate phase.

SOLID-SOLID INTERACTION

For low volume fraction it can be neglected as in the present problem.

For moderate volume fraction , interaction is of collisional nature.

collision For high volume fraction , interaction is of

frictional type.

friction

FLUID SOLID INTERACTION

It is primarily the drag exerted by one phase over another .

There are various formulae’s to represent drag depending on the volume fraction of particulate phase. For current case drag is takes as :-

VISCOSITY

Since both dispersed and continuous phase are considered as continuum , the viscosity taken is mixture viscosity represented as :-

RESULTS AND SOLUTIONS

The above set of equations is solved using analytical methods and perturbation.

PHENOMENON IN PERISTALTIC FLOW

Trapping - The phenomenon of trapping, whereby a bolus (defined as a volume of fluid bounded by closed streamlines in the wave frame) is transported at the wave speed.

Shapiro, Jaffrin, & Weinberg. (1969). Peristaltic pumping with long wavelengths at low reynolds number. Journal of fluid mechanics , 799-825.

REFLUX PHENOMENON

There are two ways in which reflux has been defined in literature :

Net Negative Velocity : The velocity in the larger cross-section of the peristaltic wave is in the direction of the flow and , the velocity in the smaller cross-section of the wave is in direction opposite to that of wave.

Net negative displacement : The net displacement of the particle over one cycle of wavelength is negative. The particle close the axis of the flow is positive displacement and that close to wall has a net negative displacement under certain conditions.

Shapiro, Jaffrin, & Weinberg. (1969). Peristaltic pumping with long wavelengths at low reynolds number. Journal of fluid mechanics , 799-825.

REFERENCES

1. en.wikipedia.org/wiki/Peristaltic

2. Kh. S.Mekheimer, E. F. (1988). Peristaltic Motion of a Particle Fluid Suspension in a planar channel. International Journal theoritical physics, 2895-2920.

3. Shapiro, Jaffrin, & Weinberg. (1969). Peristaltic pumping with long wavelengths at low reynolds number. Journal of fluid mechanics , 799-825.