Peristaltic Pumping Steven A. Jones BIEN 501 Friday, April 18, 2008 Louisiana Tech University Ruston, LA 71272

Geometry a – mean distance of wall from centerline b a h(t) R z a – mean distance of wall from centerline b – amplitude of wall motion Louisiana Tech University Ruston, LA 71272

Boundary Conditions for vZ(R,Z) h(t) R Z vZ(h,Z) = 0, vy(h,Z) = dh(t)/dt dvZ(0,Z)/dZ = 0, vy(0,Z) = 0 Louisiana Tech University Ruston, LA 71272

Conservation of Mass Conservation of mass: Symmetry Incompressible Louisiana Tech University Ruston, LA 71272

r-momentum symmetry Conservation of Momentum (r-component): Louisiana Tech University Ruston, LA 71272

Tangential Momentum vq = 0 Gravity is balanced by pressure gradient. Conservation of Momentum ( -component): vq = 0 Gravity is balanced by pressure gradient. Louisiana Tech University Ruston, LA 71272

Axial Momentum Louisiana Tech University Ruston, LA 71272

Summary of Equations The three equations: Are more complicated than we would like them to be. However, a dimensional analysis will help. Louisiana Tech University Ruston, LA 71272

Analysis of Order If vr and d/dz (hence d/dt) are small because wall motion is small and has large wavelength, we can get rid of terms of second order. We can also get rid of terms on the left of the z-momentum equation if we assume low Re. Louisiana Tech University Ruston, LA 71272

Summary of Equations The three equations now reduce to: Note that vr is on the order of bc. Louisiana Tech University Ruston, LA 71272