الفهرس 
Unsteady Flow of Non-Newtonian Fluid in a Tube
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Abstract
In the present work, the isothermal unsteady axial (IUA) {uFB02}ows of incompress-ible viscoelastic upper-convected maxwell (UCM) {uFB02}uid through a straight tube of circular cross-section are examined analytically and numerically in the absence of external forces. The {uFB02}ow is considered initially at rest, and the {uFB02}ow pattern is investigated through the velocity pro{uFB01}les with regard to four cases of pressure-gradient {uFB01}eld: (i) Constant pressure-gradient, (ii) Exponentially ris- ing pressure-gradient with time, (iii) Exponentially falling pressure-gradient with time, and (iv) Periodic pressure-gradient. The isothermal dynamical equation has been obtained, and fourier-bessel series solution is assumed in a general form for the velocity {uFB01}eld. Finite hankel integral-transform and the inverse derivative technique are used to obtain the analytical solutions of the considered initial-boundary value problem. The velocity pro{uFB01}les have been determined exactly. The limited {uFB02}ow at zero relaxation-time is the New- tonian {uFB02}uids. A comparison is made between UCM and Newtonian {uFB02}uids for each case-study, and found to have common speeds at certain evolution- times called traverse-times which are dependent on the distance from the tube wall. Finite element method using galerkin-newmark combination algorithm is developed and used to obtain numerical solutions of the considered initial-boundary value problem for each case-study. The numerical solutions show good agreements with the analytical ones when using linear interpolation functions. The iteration-errors of galerkin-newmark algorithm depend on the functional form of time-dependent pressure-gradient. The numerical ve- locity pro{uFB01}les for UCM {uFB02}uid become highly convergent at long times of the {uFB02}ow which establishes the stability of galerkin-newmark combination itera-tion method