Iterative Methods for Solving the Fractional Form of Unsteady Axisymmetric Squeezing Fluid Flow with Slip and No-Slip Boundaries

Abstract

An unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates is proposed with slip and no-slip boundaries. Using similarity transformation, the system of nonlinear partial differential equations of motion is reduced to a single fourth-order nonlinear ordinary differential equation. By using the basic definitions of fractional calculus, we introduced the fractional order form of the fourth-order nonlinear ordinary differential equation. The resulting boundary value fractional problems are solved by the new iterative and Picard methods. Convergence of the considered methods is confirmed by obtaining absolute residual errors for approximate solutions for various Reynolds number. The comparisons of the solutions for various Reynolds number and various values of the fractional order confirm that the two methods are identical and therefore are suitable for solving this kind of problems. Finally, the effects of various Reynolds number on the solution are also studied graphically.

Analytical techniques for solving the equation governing the unsteady flow of a polytropic gas with time-fractional derivative

“…To illustrate the basic idea of this method, proposed first by Gejji and Jafari, consider the following general functional equation [18][19][20][21][22][23][24][25]:…”

“…(21) [30] which is unique in view of the Banach fixed point theorem [eq31]. The convergence of the NIM has been proved in [18,25].…”

“…However, it is still very difficult to obtain closed-form solutions for most models of real-life problems. A broad class of analytical and numerical methods were used to handle such problems such as variational iteration method [1][2][3][4][5][6], Adomian decomposition method [7][8][9][10], homotopy perturbation method [11][12][13][14][15][16][17], new iterative method [18][19][20][21][22][23][24][25] and integral iterative method [26,27]. It is worth mentioning that the new iterative, homotopy perturbation and integral iterative methods are applied without any discretization, restrictive assumption or transformation and are free from round off errors.…”

Solution of the Fractional Form of Unsteady Squeezing Flow through Porous Medium

“…The longitudinal and normal velocity components in radial and axial directions are ( , , ) and ( , , ), respectively. For more physical explanation, see [14][15][16].…”

“…Also, the homotopy perturbation method (HPM) has been developed by Qayyum et al [15], to model and analyze the unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates passing through a porous medium channel with slip boundary condition. The new iterative and Picard methods had been used by Hemeda and Eladdad in [16] for solving the fractional form of unsteady axisymmetric flow of a nonconducting, Newtonian fluid squeezed between two circular plates with slip and no-slip boundaries. For more studies on squeezing flow through porous medium and also for more theoretical and experimental studies on squeezing flow, you can see [17][18][19][20][21][22][23][24][25].…”