The time-fractional advection-diffusion equation with Caputo-Fabrizio fractional derivatives (fractional derivatives without singular kernel) is considered under the time-dependent emissions on the boundary and the first order chemical reaction. The non-dimensional problem is formulated by using suitable dimensionless variables and the fundamental solutions to the Dirichlet problem for the fractional advection-diffusion equation are determined using the integral transforms technique. The fundamental solutions for the ordinary advection-diffusion equation, fractional and ordinary diffusion equation are obtained as limiting cases of the previous model. Using Duhamel’s principle, the analytical solutions to the Dirichlet problem with time-dependent boundary pulses have been obtained. The influence of the fractional parameter and of the drift parameter on the solute concentration in various spatial positions was analyzed by numerical calculations. It is found that the variation of the fractional parameter has a significant effect on the solute concentration, namely, the memory effects lead to the retardation of the mass transport.

This article explores the Jeffery-Hamel flow of an incompressible non-Newtonian fluid inside non-parallel walls and observes the influence of heat transfer in the flow field. The fluid is considered to be micropolar fluid that flows in a convergent/divergent channel. The governing nonlinear partial differential equations (PDEs) are converted to nonlinear coupled ordinary differential equations (ODEs) with the help of a suitable similarity transformation. The resulting nonlinear analysis is determined analytically with the utilization of the Taylor optimization method based on differential evolution (DE) algorithm. In order to understand the flow field, the effects of pertinent parameters such as the coupling parameter, spin gradient viscosity parameter and the Reynolds number have been examined on velocity and temperature profiles. It concedes that the good results can be attained by an implementation of the proposed method. Ultimately, the accuracy of the method is confirmed by comparing the present results with the results obtained by Runge-Kutta method.

This study deals with the numerical investigation of Jeffery-Hamel flow and heat transfer in Eyring-Powell fluid in the presence of an outer magnetic field by using Haar wavelet method. Jeffery-Hamel flows occur in various practical situations involving flow between two non-parallel walls. Applications of such fluids in biological and industrial sciences brought a great concern to the investigation of flow characteristics in converging and diverging channels. A suitable similarity transformation is applied to transform the nonlinear coupled partial differential equations (PDEs) into nonlinear coupled ordinary differential equations (ODEs), which govern the momentum and heat transfer properties of the fluid. Due to the high nonlinearity of resulting coupled ODEs, the exact solution is unlikely. Thus, the solution is approximated using a numerical scheme based on Haar wavelets and the results are verified by comparing with 4th order Runge-Kutta results.

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The helical flows of couple-stress fluids in a straight circular cylinder are studied in the framework of the newly developed, fully determinate linear couple-stress theory. The fluid flow is generated by the helical motion of the cylinder with time-dependent velocity. Also, the couple-stress vector is given on the cylindrical surface and the nonslip condition is considered. Using the integral transform method, analytical solutions to the axial velocity, azimuthal velocity, nonsymmetric force-stress tensor, and couple-stress vector are obtained. The obtained solutions incorporate the characteristic material length scale, which is essential to understand the fluid behavior at microscales. If characteristic length of the couple-stress fluid is zero, the results to the classical fluid are recovered. The influence of the scale parameter on the fluid velocity, axial flow rate, force-stress tensor, and couple-stress vector is analyzed by numerical calculus and graphical illustrations. It is found that the small values of the scale parameter have a significant influence on the flow parameters.

Starting flows of a viscous incompressible fluid, modeled by the time-fractional derivatives, within a rotating channel due to an impulsive pressure gradient are studied. Using the eigenfunction expansion, the analytic solutions in series form are obtained. The flow of the ordinary fluid is studied as a special case of the time-fractional problem. The convergence of series solutions is proved. In addition, using the classical analytical method, coupled with the Laplace transform and Stehfest’s algorithm, an approximate solution is found. The flow rates in x- and y-directions are determined. In the case of the ordinary fluid, the steady-state and transient components of velocities are obtained. The numerical calculations are carried out by using the Mathcad software. It is found that, for fractional fluids, the reversal flow is much attenuated if the values of the fractional parameter are less than 1.