Some Applications Of Fractional Calculus In Anomalous Mass And Heat Transfer Problems

In this work,the applications of fractional calculus in fluid electroosmotic flows and laser heating field are mainly studied.The article includes five parts.In the first chapter,the basic knowledge of fractional calculus,integral transformation methods,electroosmot-ic flow,fractional generalized second grade fluid and heat conduction model are briefly introduced.With the development of microfluidics and nanofluidics,electroosmosis has become one of the most attractive fields.Numerous theoretical,experimental and numerical studies of electroosmotic flows in micro-and nano-channel have been reported.However,Lab-on-a-chip devices are usually used to analyze biofluids,such as blood,saliva and DNA solution.These fluids can not be treated as Newtonian fluids.And different rheological models have been proposed to describe their non-Newtonian flow behaviour.In view of this,many researchers have recently focused on the non-Newtonian fluid behavior in electrokinetically driven microflows.On the other hand,constitutive equations with fractional derivatives have been proved to be useful for describing viscoclastic bchavior in materials.It is worth not-ing that blood,as a typical biofluid,is essentially a,shear-thinning fluid and the modified second-grade fluid is the simplest constitutive model that can describe shear-thinning(or shear-thicking)and normal stress differences.Based on the above discussion,we study the electroosmotic slip flow of generalized second grade fluid.In the second chapter,the transient,slip flow of a fractional generalized second grade fluid in a.slit microchannel under the combined influence of electro-osmotic and pres-sure gradient forcings is studied.The analytical solution for velocity distribution of the electroosmotic flow is determined by employing the Debye-Hiickel approximation and the Laplace and cos-Fourier transforms.The corresponding solutions of classical Newtonian fluid and second grade fluid are derived as the limiting cases of our general results.These solutions are drawn as a sum of steady part and unsteady part.The effects of slip bound-ary conditions,fluid rheology,electroosmotic and pressure gradient forcings on the fluid velocity distribution are also discussed in terms of the pertinent dimensionless parameters graphically.Finally,the influence of dimensionless parameters on the velocity and flow rate of Newtonian,second grade fluids and fractional second grade fluids is compared.In the third chapter,the unsteady electroosmotic slip flow of viscoelastic fluid through a parallel plate micro-channel under combined influence of electroosmotic and pressure gradient forcings is analytically and numerically investigated.The generalized second grade fluid with fractional derivative is used for the constitutive equation.The Navier slip with different slip coefficients at both walls and asymmetric zeta potentials at the walls is also considered.By employing the Debye-Huckel linearization and the Laplace and sin-cos-Fourier transforms,the analytical solutions for the velocity distribution are derived.And the numerical results are given by finite difference method.Finally,the influence of pertinent parameters on the generation of flow is presented graphically.Our results may be useful for the study of the flow of non-Newtonian fluids.In the fourth chapter,we investigate the non-fourier heat conduction behavior in a finite medium heated by short pulse laser heating based on fractional dual-phase-lag model.Considering the application of dual-phase-lag model in the heat transfer field,the fractional dual-phase-lag model and the corresponding fractional heat conduction equation for short pulse laser heating is established.The semi-analytical solution for the temperature distribution is obtained by using the Laplace and cos-Fourier transform method.Finally,the influence of fractional parameter on the temperature distribution is studied graphically.Our results can provide the new non-Fourier heat conduction model for further study on laser pulse heating.Finally,In the fifth chapter,we summarize the main conclusions and discuss the development of fractional calculus.