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Many researchers study the Casson fluid flow in the cylinder since it imitates human blood flow in the small arteries. However, only a few researchers considered slip velocity at the boundary. The slip velocity is crucial in blood flow study due to naturally occurs during stretchable movement of the arteries. Hence, the study aims to obtain analytical solutions and understand the fluid flow behaviour with the slip velocity effect for the unsteady free convection flow of Casson fluid in a cylinder. The analytical solutions are obtained by using the joint methods of the finite Hankel transform and the Laplace transform. All initial and boundary conditions are satisfied by the analytical solutions that were obtained. The behaviour of velocity and temperature profiles are plotted and discussed graphically. It is evident from the results that increasing the slip velocity, Grashof number and time will enhance blood velocity while increasing the Casson parameter causes a decrement of blood velocity. Besides, the Prandtl number increases resulting in blood velocity and the blood temperature falling. Lastly, the obtained analytical solution is validated by comparing it with the previous study and found to be in good mutual agreement. The obtained analytical solution is significant to check the accuracy of the numerical solutions

The Riga plate is a substantial alteration in the world of engineering. Mainly used in submarines to regulate water flow, studying the behaviour of fluid flowing over a Riga plate is very advantageous. Although there are ample studies on fluid flowing over a Riga plate, the introduction of fractional derivatives, coupled with a non-Newtonian fluid, has yet to be done. Within the field of fluid mechanics, specifically boundary layer flow, fractional derivatives do not have a proven geometrical representation. However, analytical solutions would be useful in aiding experimental researches in the future. Thus, this study aims to present an analytical function for a Caputo-Fabrizio fractional derivative on an unsteady Casson fluid flowing over an accelerating vertical Riga plate by using the Laplace transform method. The parametric effects considered in this study is elucidated. Through observation of obtained graphical results generated via the obtained analytical solutions, it is found that amplification of the fractional parameter and modified Hartmann number increases the fluid velocity with an average increment of 42.05% and 1.56%, respectively. While amplification of the Casson parameter and Prandtl number dampens the fluid velocity by an average of 45.09% and 43.56%, respectively

Recently, fluid with fractional-order derivative model attracted many researchers to further study compared with the classical fluid mode since it is more precise and realistic. To imitate the applications of blood flow in narrow arteries, researchers focused on the fractional Casson fluid flow in the cylinder. However, most researchers solved the problems numerically and without considering the slip effect at the boundary. Thus, obtaining solutions analytically to the unsteady fractional Casson fluid flow in the slip cylinder with free convection is the goal of this study. The Caputo-Fabrizio fractional derivative approach is utilized to model this problem. By joining the approach of the Laplace transform and finite Hankel transform, the fractional governing equations are solved, and analytical solutions to the velocity and temperature profiles are gained. The fluid velocity rises as the slip velocity and Grashof number increase and it declines with the increment of the Casson parameter and Prandtl number. Increasing the fractional parameter will result in an increase in fluid velocity and temperature for a large time interval. The slip velocity effect influenced fluid flow, especially at the cylinder’s wall. These findings are beneficial to explore the more fractional-order derivative model and for studying the problems in biomedical engineering.

One of non-Newtonian fluid, Casson fluid, has a diversify application in the manufacturing and engineering sector due to its elasticity behaviour. This study provides the analytical solution of unsteady Casson fluid with the presence of thermal radiation and chemical reaction in a porous medium. Formulation of this fluid model is initiated with the partial differential equation (PDE) of the momentum and energy equations as well as the concentration equation. By applying the proper non-dimensional variables, these equations are then converted into dimensionless form. Subsequently, the derivation of the exact solutions for the concentration, temperature and velocity profiles is conducted with the operation of the Laplace transform method that satisfies both initial and boundary equations. Graphical illustrations, which portray the various effects on this study are generated. It has been revealed that as the thermal radiation, chemical species and porosity increases, so does the velocity profile. However, MHD and Casson parameter shown opposite behaviour in the velocity profiles.

Flow over a pipe or an elongated cylinder is widely applied in many engineering processes like wire coating and pipe coating. This encourages the present study to examine the fluid flow and heat transfer over a horizontal stretching cylinder with the impact of temperature-reliant thermal conductivity and thermal radiation. The influence of heat generation is also considered. The Carreau rheology model is applied to represent the liquid coating. The similarity technique is used to simplify the developed governing equations and then solved by the homotopy analysis method. The effects of the pertinent parameters such as the thermal conductivity parameter and Weissenberg number on the fluid field and heat transfer are studied by applying the calculated series of analytical solutions, which are scrutinized through graphs and tables. The Nusselt number has a negative function with the radiation and thermal conductivity parameters. Furthermore, the Weissenberg number affects the velocity and temperature profiles differently in conditions n < 1 and n ≥ 1, respectively. The present results are essential in optimizing the pipe coating process.

Driven by technological advancement, the Riga plate can be seen as a key feature in developing the engineering world. As such, this study aims to investigate the effects of an accelerating semi-infinite Riga plate over a convective flow of MHD Casson fluid incorporated with the Caputo fractional derivative. The obtained governing PDEs are converted in dimensionless form and reduced to systems of ODEs via Laplace transform. Zakian's method of inverse Laplace transform is then utilised to generate graphical results in the time domain. Variations of parameter such as Casson, modified Hartmann number, Grashof number, magnetic parameter and fractional parameters are investigated for velocity profiles. Skin friction coefficient is also calculated and presented numerically. Study shows that Riga plate aids in fluid flow, hence increasing its velocity.

Carreau fluid flows past a stretching cylinder is elucidated in the present study. The transformed self-similarity and dimensionless boundary layer equations are solved by using the Homotopy analysis method. A convergence study of the method is illustrated explicitly. Series solutions of the highly nonlinear differential equations are computed and it is very efficient in demonstrating the characteristic of the Carreau fluid. Validation of the series solutions is achieved via comparing with earlier published results. Those results are obtained by using the Keller-Box method. The effects of the Weissenberg number and curvature parameter on the velocity profiles are discussed by graphs and tabular. The velocity curves have shown different behavior in and for an increase of the Weissenberg number. Further, the curvature parameter K does increase the velocity profiles.

Casson fluid is a non-Newtonian fluid with its unique fluid behaviour because it behaves like an elastic solid or liquid at a certain condition. Recently, there are several studies on unsteady Casson fluid flow through a cylindrical tube have been done by some researchers because it is related with the real-life applications such as blood flow in vessel tube, chemical and oil flow in pipelines and others. Therefore, the main purpose of the present study is to obtain analytical solutions for unsteady flow of Casson fluid pass through a cylinder with slip velocity effect at the boundary condition. Dimensional governing equations are converted into dimensionless forms by using the appropriate dimensionless variables. Dimensionless parameters are obtained through dimensionless process such as Casson fluid parameters. Then, the dimensionless equations of velocity with the associated initial and boundary conditions are solved by using Laplace transform with respect to time variable and finite Hankel transform of zero order with respect to the radial coordinate. Analytical solutions of velocity profile are obtained. The obtained analytical result for velocity is plotted graphically by using Maple software. Based on the obtained result, it can be observed that increasing in Casson parameter, time and slip velocity will lead to increment in fluid velocity. Lastly, Newtonian fluid velocity is uniform from the boundary to the center of cylinder while Casson fluid velocity is decreased when approaching to the center of cylinder. The present result is validated when the obtained analytical solution of velocity is compared with published result and found in a good agreement.

One major kind of arterial disease in blood flow that attracted many researchers is arterial stenosis. Arterial stenosis occurs when a lumen of artery is narrowed by the accumulation of fats, cholesterols and lipids plaques at the inner layer of the wall of an artery. To treat this arterial disease, the drug (solute) is injected into the blood vessels. Injection of the drug into the blood vessel cause the occurrence of chemical reaction between the drug and blood proteins and it affects the effectiveness of the solute transportation in blood flow. Hence, this study examines the unsteady dispersion of solute with the influence of chemical reaction and stenosis height through a very narrow artery with a cosine-curved stenosis. The blood is treating as Herschel-Bulkley (H-B) fluid. The momentum and constitutive equations are solved analytically to gain velocity of H-B blood flow. The convective-diffusion equation is solved by applying the generalized dispersion model to gain the dispersion function of solute. The influence of chemical reaction, power-law index, plug flow radius and stenosis height on the solute dispersion process is investigated. The results are validated with the previous solution without the effect of chemical reaction and stenosis. The results showed a good conformity between the two solutions. An increase in the chemical reaction coefficient, stenosis height, power-law index and plug flow radius reduces the dispersion function. It is observed that the solute dispersion in blood flow is affected by chemical reaction and stenosis height. H-B fluid is an appropriate fluid to investigate the blood velocity and transportation of the drug in blood flow to the targeted stenosed region through a very narrow artery for the treatment of arterial diseases. The results of the present study can potentially be used to predict the changes of blood flow behavior and dispersion process in blood flow.

The effect of the thermal radiation on the MHD Casson fluid along with the fractional derivative in an oscillating vertical plate is elucidated. More exactly, the Caputo fractional model is utilized in developing the governing equations. Besides, the influence of the buoyancy force due to the temperature gradient has also been considered. The derived fractional partial differential equations are converted into ordinary differential equations by using the Laplace transform technique and then are solved for analytical solutions via the characteristic method. The inversion of the Laplace transformation is obtained through the numerical approach of Zakian. The effects of various physical parameters on the velocity and temperature profiles, Nusselt number, and skin friction have been analyzed and depicted in graphs and tables. The distribution of the velocity and temperature either in viscous or Casson fluid do enhance by the fractional parameter.

The effects of radiation and magnetohydrodynamic on unsteady Casson fluid through an accelerated plate is analysed. The problem is formulated in the form of Partial Differential Equations (PDE) with imposed initial and boundary conditions. The Partial Differential Equations are transformed into dimensionless form by introducing suitable non-dimensional variables. Laplace transform method is used to derive the exact solutions for temperature and velocity profiles, fulfilling all initial and boundary conditions. The effects of parameters are depicted and illustrated graphically for radiation, Casson fluid and time, as well as Magnetohydrodynamics (MHD). It is found that the thermal radiation rises due to an increase in temperature. Besides, the increasing of Casson fluid and MHD parameter has decreasing effect on velocity. Finally, the influence of time will increase the velocity of the fluid.

Fractional derivative has perfectly adopted to model few physical phenomena such as viscoelasticity of coiling polymers, traffic construction, fluid dynamics and electrical networks. However, the application of the fractional derivatives for describing the physical characteristics of non-Newtonian fluid over a moving plate is still rare. In the present study, the effect of the Caputo fractional derivative on the Casson fluid flow which is induced by an accelerated plate is analytically analysed. The governing equations are initially transformed into dimensionless expressions by using suitable dimensionless variables. Then the Laplace transform method is utilized to calculate the exact solutions for the fractional governing partial differential equations. The obtained solutions are validated by comparing the results for specific case with the existing solutions in the literature. The impact of fractional parameter, Prandtl number, and time on the velocity and temperature profiles are graphically showed and discussed. The results depict that the temperature and velocity increase with the increment of fractional parameter and time. Interestingly, the velocity decreases at region near the plate but is enhanced at the area far away from the plate when the Casson fluid parameter is increased. This study is essential in understanding the factional non-Newtonian fluid flows which is more realistic in nature.

In this paper, free convection of nanofluids flow with oscillating vertical parallel plates and mass diffusion were considered. Obtained equations were converted into ordinary differential equations with appropriate transformations. Method of Laplace transform was used to find the exact solutions of velocity, temperature and concentration profiles from the dimensionless governing equations. Discussion of graph pertaining to different embedded parameters such as Prandtl number, Schmidt number, Grashof and mass Grashof number, oscillating parameter and nanoparticles volume fraction parameter was also added. Skin friction, Nusselt and Sherwood number were also discussed and deliberated.