In this paper, analytical expressions correlating the volumetric flow rate to the pressure drop are derived for the flow of Carreau and Cross fluids through straight rigid circular uniform pipes and long thin slits. The derivation is based on the application of Weissenberg-Rabinowitsch-Mooney-Schofield method to obtain flow solutions for generalized Newtonian fluids through pipes and our adaptation of this method to the flow through slits. The derived expressions are validated by comparing their solutions to the solutions obtained from direct numerical integration. They are also validated by comparison to the solutions obtained from the variational method which we proposed previously. In all the investigated cases, the three methods agree very well. The agreement with the variational method also lends more support to this method and to the variational principle which the method is based upon.

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Analytical Solutions for the Flow of Carreau and Cross Fluids in Circular Pipes and Thin Slits

This paper investigates the mobility of cilia in a non-uniform tapered channel in the presence of an induced magnetic field and heat transfer. Thermal radiation effects are included in the heat transfer analysis. The Jeffrey model is a simpler linear model that uses time derivatives rather than convected derivatives as the Oldroyd-B model does; it depicts rheology other than Newtonian. The Jeffrey fluid model is used to investigate the rheology of a fluid with cilia motion. The proposed model examines the behavior of physiological fluids passing through non-uniform channels, which is responsible for symmetrical wave propagation and is commonly perceived between the contraction and expansion of concentric muscles. To formulate the mathematical modeling, the lubrication approach is used for momentum, energy, and magnetic field equations. The formulated linear but coupled differential equations have been solved analytically. Graphs for velocity profile, magnetic force function, induced magnetic field, current density, pressure rise, and heat profile are presented to describe the physical mechanisms of significant parameters. It is found that the eccentricity parameter of the cilia equations opposes the velocity and the magnetic force functions. The thermal radiation decreases the temperature profile while it increases for Prandtl and Eckert numbers. A promising impact of the magnetic Reynolds number and electric field on the current density profile is also observed.

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Insight in Thermally Radiative Cilia-Driven Flow of Electrically Conducting Non-Newtonian Jeffrey Fluid under the Influence of Induced Magnetic Field

Bioconvection phenomena for MHD Williamson nanofluid flow over an extending sheet of irregular thickness are investigated theoretically, and non-uniform viscosity and thermal conductivity depending on temperature are taken into account. The magnetic field of uniform strength creates a magnetohydrodynamics effect. The basic formulation of the model developed in partial differential equations which are later transmuted into ordinary differential equations by employing similarity variables. To elucidate the influences of controlling parameters on dependent quantities of physical significance, a computational procedure based on the Runge–Kutta method along shooting technique is coded in MATLAB platform. This is a widely used procedure for the solution of such problems because it is efficient with fifth-order accuracy and cost-effectiveness. The enumeration of the results reveals that Williamson fluid parameter λ, variable viscosity parameter Λμ and wall thickness parameter ς impart reciprocally decreasing effect on fluid velocity whereas these parameters directly enhance the fluid temperature. The fluid temperature is also improved with Brownian motion parameter Nb and thermophoresis parameter Nt. The boosted value of Brownian motion Nb and Lewis number Le reduce the concentration of nanoparticles. The higher inputs of Peclet number Pe and bioconvection Lewis number Lb decline the bioconvection distribution. The velocity of non-Newtonian (Williamson nanofluid) is less than the viscous nanofluid but temperature behaves oppositely.

MHD Williamson Nanofluid Flow over a Slender Elastic Sheet of Irregular Thickness in the Presence of Bioconvection.

The main purpose of the current study is to invetigate the influence of Brownian motion and thermophoresis diffusion in non-Newtonian Williamson fluid flow through exponentially stretching sheet with the effects of thermal radiation and the bioconvection of microorganisms. For this purpose, similarity functions are involved to transmute partial differential equations to corresponding ordinary differential equations. Then Runge–Kutta method with shooting technique is hired to evaluate the desired findings with utilization of MATLAB script. The fluid velocity becomes slow against strength of magnetic parameter and it boosts with mixed convection. The temperature rises with parameter of Brownian motion and thermophoresis. The bioconvection Lewis number diminishes the velocity field. Compared with the existing literature, the results show satisfactory congruence’s. 

Impact of activation energy and MHD on Williamson fluid flow in the presence of bioconvection

In the present investigation, the impact of variable transport properties on the magnetized three dimensional motion of Casson–Williamson nanofluids with nonlinear radiation influence past a Riga plate is discussed. For proper utilization of nanoliquids, a good account of distinct fluid transport variables in any flow geometry is essential. The governing flow equations of a stretching Riga plate of variable viscosity, mass diffusivity and thermal conductivity are obtained after imposing an appropriate similarity variables. The spectral quasi-linearization method (SQLM) was employed to get good account of all pertinent flow parameters. Apart from excellent agreement with earlier known work in literature, the method performance (numerical estimation tools) are discussed via CPU time, errors ( L ∞ and L 1 ), and comparison with Galerkin weighted residual method. Also engineering dimensionless parameters were presented for the present problem. The results therein established that flow resistivity of Williamson fluid is higher when compared to the Casson fluid. In addition, Casson fluid diffuses and conduct lesser in respect to Williamson fluid. With opposing characterization of viscosity parameter, variable thermal conductivity and mass diffusivity enhances the species and thermal boundary layer, while the viscosity property appreciated the momentum(s) profiles. This study is found applicable while forecasting the right fluid rheology to use, proffer solution to noise/turbulence effect, to safeguard boundary layer isolation, and establishing a force parallel to the surface for both industrial, engineering and biomedical use.

A comparative study of three dimensional flow of Casson–Williamson nanofluids past a riga plate: Spectral quasi-linearization approach

A thorough and detailed investigation of an unsteady free convection boundary layer flow of an incompressible electrically conducting Williamson fluid over a stretching sheet saturated with a porous medium has been numerically carried out. The partial governing equations are transferred into a system of non-linear dimensionless ordinary differential equations by employing suitable similarity transformations. The resultant equations are then numerically solved using the spectral quasi-linearization method. Numerical solutions are obtained in terms of the velocity, temperature and concentration profiles, as well as the skin friction, heat and mass transfers. These numerical results are presented graphically and in tabular forms. From the results, it is found out that the Weissenberg number, local electric parameter, the unsteadiness parameter, the magnetic, porosity and the buoyancy parameters have significant effects on the flow properties.

https://www.mdpi.com/2079-3197/8/2/55/pdf

On the Numerical Analysis of Unsteady MHD Boundary Layer Flow of Williamson Fluid Over a Stretching Sheet and Heat and Mass Transfers

This work reports the Carreau–Yasuda nanofluid flow over a non-linearly stretching sheet viscous dissipation and chemical reaction effects. The coupled system of non-linear partial differential equations are changed into a system of linear differential equations employing similarity equations. The spectral quasi-linearization method was used to solve the linear differential equations numerically. Error norms were used to authenticate the accuracy and convergence of the numerical method. The effects of some thermophysical parameters of interest in this current study on the non-dimensional fluid velocity, concentration and temperature, the skin friction, local Nusselt and Sherwood numbers are presented graphically. Tables were used to depict the effects of selected parameters on the skin friction and the Nusselt number.

https://www.mdpi.com/2227-7390/8/7/1148/pdf

On Numerical Analysis of Carreau–Yasuda Nanofluid Flow over a Non-Linearly Stretching Sheet under Viscous Dissipation and Chemical Reaction Effects

This study investigates the effects of viscous dissipation and a heat source or sink on the magneto-hydrodynamic laminar boundary layer flow of a Jeffrey fluid past a vertical plate. The governing boundary layer non-linear partial differential equations are reduced to non-linear ordinary differential equations using suitable similarity transformations. The resulting system of dimensionless differential equations is then solved numerically using the bivariate spectral quasi-linearisation method. The effects of some physical parameters that include the Schmidt number, Eckert number, radiation parameter, magnetic field parameter, heat generation parameter, and the ratio of relaxation to retardation times on the velocity, temperature, and concentration profiles are presented graphically. Additionally, the influence of some physical parameters on the skin friction coefficient, local Nusselt number, and the local Sherwood number are displayed in tabular form.

MHD Laminar Boundary Layer Flow of a Jeffrey Fluid Past a Vertical Plate Influenced by Viscous Dissipation and a Heat Source/Sink

A theoretical analysis has been carried out to investigate the influence of unsteadiness on the laminar two-phase magnetohydrodynamic nanofluid flow filled with porous medium under the combined effects of Brownian motion and thermophoresis. Thermal variable conductivity, thermal radiation and viscous dissipation effects are also considered in this numerical study. The highly nonlinear partial differential equations are transformed into a set of coupled nonlinear ordinary differential equations through suitable similarity transformations. The resultant ordinary differential equations are then numerically solved using the spectral quasilinearization method. The effects of the pertinent physical parameters over the fluid velocity, temperature, concentration, skin friction, Nusselt and Sherwood numbers for both Blasius and Sakiadis flows are presented in graphical and tabular forms and discussed. It is found that the two types of flows behave differently when the physical parameters are varied.

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Unsteady mhd blasius and sakiadis flows with variable thermal conductivity in the presence of thermal radiation and viscous dissipation

Abstract In this paper, a bivariate spectral quasi-linearization method is used to solve the highly non-linear two dimensional Bratu problem. The two dimensional Bratu problem is also solved using the Chebyshev spectral collocation method which uses Kronecker tensor products. The bivariate spectral quasi-linearization method and Chebyshev spectral collocation method solutions converge to the lower branch solution. The results obtained using the bivariate spectral quasi-linearization method were compared with results from finite differences method, the weighted residual method and the homotopy analysis method in literature. Tables and graphs generated to present the results obtained show a close agreement with known results from literature.

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Hillary Muzara

Papers

On the Numerical Analysis of Unsteady MHD Boundary Layer Flow of Williamson Fluid Over a Stretching Sheet and Heat and Mass Transfers

On Numerical Analysis of Carreau–Yasuda Nanofluid Flow over a Non-Linearly Stretching Sheet under Viscous Dissipation and Chemical Reaction Effects

MHD Laminar Boundary Layer Flow of a Jeffrey Fluid Past a Vertical Plate Influenced by Viscous Dissipation and a Heat Source/Sink

Unsteady mhd blasius and sakiadis flows with variable thermal conductivity in the presence of thermal radiation and viscous dissipation

On the bivariate spectral quasi-linearization method for solving the two-dimensional Bratu problem