Partial Differential Equations In Nature Research Articles

Unsteady MHD squeezing viscous Casson fluid flow in upright channel with cross-diffusion and thermal radiactive effects

The purpose of the present work is to analyse two-dimensional unsteady squeezing Casson fluid flow between porous channel along with cross-diffusion, viscous dissipation, radiation, and chemical reaction effects. To get the results of governing equations which are partial differential equations in nature, we have converted them into ordinary differential equations by utilizing suitable similarity transformation and then applied the optimal homotopy analysis method. The effects of various dimensionless parameters ascended in the velocity, temperature and concentration fields are discussed through graphs. Values of skin friction coefficient, Nusselt number, and Sherwood coefficients are derived and presented in tabular form. Results indicate that with enhancement in the values of the Casson parameter and magnetic parameter, the velocity decreases near the lower plate while it shows reverse behaviour near the upper one. The temperature of the fluid increases with an increase in both radiation parameter as well as with Eckert number while it decreases by augmenting values of the Casson parameter. Soret and Dufour number affect mass and heat transfer rate in reverse manner. Also, concentration decreases when chemical reaction parameter increases.

Unsteady Tangent Hyperbolic Fluid on Radiated Exponentially Porous Surface

This analysis refers the radiation and porosity effects on unsteady hyperbolic tangent fluid over exponentially stretching sheet. Using similarity transformation the governing equations which are partial differential equations in nature have been modified into nonlinear differential equations (ODE), and then we obtained the solution by shooting technique along with Runge-Kutta method. The dimensional less velocity and temperature have been represented graphically. We have found local Nusselt number and friction factor for distant dimensional less physical parameter values, and they displayed in table. At the end of the analysis we conclude that magnetic field decreases velocity of hyper tangent fluid and porosity effect decreases velocity of hyper tangent fluid. Further the radiation enriches temperature profile in injection case than suction case. This conclusion tells us injection case is useful in temperature transportation than suction case.

Unsteady Tangent Hyperbolic Fluid on Radiated Exponentially Porous Surface

This analysis refers the radiation and porosity effects on unsteady hyperbolic tangent fluid over exponentially stretching sheet. Using similarity transformation the governing equations which are partial differential equations in nature have been modified into nonlinear differential equations (ODE), and then we obtained the solution by shooting technique along with Runge-Kutta method. The dimensional less velocity and temperature have been represented graphically. We have found local Nusselt number and friction factor for distant dimensional less physical parameter values, and they displayed in table. At the end of the analysis we conclude that magnetic field decreases velocity of hyper tangent fluid and porosity effect decreases velocity of hyper tangent fluid. Further the radiation enriches temperature profile in injection case than suction case. This conclusion tells us injection case is useful in temperature transportation than suction case.

Impact of generalized Fourier’s law and Fick’s law for MHD flow of Ag‒H2O and TiO2‒H2O nanomaterials

Purpose The purpose of this paper is to investigate the effect of inclined magnetic field, variable viscosity and Cattaneo–Christov heat and mass flux theories on the steady MHD free convective boundary layer flow of viscous, incompressible and electrically conducting water-driven silver and titanium-oxide nanofluids over a vertical stretching sheet. Design/methodology/approach The boundary layer equations of momentum, energy and nanoparticle concentration are partial differential equations in nature, which are reduced to nonlinear ordinary differential equations by means of similarity transformations. The resulting nonlinear equations are solved analytically by means of optimal homotopy analysis method. Findings Assessments with numerical results are performed and are found to be in an excellent agreement. Numerical results of the skin friction factor, the local Nusselt number and the local Sherwood number are obtained through tables. The effects of various physical parameters on the velocity, temperature and nanoparticles fraction are incorporated through graphs. The study analyzes the efficiency of heat transfer of nanofluids in cooling plants and rubber sheets. Originality/value No research works have been conducted to evaluate the effects of various physical phenomena on the copper and titanium nanofluids flow.

Influence of partial slip flow and thermal jump on mixed convection from an exponentially stretching surface

PurposeThis paper aims to provide a detailed study on the influence of slip flow and thermal jump over mixed convection flow along an exponentially stretching surface. Also, impacts of suction/blowing, volumetric heat source/sink and velocity ratio parameter will be studied in this analysis.Design/methodology/approachThe modeled governing equations for the assumed problem are dimensional nonlinear partial differential equations in nature. To reduce these equations, non-similar transformations are used to get the dimensionless nonlinear partial differential equations. Then, quasi-linearization technique is used to linearize these non-dimensional nonlinear partial differential equations. Finally, an implicit finite difference scheme is used to discretize the resulting equations.FindingsThe physical explanations are provided for the variations of various non-dimensional governing parameters over the velocity and temperature profiles. Also, the effects of these dimensionless parameters on skin friction coefficient and heat transfer rate are scrutinized in a manner which highlights their physical interpretation. The detailed discussion exhibits the fact that the streamwise co-ordinate velocity ratio parameter, partial slip parameter and the thermal jump parameter have significant influence over the flow and thermal fields.Originality/valueThis work has not been reported in the literature to the authors’ best of knowledge.

Unsteady mixed convection along a permeable exponentially stretching surface

PurposeThe purpose of this paper is to consider the influence of slip flow and thermal jump and to investigate its effects on unsteady mixed convection along an exponentially stretching surface. It is also intended to explore the influence of suction/injection and volumetric heat source/sink on the fluid flow.Design/methodology/approachThe assumed problem is modelled into governing equations which are dimensional non-linear partial differential equations in nature. To obtain solutions, initially the governing equations were made non-dimensional by the suitable non-similar transformations. Then, the dimensionless non-linear partial differential equations are linearized with the aid of Quasilinearization technique. The so obtained equations are discretized by the implicit finite difference method.FindingsThe detailed analysis of the considered problem displays that the non-similarity variable reduces the velocity and temperature profiles. For higher values of mixed convection parameter, the magnitude of velocity profile as well as the Nusselt number increase. The unsteady variable diminishes the fluid flow. The higher values of velocity ratio parameter reduce the skin-friction coefficient. Further, the magnitude of skin-friction coefficient and heat transfer rate are to minimize for increasing values of partial slip and thermal jump parameters, respectively. Volumetric heat source and injection parameters are to rise the flow behavior within the momentum and thermal boundary layers significantly.Originality/valueTo the best of authors’ knowledge, no such investigation has been found in the literature.

Electrical and mechanical fully coupled theory and experimental verification of Rosen-type piezoelectric transformers

A piezoelectric transformer is a power transfer device that converts its input and output voltage as well as current by effectively using electrical and mechanical coupling effects of piezoelectric materials. Equivalent-circuit models, which are traditionally used to analyze piezoelectric transformers, merge each mechanical resonance effect into a series of ordinary differential equations. Because of using ordinary differential equations, equivalent circuit models are insufficient to reflect the mechanical behavior of piezoelectric plates. Electromechanically, fully coupled governing equations of Rosen-type piezoelectric transformers, which are partial differential equations in nature, can be derived to address the deficiencies of the equivalent circuit models. It can be shown that the modal actuator concept can be adopted to optimize the electromechanical coupling effect of the driving section once the added spatial domain design parameters are taken into account, which are three-dimensional spatial dependencies of electromechanical properties. The maximum power transfer condition for a Rosen-type piezoelectric transformer is detailed. Experimental results, which lead us to a series of new design rules, also are presented to prove the validity and effectiveness of the theoretical predictions.