Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls
Abstract
The self-similar two-dimensional steady boundary-layer flow induced by a permeable surface stretching with velocity Uw(x)=A·xm in a quiescent fluid in the presence of suction or injection with velocity Vw(x)=a·x(m−1)/2 is considered for A>0 and m>−1. The exact analytic solutions of this problem are given for m=−1/3 and m=−1/2 and the mechanical characteristics of the corresponding flows are discussed in detail. Boundary layers of the same thickness corresponding to different lateral mass fluxes are described. It is shown that to the smallest entrainment velocity, there corresponds a vanishing skin friction, i.e. a `dragless motion' of the fluid. The exact results are also compared with the results of the analytical approximations reported recently by other authors (in a physically different but mathematically identical context) in this journal.
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Convective heating and mass transfer in Buongiorno model of nanofluid using spectral collocation method of shifted Chebyshev polynomial
2023, International Journal of ThermofluidsIn this article, the investigation is made to shed a light on the influence of convective boundary conditions on the boundary layer flow over a linearly stretching flat surface. The equations (partial differential equations) describing the model are transformed into system of nonlinear ordinary differential equations using similarity transformations. Equations contain various non-dimensional flow characterizing numbers viz. Prandtl number Pr, Lewis number Le, Biot number Bi, Brownian motion parameter Nb and thermophoresis parameter Nt. The influence of these parameters on thermal boundary layer, concentration distribution and temperature are analyzed in detail, by solving the equations using novel Shifted Chebyshev collocation method. The computed results, reduced Nusselt number, reduced Sherwood number, surface temperature and concentration profiles as functions of dimensionless numbers are validated by comparing the predicted results with available earlier findings (using other methods). To assert the convergence and stability of the scheme used (for much larger, but moderate, parameters values), predicted results are presented in various tabular forms. For presenting finer details of the computed values some results are also given graphically. The innovative semi-numerical scheme is robust and efficient compared with other conventional methods, used in previous studies and enables the analysis of the complex problem adequately.
The improved thermal performance of recently discovered hybridized nanofluids has become essential in large scale thermal processes. In fact, this is highly efficient technique to introduce the thermal efficiency of tranditional heat transferring fluids. The behavior of the nanofluid can be significantly impacted by the unsteady heating and magnetic field effects that may be present in many applications. Therefore, the current study investigat the unsteady magnetized flow of hybrid nanofluid with heat transport characteristics subject to thermal radiation and slip at the surface wall. The shrinking/stretching surface is chosen as a flow source, which is frequently occure in polymer technology, which deals with the deformability of elastic sheets, and in metallurgy, where continued strips are cooled. The novel form of shrinking surface flow is fundamentally a reverse flow and exhibits physical characteristics that differ significantly from the channel flow scenario. The distinctive features of this scruinity is the use of empirical relations to approximate the optimum thermophysical attributes of a water hybrid nanofluid in order to model the 2-dimensional flow past a flat shrinking/stretching sheet under the action of radiation, Lorentz forces and realastic boundary condition responses. The governing system of modelled equation are assembled using the Tiwari-Das model in conjunction with a hybrid mass-based nanofluid model. The bvp4c algorithm is employed within the computer MATLAB programme. The hybrid nanofluid flow shows conclusive improvement in the frictional coefficient and heat transport performance. However, the effectiveness the unsteadiness parameter deteriorates the heat transmission. In the contiguity of a suction parameter, multiple outcomes appear to arise for both stretched and shrinking instances. The coefficient of energy transport improves as the magnetic factor is augmented, however the skin coefficient of friction exhibits dual behavior for the second solutions. A time-dependence investigation is undertaken to figure out the reliability of the twin solutions, and it is discovered that merely one of them remains stable and aesthetically credible.
An exact solution for two-dimensional laminar boundary layer flows in porous media under stretching/shrinking boundary with power-law velocity
2023, Journal of the Taiwan Institute of Chemical EngineersThe boundary layer flow past stretching/shrinking sheet is widely discussed due to their significance in chemical and engineering applications. A class of laminar boundary layer flows driven by a permeable stretching/shrinking boundary with power-law velocities is under investigation as well as in the presence of mass transpiration and Darcy-Brinkman porous media.
The fluid flow modelled into coupled highly non-linear partial differential equations and these are mapped into nonlinear ordinary differential equations via similarity transformations, which as a consequence are analytically solved.
The domain of solution so derived is a function of mass transpiration, stretching/shrinking boundary, Brinkman viscosity ratio and power-law index. The associated nonlinear equations exhibit lower- and upper-branch solutions that reveal very interesting axial and transverse profiles for various physical parameters under different cases of power-law indices. The boundary-layer effect of various power-law indices over the moving surface is thoroughly revisited in the present study to include the influence of mass transpiration and porous media. In the mass suction case, velocity is a monotonically decreasing function and highest velocity happens at the wall. In the mass injection case, the highest velocity does happen in the fluid region.
Non-similar investigation of magnetized boundary layer flow of nanofluid with the effects of Joule heating, viscous dissipation and heat source/sink
2023, Journal of Magnetism and Magnetic MaterialsThe main objective of this numerical study is to investigate the steady, incompressible and two-dimensional magnetohydrodynamic convection flow of a nanofluid across a stretched sheet with the influence of viscous dissipation, Joule heating, and slip condition. Also, the effects of convective boundary condition, mass flux and heat source are considered. The nanofluid under consideration is a combination of water and copper nanoparticles. In this analysis, appropriate non-similar transmutations are applied to transform partial differential equations (PDEs) into dimensionless PDEs. The dimensionless PDEs are converted into ordinary differential equations (ODEs) using the local non-similarity method. ODEs are solved through the use of well-known method (MATLAB-based bvp4c). The different results of velocity profile and temperature profile are conferred through graphs for several dimensionless parameters including Hartmann number, slip parameter, porosity parameter, Biot number and Eckert number. Physical quantities like Nusselt number and surface drag force are also estimated in tabular form. From the obtained results, it is noted that enhancing the values of volume fraction, slip parameter cause an increment in the values of skin friction coefficient while enhancing the values of Hartman number, porosity parameter causes a reduction in the values of skin friction coefficient. Also, by increasing the values of volume fraction, slip parameter, Eckert number causes a reduction in the values of local Nusselt number while the values of Nusselt number enhances by increasing the values of Hartman number, porosity parameter and heat source parameter.
In this paper, the boundary layer flow of viscous incompressible fluid over an inclined stretching plate in porous media with body force and heat transfer has been studied. To solve this problem, we develop a suitable spline method which is used to calculate the velocity function of the flow problem. We proceed as follows:
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With a suitable stream function, the concerned boundary layer equation is converted into non-linear third order ordinary differential equation together with appropriate boundary conditions in an infinite domain which has been further linearized by using quasi-linearization method.
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Then, we develop a non polynomial quintic spline technique which has been used to find the numerical values of the velocity function of the flow problem. The convergence analysis of the developed spline technique has been discussed.
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Later, the method developed so far has been applied to solve nonlinear boundary value problem for different angles of inclination and Froude number. The values obtained so far have been used to study heat flow problem. Finally, skin friction has been discussed.
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Generalized heat and mass transport features of MHD Maxwell nanofluid flows past a linearly Bi-stretching surface in the presence of motile microorganisms and chemical reaction
2023, South African Journal of Chemical EngineeringRecent studies have shown that non-Newtonian fluid flows over stretching sheets have received a lot of interest. These fluids have extensive practical applications in various fields of engineering and industries. The significance of the present study is that it is used in copper wire thinning and annealing, aerodynamic emission of plastic films, the liquid film compression process, etc. In this article, a semi-analytical simulation of the magnetohydrodynamic Maxwell nanofluid flow containing nanoparticles and microorganisms over a stretching surface has been addressed. The nanofluid flow is examined through a bi-directional linearly stretched surface. The impacts of chemical reaction, Soret, and Dufour numbers on Mawell nanofluid flow are taken into account. The aim of the existing problem is to deliberate the Cattaneo-Christov heat and mass flux model in order to examine the thermal and mass transport phenomena. In the methodology section, the computation of the higher-order ODEs are solved by means homotopy analysis method. Some interesting results are found here. The increasing Soret number and chemical reaction have reduced the temperature distribution, whereas the greater Dufour number has increased the thermal profile. In the case of the Fourier law, the temperature is greater, while in the case of the Cattaneo-Christov heat flux model, the temperature is smaller. In the case of the Fick model, the concentration is greater, while in the case of the Cattaneo-Christov mass flux model, the concentration is smaller. The greater Dufour number and reducing Soret number or the greater Soret number and reducing Dufour number have significantly declined the heat rate transport. The greater Dufour number and reducing Soret number have increased the mass transfer rate, while the greater Soret number and reducing Dufour number have reduced the mass transfer rate.