Prandtl Boundary Layer Equations Research Articles

Two-phase numerical simulation of thermal and solutal transport exploration of a non-Newtonian nanomaterial flow past a stretching surface with chemical reaction

Abstract Non-uniform heat sources and sinks are used to control the temperature of the reaction and ensure that it proceeds at the desired rate. It is worldwide in nature and may be found in all engineering applications such as nuclear reactors, electronic devices, chemical reactors, etc. In food processing, heat is used to cook such as microwave ovens, pasteurize infrared heaters, and sterilize food products. Non-uniform heat sources are mainly used in biomedical applications, such as hyperthermia cancer treatment, to target and kill cancer cells. Because of its ubiquitous nature, the idea is taken as our subject of study. Heat and species transfer analysis of a non-Newtonian fluid flow model under magnetic effects past an extensible moving sheet is modelled and examined. Homogeneous chemical reaction inside the fluid medium is also investigated. This natural phenomenon is framed as a set of Prandtl boundary layer equations under the assumed convective surface boundary constraint. Self-similarity transformation is employed to convert framed boundary layer equations to ordinary differential equations. The resultant system is solved using the efficient finite difference utilized Keller box method with the help of MATLAB programming. The influence of various fluid-affecting parameters on fluid momentum, energy, species diffusion and wall drag, heat, and mass transfer coefficients is studied. Accelerating the Weissenberg number decelerates the fluid velocity. The temperature of the fluid rises due to variations in the non-uniform heat source and sink parameters. Ohmic dissipation affects the temperature profile significantly. Species diffusion reduces when thermophoresis parameter and non-uniform heat source and sink parameters vary. The Eckert number enhances the heat and diffusion transfer rate. Increasing the chemical reaction parameter decreases the shear wall stress and energy transmission rate while improving the diffusion rate. The wall drag coefficient and Sherwood number decrease as the thermophoretic parameter increases whereas the Nusselt number increases. We hope that this work will act as a reference for future scholars who will have to deal with urgent problems related to industrial and technical enclosures.

Impact of magnetic induction on the flow of self-rewetting power-law fluid over a disk surface: Onset of Marangoni convection

In the present investigation, the onset of Marangoni convection in the flow of self-rewetting fluid over a stretching disk has been discussed. The self-rewetting fluid is considered to be electrically conducting and the flow problem is discussed under the influence of an induced magnetic field. The fluid flow is mainly induced due to two different mechanisms: (i) onset of Marangoni convection due to presence of temperature gradient along the liquid-gas interface and (ii) induction of magnetic field due to presence of an external magnetic field, which modifies the original field. The governing partial differential equations are written, following Navier-stokes equations and Prandtl boundary layer equation. With the use of similarity transformations, the governing equations are turned into a system of ordinary differential equations. Nonlinear systems of ordinary differential equations are solved by the bvp4c technique. The reciprocal of the magnetic Prandtl number diminishes the induced magnetic profile.

Fractional Boundary Layer Flow: Lie Symmetry Analysis and Numerical Solution

In this paper, we present a fractional version of the Sakiadis flow described by a nonlinear two-point fractional boundary value problem on a semi-infinite interval, in terms of the Caputo derivative. We derive the fractional Sakiadis model by substituting, in the classical Prandtl boundary layer equations, the second derivative with a fractional-order derivative by the Caputo operator. By using the Lie symmetry analysis, we reduce the fractional partial differential equations to a fractional ordinary differential equation, and, then, a finite difference method on quasi-uniform grids, with a suitable variation of the classical L1 approximation formula for the Caputo fractional derivative, is proposed. Finally, highly accurate numerical solutions are reported.

Heat transfer characteristics of a Williamson fluid flow through a variable porosity regime

A theoretical framework pertaining to the study of boundary layer flow of a Williamson fluid over a moving vertical cylinder with variable porosity has been discussed. The significance of this analysis is to investigate the heat transfer enhancement for a shear thinning flow over a cylinder entrenched in a variable permeability of the porous medium. Prandtl boundary layer equations with appropriate conditions are solved numerically through the utility of a Crank Nicholson method. The impressions of the various morphological and rheological characteristics involving parameters like variable porosity parameter, Williamson parameter, Nusselt number and Sherwood number have been scrutinized and presented graphically. Low-velocity profile develops for the variable porosity parameter with high intensity. It has been discerned that the ferocity of heat and mass transfer rates is high, when there is less variation supervenes in the shear-thinning fluids. Moreover, the velocity boundary layer will override the thermal boundary layer for a substantial increase in the Prandtl number. Further, the validation of the results was verified through an extensive comparative study with the available results in the literature.

Numerical and analytical solutions of new Blasius equation for turbulent flow

The Blasius equation for laminar flow comes from the Prandtl boundary layer equations. In this article, we establish a new and generic Blasius equation for turbulent flow derived from the turbulent boundary layer equation that can be used for turbulent as well as laminar flow. The analytical and numerical solutions have been investigated under specific conditions to the developed new Blasius equation. The analytical and numerical results have been compared through tables and graphs to validate the established model. In fluid dynamics, analytical solutions to complicated systems are tedious and time-consuming. Changing one or more constraints can introduce new challenges. In this case, symbolic computation software provides an easier and more flexible solution for fluid dynamical systems, even if boundary conditions are adjusted to explain reality. Therefore, the MATLAB code is used to investigate the new third-order Blasius equation. The comparison and graphical representations demonstrate that the achieved results are encouraging.

Comparative Study of Two Dimensional Boundary Layer over NACA 23012 and NACA 23021 Airfoils by Using Keller’s Box Method

This paper begins with general description onvarious methods for solving boundary layer equation. In particular, Prandtl's boundary layer equation is a simplified version of Navier-Stokes equation thatis solved integrally. Order of magnitudeanalysiscan be used to solve laminar and turbulent flow problems along the boundary layer.A more complex technique based onKeller’sbox method isalso discussed.This study aims atutilising a developed computercode which allows one to predict the development of boundary layer characteristics along airfoils’ surface, andhavea comparative understanding on flow behavioursbetween NACA 23012 and NACA 23021 airfoils, as well as 0⁰ and 5⁰angles of attack. The ability of Keller’s box method to solve the boundary layer equation prevails through correctskin friction,momentum thickness, and shape factor distributions over the surfaces. It was found that skin frictionin boundary layer over NACA23021 is higher than that over NACA23012 for about 10% of the chord length from the leading edge. However, momentum thickness of the boundary layer over NACA23021is lessthan the boundary layer momentum thickness over NACA23012for the whole chord length. Similarly, the shape factor Hfor the boundary layer over the formeris smallerin comparison to the latterfor about 30% of the chord length.

Well-posedness and Blowup of the Geophysical Boundary Layer Problem

The proposal of this paper is to study the local existence of analytic solutions, and blowup of solutions in a finite time for the geophysical boundary layer problem. In contrast with the classical Prandtl boundary layer equation, the geophysical boundary layer equation has an additional integral term arising from the Coriolis force. Under the assumption that the initial velocity and outer flow velocity are analytic in the horizontal variable, we obtain the local well-posedness of the geophysical boundary layer problem by using energy method in the weighted Chemin-Lerner spaces. Moreover, when the initial velocity and outer flow velocity satisfy certain condition on a transversal plane, for any smooth solution decaying exponentially in the normal variable to the geophysical boundary layer problem, it is proved that its $$W^{1,\infty }-$$ norm blows up in a finite time. Comparing with the blowup result obtained in Kukavica et al. (Adv Math 307:288–311, 2017) for the classical Prandtl equation, we find that the integral term in the geophysical boundary layer equation triggers the formulation of singularities earlier.

Viscous dissipation impact on electrical resistance heating distributed Carreau nanoliquid along stretching sheet with zero mass flux

The present paper envisages on the electric resistance heating distribution on three-dimensional Carreau dissipated NF along a nonlinear stretching sheet. The characteristics of heat and mass transfer are conferred by utilizing nonlinear radiation and zero mass flux. The passive control on NPhas its own application in drug targeting therapy. Convectively hot fluid is placed near the stretching sheet. The governing Prandtl boundary layer equations are modeled using relative laws and transformed to highly nonlinear ordinary differential equations with similarity conversion variables. The dependent variables in governing equation are solved by shooting method with R-K scheme. A comparative study of Pseudoplastic and Dilatant fluids is deliberated in this study. Varied physical parameters, whose behaviors on the velocity, energy and species concentration are analyzed. Shear thickening fluid nature superiors the shear thinning fluid nature when the fluid flow swifts, whereas energy exchange from the system is more in Dilatant fluid. Heat transfer rate is higher when the fluid flow swifts. Heat transfer from the fluid to the surface is slow as the Eckert number along both x–y directions. The examination of present outcomes with the existing work has been made, which is good agreed. The present study reveals that the liquid stream velocity declines for the larger values of ratio of stretching rates parameter c and conflicting behavior is detected in tangential velocity.

Stratified Casson Fluid Flow Past a Riga-plate with Generative/Destructive Heat Energy

The combination of heat generation or absorption and cross-bedding effects of a chemically reacting and radiating Casson fluid flow on a permeable Riga-plate is numerically modelled. Solutions of the physical problem which is governed by the Prandtl boundary layer equations are attained using an implicit finite difference scheme. The objective of this analysis is to signify the impact of EMHD Lorentz force, stratification and heat source/sink on the rheology of the fluid flow. The improved modified Hartmann number and heat generation enhances the flow speed. Also, contour slices of velocity depict the precise perception of the flow. The intensity of shear stress near the surface is high owing to double stratification which is strengthened alongside of the plate surface. Reliability of the results assured by means of comparison study.

“The joy that engineers and mathematicians have come together.”1

“The joy that engineers and mathematicians have come together.”¹

Comparative analysis on non-linear radiative heat transfer on MHD Casson nanofluid past a thin needle

The aim of this work is scrutinizing the consequences of non-linear radiation on MHD Casson nanofluid along thin needle. The situation has been mathematically modelled taking into account the thermo-diffuso and diffuso-thermo effects. Here two types of surfaces are dealt; one is fixed needle and other is moving needle. The Prandtl boundary layer equations are enclosed and solved numerically using similarity variables. Impact of different material parameters on the momentum, temperature and species concentration along with the quantities related engineering aspects like skin friction coefficient, rate of energy transfer and Sherwood number are obtained and illustrated through graphs. A comparison examination is made between studied Casson nanoflow usage makes the environment cool, reduces the friction at the surface. But Newtonian nanofluid is good for species diffusion. Numerical obtained solutions are contrasted with the published literature and found to be in nice agreement. The present exploration exhibits the prominent features in hybrid solar magneto-hydrodynamic nanofluid systems and aircraft technology.

Combined Effects of Inclined Lorentzian forces and Melting heat transfer on Radiating Flow of Carbon Nanofluid past a Stretching Cylinder

An analysis was made to study the effects of non-uniform heat source or sink and aligned magnetic field effect on boundary layer flow of carbon nanofluid past a stretching cylinder with melting heat transfer and radiation. The Prandtl boundary layer equations are transformed into highly nonlinear ordinary differential equations utilizing similarity variables. The final resolved system is explained with shooting method. The upshots of governing factors on the velocity, temperature, surface skin friction and rate of heat transfer are discoursed with the help of graphs. MWCNT performance is better than SWCNT in case of fluid flow. SWCNTs transfers more heat from the fluid rather than multi wall tubes.Comparison of our numerical results with available literature works and shown a good agreement.

Long time well-posedness of Prandtl equations in Sobolev space

In this paper, we study the long time well-posedness for the nonlinear Prandtl boundary layer equation on the half plane. While the initial data are small perturbations of some monotonic shear profile, we prove the existence, uniqueness and stability of solutions in weighted Sobolev space by energy methods. The key point is that the life span of the solution could be any large T as long as its initial datum is a perturbation around the monotonic shear profile of small size e−T. The nonlinear cancellation properties of Prandtl equations under the monotonic assumption are the main ingredients to establish a new energy estimate.

Potential Systems and Nonlocal Conservation Laws of Prandtl Boundary Layer Equations on the Surface of a Sphere

Abstract: The potential systems and nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere have been investigated. The multiplier approach yields two local conservation laws for the Prandtl boundary layer equations on the surface of a sphere. Two potential variables ψ and ϕ are introduced corresponding to first and second conservation law. Moreover, another potential variable p is introduced by considering the linear combination of both conservation laws. Two level one potential systems involving a single nonlocal variable ψ or ϕ are constructed. One level two potential system involving both nonlocal variables ψ and ϕ is established. The nonlocal variable p is utilised to derive a spectral potential system. The nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere are derived by computing the local conservation laws of its potential systems. The nonlocal conservation laws are utilised to derive the further nonlocally related systems.

A well-posedness theory for the Prandtl equations in three space variables

The well-posedness of the three space dimensional Prandtl equations is studied under some constraint on its flow structure. Together with the instability result given in [28], it gives an almost necessary and sufficient structural condition for the stability of the three-dimensional Prandtl equations. It reveals that the classical Burgers equation plays an important role in determining this type of flow with special structure, that avoids the appearance of the secondary flow, an unstabilizing factor in the three-dimensional Prandtl boundary layers. And the sufficiency of the monotonicity condition on the tangential velocity field for the existence of solutions to the Prandtl boundary layer equations is illustrated in the three-dimensional setting. Moreover, it is shown that this structured flow is linearly stable for any smooth three-dimensional perturbation.

The van Dommelen and Shen singularity in the Prandtl equations

In 1980, van Dommelen and Shen provided a numerical simulation that predicted the spontaneous generation of a singularity in the Prandtl boundary layer equations from a smooth initial datum, for a nontrivial Euler background. In this paper we prove this numerical conjecture by rigorously establishing the finite time blowup of the boundary layer thickness.

Gevrey Class Smoothing Effect for the Prandtl Equation

It is well known that the Prandtl boundary layer equation is instable, and the well-posedness in Sobolev space for the Cauchy problem is an open problem. Recently, under the Oleinik's monotonicity assumption for the initial datum, [1] have proved the local well-posedness of Cauchy problem in Sobolev space (see also [21]). In this work, we study the Gevrey smoothing effects of the local solution obtained in [1]. We prove that the Sobolev's class solution belongs to some Gevrey class with respect to tangential variables at any positive time.

Almost Global Existence for the Prandtl Boundary Layer Equations

We consider the Prandtl boundary layer equations on the half plane, with initial datum that lies in a weighted $H^1$ space with respect to the normal variable, and is real-analytic with respect to the tangential variable. The boundary trace of the horizontal Euler flow is taken to be a constant. We prove that if the Prandtl datum lies within $\varepsilon$ of a stable profile, then the unique solution of the Cauchy problem can be extended at least up to time $T_\varepsilon \geq \exp(\varepsilon^{-1}/ \log(\varepsilon^{-1}))$.

Local Well-Posedness of Prandtl Equations for Compressible Flow in Two Space Variables

In this paper, we consider the local well-posedness of the Prandtl boundary layer equations that describe the behavior of the boundary layer in the small viscosity limit of the compressible isentropic Navier--Stokes equations with nonslip boundary condition. Under the strictly monotonic assumption on the tangential velocity in the normal variable, we apply the Nash--Moser--Hörmander iteration scheme and further develop the energy method introduced in [R. Alexander et al., J. Amer. Math. Soc., DOI:S0894-0347(2014)00813-4] to obtain the well-posedness of the equations locally in time.

Well-posedness issues for the Prandtl boundary layer equations

These notes are an introduction to the recent paper [7], about the well-posedness of the Prandtl equation. The difficulties and main ideas of the paper are described on a simpler linearized model.