Prandtl Boundary Layer Research Articles

Hydrodynamic stability of magnetic boundary layer flow of viscoelastic Walters' liquid B embedded in a porous medium

The present study investigates the linear stability of stagnation boundary layer flow of viscoelastic Walters' liquid B in the presence of magnetic field and porous medium by solving modified Orr–Sommerfeld equation numerically using the Chebyshev collocation method. The model is characterized mainly by the elasticity number (E), the magnetic number (Q), and the permeability parameter (K) in addition to the Reynolds number(Re). The Prandtl boundary layer equations derived for the present model are converted through appropriate similarity transformations, to an ordinary differential equation whose solution describes the velocity, which has oscillatory behavior. The solution of generalized eigenvalue problem governing the stability of the boundary layer has an interesting eigenspectrum. The spectra for different values of E, K, and Q are shown to be a continuation of Newtonian eigenspectrum with the instability belongs to viscoelastic wall mode for certain range of parameters. It is shown that the role of elasticity number is to destabilize the viscoelastic boundary layer flow, whereas both magnetic field and porous medium have the stabilizing effect on the flow. These interesting features are further confirmed by performing the energy budget analysis on the perturbed quantities. Region of negative production due to the Reynolds stress as well as production due to viscous dissipation and viscoelastic contributions in the positive region, and there is reduction in the growth rate of kinetic energy that causes stability. Other physical mechanisms related to flow stability are discussed in detail.

Prandtl boundary layer expansion with strong boundary layers for inhomogeneous incompressible magnetohydrodynamics equations in Sobolev framework

We consider the validity of Prandtl boundary layer expansion of solutions to the initial boundary value problem for inhomogeneous incompressible magnetohydrodynamics equations in the half-plane when both viscosity and resistivity coefficients tend to zero, where the no-slip boundary condition is imposed on velocity while the perfectly conducting condition is given on magnetic field. Since there exist strong boundary layers, the essential difficulty in establishing the uniform L∞ estimates of the error functions comes from the unboundedness of vorticity of strong boundary layers. Under the assumptions that the viscosity and resistivity coefficients take the same order of a small parameter and the initial tangential component of magnetic field has a positive lower bound near the boundary, we prove the validity of Prandtl boundary layer ansatz in L∞ sense in Sobolev framework. Compared with the homogeneous incompressible case considered in [33], there exists a strong boundary layer of density. Consequently, some suitable functionals should be designed and the elaborated co-normal energy estimates will be involved in analysis due to the variation of density and the interaction between the density and velocity.

The steady Prandtl boundary layer expansions for non-shear Euler flow

We continue the study on the validity of the Prandtl boundary layer expansions in [Gao et al., Sci. China Math. 66, 679–722 (2023)], whereby estimating the stream-function of the remainder, we proved the case when the Euler flow is the perturbation of shear flow in a narrow domain. In this paper, we obtain a new derivatives estimate of stream-function away from the boundary layer and then prove the validity of expansions for any non-shear Euler flow, provided that the width of the domain is small.

Formation and local stability of a two-dimensional Prandtl boundary layer system in fluid dynamics

By the linearized method, Oleǐnik-Samokhin had constructed solutions to distinguish between two cases of the behavior of the fluid at the initial stage of its motion past the surface in Oleinǐk and Samokhin (1999). In this paper, we consider the development of the boundary layer about a body that gradually starts to move in a resting fluid. In the analytical frame, the formation of the layer shows that, when the layer starts to move, the local solutions of the Prandtl boundary layer system is positive. By the Crocco transformation, the Prandtl boundary layer system reduces a degenerate parabolic equation with a nonlinear boundary value condition. Then by the reciprocal transformation, we can obtain a divergence type parabolic equation. We quote a new kind of BV entropy solution matching up with this divergence type parabolic equation when t≥t0, where t0 is a small enough positive constant. By imposing some close connections between the velocity of the outflow and the geometric characteristic of the spatial domain, using Kružkov’s bi-variables method, the local stability of BV entropy solutions is proved. Thus, by the Crocco inverse transformation, in the analytical frame, we obtain the existence and the uniqueness of the global solution of the Prandtl boundary layer system for a regular spatial domain.

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.

On nonlinear instability of Prandtl's boundary layers: The case of Rayleigh's stable shear flows

In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of Navier Stokes equations near a boundary as the viscosity goes to 0. His Ansatz has later been justified for analytic data by R.E. Caflisch and M. Sammartino. In this paper, we prove that his expansion is false, up to O(ν1/4) order terms in L∞ norm, in the case of solutions with Sobolev regularity, even in cases where the Prandlt's equation is well posed in Sobolev spaces.In addition, we also prove that monotonic boundary layer profiles, which are stable when ν=0, are nonlinearly unstable when ν>0, provided ν is small enough, up to O(ν1/4) terms in L∞ norm.

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.

Compatibility and generalized group approach for novel solutions in nonlinear fluid dynamics: Applications to remarkable Prandtl boundary layer equations

Compatibility and generalized group approach for novel solutions in nonlinear fluid dynamics: Applications to remarkable Prandtl boundary layer equations

Performance of copper nanofluid on changing width of stretching sheet using thermal radiation and heat flux

The phenomenon of nanomaterial carrying fluid flow over a non-linear elongated sheet with changing thickness is investigated in this study. The present research focuses at the flow of magnetic nanofluid over non-linear elongated sheets of different thicknesses, with an emphasis on copper nanoparticles distributed in water. The phenomena, which is significant in many industrial applications including paper manufacturing and atomic reactors, is analysed using Prandtl boundary layer theory and Navier-Stoke's equation. The study covers a wide range of physical features. There have been fewer works on the mathematical modelling of stretching sheets with varying widths. The main aim of investigation is to analyse the effect of EMHD, as well as other characteristics like Eckert number, Boit number, radiation effect and absorption factor. Paper manufacturing, dye and filament extrusion, atomic reactors, and metallurgical processes all rely heavily on these sheets of varied widths. The problem employs Navier-Stokes' equation and Prandtl boundary layer theory. The similarity transformation converts partial differential equations into ordinary differential equations. MATLbvp4c programme is used to obtain numerical solutions. The present study helps to achieve desired quality of stretching sheet.

Long-time existence of Gevrey-2 solutions to the 3D Prandtl boundary layer equations

Long-time existence of Gevrey-2 solutions to the 3D Prandtl boundary layer equations

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.

On the steady Prandtl boundary layer expansions

In this paper, we consider the zero-viscosity limit of the 2D steady Navier-Stokes equations in (0,L) × ℝ+ with no-slip boundary conditions. By estimating the stream-function of the remainder, we justify the validity of the Prandtl boundary layer expansions. Specially, we show the global stability under the concavity condition of the Prandtl profile for an arbitrarily large constant L when the Euler flow is shear.

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.

Computational study of consequence of effect of velocity slip on nanofluids with suspended CNTs

In this article, we have discussed two-dimensional steady, incompressible, and laminar hydromagnetic flow along with the transfer of heat above a plane plate that contains water and ethylene glycol-based nanofluids with multi- and single-wall CNTs. The velocity slip boundary condition is included in this research article to evaluate the nature of flow. The plate surface is subjected to variable wall temperature. The transfer of heat has been contemplated in the company of suspended CNTs above a flat plate. Navier–Stokes and the concept of the Prandtl boundary layer have been used to determine the governing equations of heat transfer and flow. Since the attained governing equations representing energy and flow are nonlinear PDEs, they are altered into nonlinear ODEs accompanied by analogous boundary limitations by applying some appropriate similarity transformations. Hence the "RK45" function is used to solve them numerically. The influence of governing parameters on the dimensionless flow and heat transfer profiles are studied and discussed thorough graphical plots .

Validity of Prandtl layer expansions for steady magnetohydrodynamics over a rotating disk

This paper is concerned with the validity of Prandtl boundary layer expansions for 2D steady viscous incompressible magnetohydrodynamic (MHD) flows over a rotating disk {(θ, r) ∈ [0, θ0] × [R0, ∞)} with a moving curved boundary {r = R0}. We establish the validity of boundary layer expansions and convergence rates in the Sobolev sense. Then, we extend the results by Iyer [Arch. Ration. Mech. Anal. 224(2), 421–469 (2017)] for Navier–Stokes equations to the MHD flows.

Prandtl–Batchelor Flows on a Disk

For steady two-dimensional flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity the vorticity is constant in an inner region separated from the boundary layer. In this paper, by constructing higher order approximate solutions of the Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on a disk with the wall velocity slightly different from the rigid-rotation. The leading order term of the flow is the constant vorticity solution (i.e. rigid rotation) satisfying Batchelor-Wood formula.

Physical Formation Mechanisms of the Southwest China Vortex

On the basis of the Prandtl boundary layer theory and an improved perturbation method, the process of laminar flow bifurcating into the Southwest China vortex (SWV) in the Hengduan Mountains is studied. The results show that the formation of SWV is mainly determined by the speed of incoming airflow in the direction of the main axis of the Hengduan Mountains. The vortex is generated in the leeward area of the Hengduan Mountains when the speed of incoming airflow is greater than the critical velocity. Moreover, it means that the laminar flow bifurcates into a vortex. The formation position of the SWV is mainly determined by the relative position of the incoming airflow in the windward area of the Hengduan Mountains and the main axis of the Hengduan Mountains. The seasonal distribution of SWVs is determined by both the velocity of the incoming airflow and the relative position of the incoming airflow to the main axis of the Hengduan Mountains. These findings are consistent with the SWV observation facts, which not only adequately explain the physical formation mechanisms and processes of SWVs, but also present the formation location and seasonal distribution of SWVs. Meanwhile, a solution from laminar to vortex in circumflow motion is also presented.

Instability of Boundary Layers with the Navier Boundary Condition

We study the L^{infty } stability of the Navier-Stokes equations in the half-plane with a viscosity-dependent Navier friction boundary condition around shear profiles which are linearly unstable for the Euler equation. The dependence from the viscosity is given in the Navier boundary condition as partial _y u = nu ^{-gamma }u for some gamma in {mathbb {R}}, where u is the tangential velocity. With the no-slip boundary condition, which corresponds to the limit gamma rightarrow +infty , a celebrated result from E. Grenier (Comm. Pure Appl. Math. 53:1067–1091, 2000) provides an instability of order nu ^{1/4}. M. Paddick (Differ. Integral Equ. 27:893–930, 2014) proved the same result in the case gamma =1/2, furthermore improving the instability to order one. In this paper, we extend these two results to all gamma in {mathbb {R}}, obtaining an instability of order nu ^{vartheta }, where in particular vartheta =0 for gamma le 1/2 and vartheta =1/4 for gamma ge 3/4. When gamma ge 1/2, the result denies the validity of the Prandtl boundary layer expansion around the chosen shear profile.

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.