Unsteady Laminar Boundary Layer Research Articles

The Effect of Variable Heat Flux on Unsteady Laminar MHD Boundary Layer Flow and Heat Transfer Due to a Stretching Sheet

The purpose of this research is to look into the solution technique for obtaining MHD velocity and temperature profiles. In the presence of a changing heat flux, the unsteady laminar boundary layer flow and heat transfer of a viscous incompressible fluid across a stretching sheet are numerically investigated. The unsteadiness is thought to be generated by a sudden increase in the surface temperature and a time-dependent stretching velocity. The flow and heat transfer partial differential equations were numerically solved using an implicit finite difference scheme and a quasi-linearization technique. Both velocity and temperature rise with time and magnetic field, according to the findings. The computed results are compared to previous work that has been published. Variable heat flux (VHF) conditions have also been taken into account.

Heat transfer modification in an unsteady laminar boundary layer subject to free-stream traveling waves

A laminar, thermal boundary layer was forced computationally by free-stream, traveling-wave velocity fluctuations and the effects on the wall heat flux and skin friction were measured as a function of the phase speed of the disturbances and the streamwise location along the developing flow. The heat flux modification due to the flow forcing was significantly higher than the corresponding skin friction enhancement, and the dependence of these two transport properties on the phase speed was qualitatively different. The skin friction modification exhibited a maximum at an optimal phase speed, which was explained in terms of the overlap of two distinct viscous layers within the boundary layer. The heat flux modification did not exhibit this maximum, although evidence was found to suggest such a maximum may occur with sufficient boundary layer development. Because the magnitude of the wall heat-flux modification scales quadratically with wave amplitude, traveling wave disturbances pose significant challenges for thermal transport measurements in periodically perturbed environments, like turbomachinery, but also new opportunities for the control of heat transfer.

The effects of thermal radiation, internal heat generation (absorption) on unsteady MHD free convection flow about a truncated cone in presence of pressure work

The aim of present paper is to analyse the effect of radiation, internal heat generation (absorption) on unsteady laminar natural convective boundary layer flow over a truncated cone considering pressure work and magnetic field due to the variation in fluid flow behavior in various physical conditions. Suitable transformation is used to form system of coupled non linear partial differential equations is to governing the flow and heat transfer. These equations have been solved numerically by using an implicit finite difference scheme along with quasilinearization technique. According to numerical findings, the pressure work, internal heat generation (absorption), radiation, and magnetic field have a significant impact on both skin friction and heat transfer. Further, skin friction is found to decrease 12.13%, 30.3% and 41.67% for various values (from 0.0 to 0.5) of pressure work, magnetic field and heat generation (absorption) alongside the cone surface whereas heat transfer rate rises 17.19%, 17.65% and 73.51% respectively due to unsteadiness in the flow. It is observed that the thickness of boundary layer enhances for various values (from 0.0 to 0.5) of heat generation and radiation parameter but diminishes for pressure work and Prandtl number.

Numerical and asymptotic study of unsteady laminar boundary layer flow over a moving wedge

Numerical and asymptotic study of unsteady laminar boundary layer flow over a moving wedge

Analytic similarity solutions for fully resolved unsteady laminar boundary layer flow and heat transfer in the presence of radiation

In this paper, we have compared a new type of similarity transformation derived systematically by using Lie point symmetries with the existing similarity transformations for unsteady fluid flow and heat transfer in the boundary layer in the presence of radiation. It is observed that the existing transformations map the steady and marginally accelerating flows only, while the Lie similarity transformations provide solutions for all types of accelerating flows and are independent of unsteadiness in the fluid. The previous transformations are valid for a specific time interval which depends on a range of unsteadiness parameter, however the Lie similarity transformations provide valid solutions at any given time. This implies that the Lie similarity transformations yield solutions for previously unexplored ranges of unsteadiness in the fluid. Boundary layer flow physics for both types of transformations is discussed by employing the Homotopy analysis method. We show that for accelerating fluids, in the developing region, the boundary layer thickness first increases and than starts to decrease with increase in unsteadiness for fully developed flow. Detailed comparison of velocity and temperature profiles in the boundary layer is made using the tables and graphs which show that with Lie similarity transformations the region of study of the considered flow extends significantly for the unsteadiness parameter. The effect of the Prandtl number and radiation parameter on temperature distribution is also compared for both types of similarity transformations. The Lie symmetry similarity transformations are shown to explain the unsteady laminar boundary layer flow and heat transfer to an extent where the existing similarity transformations do not work.

Magnetorheological Fluid of High-Speed Unsteady Flow in a Narrow-Long Gap: An Unsteady Numerical Model and Analysis

To investigate the unsteady flow field generated by magnetorheological (MR) fluid of a high-speed unsteady laminar boundary layer flow in a narrow-long gap of the magnetorheological absorber (MRA), a new unsteady numerical model is proposed. The gap has magnetic-field-activated and inactivated regions, with MR fluid flowing as bi-viscous (non-Newtonian) and Newtonian fluid. The unsteady flow field is described by the unsteady incompressible governing partial differential equation (PDE) and initial-boundary conditions with the moving boundary. The space-time solution domain is discretized using the finite difference method, and the governing PDE is transformed into implicit partial difference equations. The volume flow rate function is constructed to solve numerical solutions of pressure gradient and fluid velocity based on mass conservation, the continuity equation, and the bisection method. The accuracy of unsteady numerical model is validated by the experiment data. The results show that the fluid acceleration profiles along the gap’s height are non-uniform distribution. Further, the volume flow rate and excitation current has a significant impact on the dynamic distribution of fluid velocity profiles, and the moving boundary makes the flow field asymmetric about the central plane. Furthermore, as the transition stress increases, the thickness of the pre-yield region in the activated region increases. There is also a transition flow phenomenon in the activated region as the volume flow rate increases. Finally, the unsteady numerical model has good stability and convergence.

Radiation and heat absorption effects on an unsteady MHD boundary layer flow along an accelerated infinite vertical plate with ramped plate temperature in the existence of slip condition

A numerical analysis of an unsteady MHD laminar boundary layer flow over an accelerated moving vertical plate along with ramped temperature was performed, in which chemical reaction, heat absorption, and radiation were considered. Slip circumstances have sparked much concern, because of its widespread application in chemical engineering and manufacturing industries. The equations of momentum, thermal, and as well as species concentration, were solved numerically by implementing the Crank–Nicholson scheme. There is a visual representation of changes in fluid velocity, temperature, and concentration, the numerical values of drag force, Nusselt, and Sherwood numbers are provided in tabular form for various values of field variables. Results show that slip condition has a considerable impact on a small intensity of λ, and that this effect is extra pronounced in the steady condition than in the ramped condition. Profiles of velocity and temperature are reducing in both cases for increased values of heat absorption and magnetic field parameters.

Waves in a coherent two-dimensional flow

A condensate in two-dimensional turbulence confined to a finite domain was predicted by Kraichnan. The most common spatial form of the condensate is a coherent vortex of radius comparable with the domain size, which is statistically steady over times much longer than its turnover time. The vortices were successively studied during the last four decades, and their time-averaged properties were recently and extensively considered in theory and measured in experiments. Here, we consider weak perturbations of the coherent flow in the vortices. Our interest lies in slow perturbations, which implies that they are homogeneous along the streamlines of the coherent flow. We show that such kind of perturbations can be considered as waves of condensate propagating in the radial direction with some dispersion law. In the present work, the dispersion law and the propagation length of the waves are found as a function of the radial position inside a vortex flow. Cases of condensate saturated due to bottom friction and of viscous condensate are different. In the first case, there are waves with low damping. In the second case, all waves are of a similar kind as the shear waves in an unsteady laminar boundary layer.

Unsteady boundary layer flow and heat transfer of a viscous fluid over a semi-infinite stationary flat plate: Blasius flow problem

In this paper, a class of similarity solution for the unsteady Navier–Stokes equations which describe the two-dimensional unsteady laminar boundary layer flow of a viscous incompressible fluid over a semi-infinite flat stationary plate surface has been derived. Here, the boundary layer is formed owing to an uniform free stream flow over this plate surface and the unsteadiness in this flow dynamics comes only from the similarity solution which is not yet available in the open literature. For this similarity solution the governing boundary layer equations (momentum and thermal) transform into a set of non-linear ordinary differential equations which conceive two physical parameters, namely, unsteadiness parameter γ and Prandtl number Pr. The impact of these two parameters on the unsteady boundary layer flows as well as on the heat transfer analysis have been discussed in detailed. The present analysis confirms the non-existence of the boundary layer solutions after a certain negative value of γ, while it continues for a large positive value of γ and specifically for the value of γ = 0, we get the steady boundary layer solution for the Blasius flow problem. Here, we have considered the whole range of the Pr values, which includes both the limiting cases Pr → 0 and Pr → ∞, in the discussion of the heat transfer analysis. For these two limiting cases, two approximated separate mathematical formulae for the heat transfer solutions have been derived in terms of the well-known bessel, error and hypergeometric functions, whose results for any given value of γ within the solution domain are well fitted with the corresponding numerical results.

Comparison of the effect of 1st order and 2nd order fluctuating temperature field on convective heat transfer coefficient in an unsteady laminar-boundary layer

An unsteady thermal boundary layer analysis over a heat source (flat plate) has revealed that first order temperature change in source and core flow temperatures affects the time average or RMS value of h’S. It also emphasize that 1st order changes are important. Analysis also indicates that a phase difference between pressure and temperature exists at the source in order to increase the RMS value of h’S. Value of h’S is directly proportional to the tan/sine of the above phase difference, This implies that entropy oscillations must exist at the heat source. The purpose of this paper is to compare the result of 2nd order analysis previously done, with the 1st order study carried out in this paper. In second order analysis time average of h”S depends upon 2nd order change due to 1st order effects. Finally, existence of phase difference between pressure and temperature, thus having entropy oscillations are evident, comparison implies although there is similarity between 1st order and 2nd order results but first order effects are more important than 2nd order effects.

Effect of Magnetic Field on the Unsteady Boundary Layer Flows Induced by an Impulsive Motion of a Plane Surface

Abstract The unsteady laminar boundary layer flow of an electrically conducting viscous fluid near an impulsively started flat plate of infinite extent is considered, with a view to examine the influence of transverse magnetic field fixed to the fluid. A new type of similarity transformation is proposed, which renews the governing partial differential equation into a linear ordinary differential equation with four physical parameters, viz. unsteadiness parameter β, magnetic parameter M, and the velocity indices (p, q). The analytic solution of this equation has been found in terms of a first kind confluent hypergeometric function for some specific parameter regimes. This solution shows the structure of a new type of boundary layer flow that includes the solution of the first Stokes problem as a special case. For non-zero values of (p, q), there is a definite range of p (either −∞ < p < 2q or 2q < p < ∞ according to β < or > 0) for which this flow problem will be valid. This analysis reveals an important relation ( p β + M 2 = q β ) $(p\beta+{M^{2}}=q\beta)$ at which separation appears inside the layer and has been detected as the separation threshold of the problem. Indeed, this relation gives us the critical value of one when the others are known. Flow separation inside the layer is delayed with an increasing value of q but cannot be completely removed whatever is the value of q (>0). The present analysis ensures that the reverse flow can be suppressed by the use of a proper amount of magnetic field M depending upon the values of p, q, and β. The obtained result provides insight into the stability of the boundary layer flows.

Heat Transfer in Hydromagnetic Flow over an Unsteady Stretching Permeable Sheet

Unsteady laminar boundary layer flow of viscous incompressible electrically conducting fluid along a continuous stretched permeable surface with the magnetic field effect is investigated. The defining characteristics of unsteady laminar boundary layer flow are governed a more than one independent variables, stretching velocity and surface temperature of the field. Governing equations are obtained for influencing parameters and transformed into ordinary differential equations by taking convenient similarity variables. Runge-Kutta fourth order method in corporation by the shooting technique is introduced to carry out numerical computations of the investigation. Velocity and temperature profiles are computed and represented graphically for the influences of suction/injection parameter, unsteadiness parameter, magnetic parameter and Prandtl number, while numerical solutions of local skin friction coefficient and local Nusselt number are discussed through tables. For non-magnetic condition, results are found in concordance with earlier research work.

Transient analysis of Casson fluid thermo-convection from a vertical cylinder embedded in a porous medium: Entropy generation and thermal energy transfer visualization

Thermal transport in porous media has stimulated substantial interest in engineering sciences due to increasing applications in filtration systems, porous bearings, porous layer insulation, biomechanics, geomechanics etc. Motivated by such applications, in this article a numerical investigation of entropy generation effects on the heat and momentum transfer in unsteady laminar incompressible boundary layer flow of a Casson viscoplastic fluid over a uniformly heated vertical cylinder embedded in a porous medium is presented. Darcy’s law is employed to simulate bulk drag effects at low Reynolds number for an isotropic, homogenous porous medium. Heat line visualization is also included. The mathematical model is derived and normalized using appropriate transformation variables. The resulting time-dependent non-linear coupled partial differential conservation equations with associated boundary conditions are solved with an efficient unconditionally stable implicit finite difference Crank Nicolson scheme. The time histories of average values of momentum and heat transport coefficients, entropy generation and Bejan number, as well as the steady-state flow variables are computed for several values of non-dimensional parameters arising in the flow equations. The results indicate that entropy generation parameter and Bejan number are both elevated with increasing values of Casson fluid parameter, Darcy number, group parameter and Grashof number. To analyze the heat transfer process in a two-dimensional domain, plotting heat lines provides an excellent approach in addition to streamlines and isotherms. The dimensionless heat function values are shown to correlate closely with the overall rate of heat transfer. Bejan’s heat flow visualization implies that the heat function contours are compact in the neighbourhood of the leading edge of the boundary layer on the hot cylindrical wall. It is observed that as the Darcy number increases, the deviations of heat lines from the hot wall are reduced. Furthermore the deviations of flow variables from the hot wall for a Casson fluid are significant compared with those computed for a Newtonian fluid and this has important implications in industrial thermal materials processing operations.

Bioconvection in Casson nanofluid flow with Gyrotactic microorganisms and variable surface heat flux

This paper presents a two-dimensional unsteady laminar boundary layer mixed convection flow heat and mass transfer along a vertical plate filled with Casson nanofluid located in a porous quiescent medium that contains both nanoparticles and gyrotactic microorganisms. This permeable vertical plate is assumed to be moving in the same direction as the free stream velocity. The flow is subject to a variable heat flux, a zero nanoparticle flux and a constant density of motile microorganisms on the surface. The free stream velocity is time-dependent resulting in a non-similar solution. The transport equations are solved using the bivariate spectral quasilinearization method. A grid independence test for the validity of the result is given. The significance of the inclusion of motile microorganisms to heat transfer processes is discussed. We show, inter alia, that introducing motile microorganisms into the flow reduces the skin friction coefficient and that the random motion of the nanoparticles improves the rate of transfer of the motile microorganisms.

Effects MHD and Heat Generation on Mixed Convection Flow of Jeffrey Fluid in Microgravity Environment over an Inclined Stretching Sheet

In this paper, Jeffrey fluid is studied in a microgravity environment. Unsteady two-dimensional incompressible and laminar g-Jitter mixed convective boundary layer flow over an inclined stretching sheet is examined. Heat generation and Magnetohydrodynamic MHD effects are also considered. The governing boundary layer equations together with boundary conditions are converted into a non-similar arrangement using appropriate similarity conversions. The transformed system of equations is resolved mathematically by employing an implicit finite difference pattern through quasi-linearization method. Numerical results of temperature, velocity, local heat transfer, and local skin friction coefficient are computed and plotted graphically. It is found that local skin friction and local heat transfer coefficients increased for increasing Deborah number when the magnitude of the gravity modulation is unity. Assessment with previously published results showed an excellent agreement.

Unsteady three-dimensional stagnation point magnetohydrodynamic flow of bionanofluid with variable properties

Unsteady three-dimensional laminar stagnation point forced convective boundary layer magnetohydrodynamic flow of a bionanofluid with variable transport properties is studied theoretically and numerically. Thermal convective and zero mass flux boundary conditions are incorporated in this study. The transport properties are assumed to be a function of nanoparticle volume fraction to get physically realistic results. The dimensional boundary layer equations along with the coupled boundary conditions are transformed via similarity transformations into a system of ordinary differential equations. The transformed equations are solved numerically using the Runge–Kutta–Fehlberg fourth-, fifth-order numerical method. The effect of selected governing parameters, namely, viscosity, thermal conductive, mass diffusivity, microorganism diffusivity, magnetic field and bioconvection Schmidt number, on the dimensionless velocity, temperature, nanoparticle volume fraction, microorganism, skin friction coefficient, heat transfer rate, mass transfer rate and microorganism transfer rate, is illustrated graphically and interpreted in detail. Comparisons with previous works are carried out for some limiting cases and found to be in good agreement.

Large-Eddy Simulation of Buoyancy-Induced Flow in a Sealed Rotating Cavity

Buoyancy-induced flows occur in the rotating cavities of gas turbine internal air systems, and are particularly challenging to model due to their inherent unsteadiness. While the global features of such flows are well documented, detailed analyses of the unsteady structure and turbulent quantities have not been reported. In this work, we use a high-order numerical method to perform large-Eddy simulation of buoyancy-induced flow in a sealed rotating cavity with either adiabatic or heated disks. New insight is given into long-standing questions regarding the flow characteristics and nature of the boundary layers. The analyses focus on showing time-averaged quantities, including temperature and velocity fluctuations, as well as on the effect of the centrifugal Rayleigh number on the flow structure. Using velocity and temperature data collected over several revolutions of the system, the shroud and disk boundary layers are analyzed in detail. The instantaneous flow structure contains pairs of large, counter-rotating convection rolls, and it is shown that unsteady laminar Ekman boundary layers near the disks are driven by the interior flow structure. The shroud thermal boundary layer scales as approximately Ra−1/3, in agreement with observations for natural convection under gravity.

On the fluid dynamics of unsteady separated stagnation-point flow of a power-law fluid on the surface of a moving flat plate

The transformation group theoretic approach is applied to perform an analysis of the unsteady separated stagnation-point (USSP) flow over a moving flat plate immersed in a non-Newtonian power-law fluid. This unsteady two-dimensional laminar boundary layer flow of a power-law fluid is governed by the classical Ostwald–de Waele power-law model. The system of governing partial differential equations reduces to an autonomous third order nonlinear ordinary differential equation via two-parameter group theoretic method. By using an efficient shooting method, we have established the existence of two different kinds of solutions, namely: (i) attached flow solution (AFS) and (ii) reverse flow solution (RFS) of this USSP flow problem under a definite negative value of the unsteadiness parameter β with a given value of the acceleration parameter a and for all values of the power-law index n considered in the present study. We have also found a unique AFS solution that is irrespective of n when β≥0. Dual solutions of the governing boundary layer equation are found for all kinds of fluids with the fixed values of a=1 and β=−1. The role of the rheological index n on the displacement thickness, skin-friction coefficient and velocity boundary layer are explored numerically. The results of this analysis reveal that the wall shear stress is always positive for AFS flow and negative for RFS flow for the above values of a and β with any given value of n. A novel result that emerges from this analysis is that the magnitude of the wall shear stress decreases and finally it approaches asymptotically to a constant value with the increase in the non-Newtonian behavior (n>1) of the fluids for both the cases of AFS and RFS.

Unsteady Flow of a Nanofluid Past a Permeable Shrinking Cylinder using Buongiorno's Model

The unsteady laminar boundary layer flow of a nanofluid and heat transfer over a permeable shrinking cylinder using the Buongiorno’s nanofluid model is investigated. Using a similarity transformation, the governing partial differential equations are transformed into a system of ordinary differential equations and then solved numerically using a shooting method. The numerical results are obtained for velocity, temperature and concentration profiles as well as the skin friction coefficient, the local Nusselt number and the local Sherwood number. Dual solutions are found to exist in a certain range of the suction and unsteadiness parameters. It is observed that suction parameter increase both the skin friction coefficient and the heat transfer rate at the surface, whereas the opposite trend is obtained for the Sherwood number. It is also observed that suction widens the range of the unsteadiness parameter for which the solution exists.

Unsteady magnetohydrodynamics flow about a stagnation point on a stretching plate embedded in porous medium

This paper presents an unsteady two-dimensional laminar boundary layer flow of an incompressible, viscous and electrically conducting fluid in a vicinity of a stagnation point towards a stretching plate embedded in porous medium with variable free stream velocity. The governing nonlinear partial differential equations of mass, momentum and energy are reduced to a system of nonlinear ordinary differential equations by using suitable similarity variables. The ordinary differential equations are tackled numerically by using Runge-Kutta Fehlberg fourth-fifth order method with shooting technique. The effect of the significant parameters on the fluid temperature and velocity are graphically presented and discussed. The numerically computed results are in good agreement with the available published results in the literature. It is found that the velocity increases with the increasing unsteadiness parameter where the velocity ratio parameter λ = 0.5 but it decreases for λ = 1.5.