Unsteady Boundary Layer Equations Research Articles

Group Classification of the Unsteady Axisymmetric Boundary Layer Equation

Unsteady equations of flat and axisymmetric boundary layers are considered. For the unsteady axisymmetric boundary layer equation, the problem of group classification is solved. It is shown that the kernel of symmetry operators can be extended by no more than four-dimensional Lie algebra. The kernel of symmetry operators of the unsteady flat boundary layer equation is found and it is shown that it can be extended by no more than a five-dimensional Lie algebra. The non-existence of the unsteady analogue of the Stepanov–Mangler transformation is proved.

Numerical Investigation of Three-Dimensional Magnetohydrodynamic Flow of Ag 􀀀 H2O Nanofluid Over an Oscillating Surface in a Rotating Porous Medium

Objective: To investigate the three-dimensional flow of a nanofluid (Ag-water) over a stretchable vertical oscillatory sheet. This study involves considering fluctuating temperatures on the sheet and comparing them to the free stream temperature. The formulation of the unsteady boundary layer equations leading to the flow of nanofluid also takes into consideration the occurrence of the heterogeneous-homogeneous chemical reaction and thermal radiation. Method: The governing equations and the boundary conditions have been derived in a dimensionless form by using the appropriate transformations, and they are then solved using an EFDS (Explicit Finite Difference Scheme) in Matlab software. The Von-Neumann stability analysis is used to determine the method’s stability requirements for constant sizes of the grid. Findings: The physical factors impact on the concentration fields, temperature distribution, and velocity distribution were obtained and are studied by graphs and described in extensive detail. Convergence and stability requirements are attained in order to achieve accurate solutions. Novelty: In this study fluctuations in the temperature and stretching velocity of sheet on three-dimensional magnetohydrodynamic flow of Ag − H2O nanofluid over an oscillating surface through rotating porous are taken into account. Impacts of porous media permeability, velocity slip, magnetic fields, nanoparticle volume fraction, heat radiation, rotation, and homogeneous and heterogeneous chemical reaction parameters had all been attempted to be determined. Keywords: Oscillatory Surface, Heat transmission, Nonlinear PDE, Explicit Finite Difference Scheme, Nanoparticle

Mathematical modelling of unsteady Oldroyd-B fluid flow due to stretchable cylindrical surface with energy transport

The unsteady boundary layer flow and thermal transportation analysis of an incompressible Oldroyd-B nanofluid driven by stretching cylinder is scrutinized with analytically approach in this study. The governing unsteady boundary layer equations (PDEs) are first established, and then these PDEs are converted into highly nonlinear ordinary differential equations (ODEs) using appropriate similarity transformations. The obtained equations are then solved in the form of series solutions via homotopy analysis method (HAM) for the velocity, temperature, and concentration fields. Additionally, the convergence analysis of the analytic series solutions is discussed. The model under consideration predicts the features of both the relaxation and retardation times effects. The aspects of thermophoresis and Brownian motion characteristics due to nanoparticles are studied by employing Buongiorno's model. The findings demonstrated that when the curvature parameter increases, the velocity, temperature, and concentration distributions near the cylinder's surface enhanced. Moreover, the velocity profile, for relaxation and the retardation times parameters, showed the opposite behavior. Also, the unsteadiness parameter enhanced the fluid velocity. The comparison of some current and old results is remarkable.

New Reductions of the Unsteady Axisymmetric Boundary Layer Equation to ODEs and Simpler PDEs

Reductions make it possible to reduce the solution of a PDE to solving an ODE. The best known are the traveling wave, self-similar and symmetry reductions. Classical and non-classical symmetries are also used to construct reductions, as is the Clarkson–Kruskal direct method. Recently, authors have proposed a method for constructing reductions of PDEs with two independent variables based on the idea of invariance. The proposed method in this work is a modification of the Clarkson–Kruskal direct method and expands the possibilities for its application. The main result of this article consists of a method for constructing reductions that generalizes the previously proposed approach to the case of three independent variables. The proposed method is used to construct reductions of the unsteady axisymmetric boundary layer equation to ODEs and simpler PDEs. All reductions of this equation were obtained.

Convective transport of thermal and solutal energy in unsteady MHD Oldroyd-B nanofluid flow

In this work, an analysis is presented for the unsteady axisymmetric flow of Oldroyd-B nanofluid generated by an impermeable stretching cylinder with heat and mass transport under the influence of heat generation/absorption, thermal radiation and first-order chemical reaction. Additionally, thermal and solutal performances of nanofluid are studied using an interpretation of the well-known Buongiorno’s model, which helps us to determine the attractive characteristics of Brownian motion and thermophoretic diffusion. Firstly, the governing unsteady boundary layer equation’s (PDEs) are established and then converted into highly non-linear ordinary differential equations (ODEs) by using the suitable similarity transformations. For the governing non-linear ordinary differential equations, numerical integration in domain [0, ∞) is carried out using the BVP Midrich scheme in Maple software. For the velocity, temperature and concentration distributions, reliable results are prepared for different physical flow constraints. According to the results, for increasing values of Deborah numbers, the temperature and concentration distribution are higher in terms of relaxation time while these are decline in terms of retardation time. Moreover, thermal radiation and heat generation/absorption are increased the temperature distribution and corresponding boundary layer thickness. With previously stated numerical values, the acquired solutions have an excellent accuracy.

Eigensolutions of the unsteady boundary-layer equations revisited (with extensions to three-dimensional modes)

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Discontinuous solutions of the unsteady boundary-layer equations for a rotating disk of finite radius

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Development and Validation of a Machine Learned Turbulence Model

A stand-alone machine learned turbulence model is developed and applied for the solution of steady and unsteady boundary layer equations, and issues and constraints associated with the model are investigated. The results demonstrate that an accurately trained machine learned model can provide grid convergent, smooth solutions, work in extrapolation mode, and converge to a correct solution from ill-posed flow conditions. The accuracy of the machine learned response surface depends on the choice of flow variables, and training approach to minimize the overlap in the datasets. For the former, grouping flow variables into a problem relevant parameter for input features is desirable. For the latter, incorporation of physics-based constraints during training is helpful. Data clustering is also identified to be a useful tool as it avoids skewness of the model towards a dominant flow feature.

The Unsteady MHD Flow Under the Action of Chemical Reaction and Thermal Radiation at the Stagnation Point of a Rotating Sphere

In the present work, we have studied the unsteady MHD flow under the action of thermal radiation and chemical reaction at the stagnation point of a rotating sphere. By using similarity transformation, the unsteady non-linear boundary layer equation obtained as a result of heat transfer and mass transfer along with momentum were changed to a set of ordinary differential equations. Through the use of MATLAB’s built in solver bvp4c, the obtained differential equations were solved. Fluid velocity profile along with temperature profile and species concentration profiles are drawn for radiation parameter ( ) a R and chemical reaction parameter ( ) r K and the obtained results are discussed.

An adjoint compressible linearised Navier–Stokes approach to model generation of Tollmien–Schlichting waves by sound

The generation of the first-mode instability through scattering of an acoustic wave by localised surface roughness, suction or heating is studied with a time-harmonic compressible adjoint linearised Navier–Stokes (AHLNS) approach for subsonic flow conditions. High Strouhal number analytical solutions to the compressible Stokes layer problem are deduced and shown to be in better agreement with numerical solutions compared to previous works. The adjoint methodology of Hill in the context of acoustic receptivity is extended to the compressible flow regime and an alternative formulation to predict sensitivity to the angle of incidence of an acoustic wave is proposed. Good agreement of the acoustic AHLNS receptivity model is found with published direct numerical simulations and the simpler finite Reynolds number approach. Parametric investigations of the influence of the acoustic wave angle on receptivity amplitudes reveal that the linearised unsteady boundary layer equations are a valid model of the acoustic signature for a large range of acoustic wave obliqueness values, failing only where the wave is highly oblique and travels upstream. An extensive parametric study of the influence of frequency, spanwise wavenumber, local Reynolds number and free-stream Mach number over the efficiency function for the different types of wall perturbation mechanisms is undertaken.

Transformations, properties, and exact solutions of unsteady axisymmetric boundary layer equations for non-Newtonian fluids

Unsteady axisymmetric boundary layer equations for power-law non-Newtonian fluids are analyzed. A number of new exact solutions containing arbitrary functions and free parameters are constructed using generalized or functional separation of variables. The solutions are obtained using a Crocco-type transformation reducing the order of the equations examined and simpler point transformations. Along with the exact solutions to axisymmetric boundary layer equations, some new exact solutions to planar boundary layer equations for non-Newtonian fluids are constructed. Several properties have been discovered that allow the exact solutions of the unsteady axisymmetric boundary layer equations to be generalized by including additional arbitrary functions therein. All results refer to an arbitrarily shaped streamlined solid of revolution.

Combined influence of fluctuations in the temperature and stretching velocity of the sheet on MHD flow of Cu-water nanofluid through rotating porous medium with cubic auto-catalysis chemical reaction

This work is devoted to examine the three dimensional flow of Cu-water nanofluid over a vertical oscillatory stretchable sheet having fluctuating temperature in comparison to the constant free stream temperature. The sheet embedded in porous medium is assumed to be rotating in a direction normal to its plane and is considered to be lying in a strong transverse magnetic field. The novel concepts of homogeneous-heterogeneous chemical reaction and thermal radiation are also incorporated while developing unsteady boundary layer equations governing the nanofluid flow. After exploiting appropriate transformations, the equations are reduced to dimensionless form which in turn are solved by employing an explicit finite difference scheme (EFDS). To obtain accurate solutions, stability and convergence conditions are also derived. Further, a comparative analysis is presented for the two cases namely fluctuating nanofluid flow (FNF) and non-fluctuating nanofluid flow (NFNF), and impacts of involved physical parameters on velocity distribution, temperature distribution and concentration fields are explored through graphs, and discussed in detail. The quantities of physical interest like surface shear stress, and heat and mass transfer rates are also computed for pertinent parameters and discussed with the help of tables. An interesting and novel result which we found is that oscillation can be utilized in delaying the boundary layer separation and moreover, in controlling the heat and mass transfer processes.

An efficient spectral solution for unsteady boundary-layer flow and heat transfer due to a stretching sheet

In this paper, an efficient spectral collocation method based on the shifted Legendre polynomials is applied to study the unsteady boundary-layer flow and heat transfer due to a stretching sheet. A similarity transformation is used to reduce the governing unsteady boundary-layer equations to a system of non-linear ordinary differential equations. Then, the shifted Legendre polynomials and their operational matrix of derivative are used for producing an analytical approximate solution of this system of non-linear ordinary differential equations. The main advantage of the proposed method is that the need for guessing and correcting the initial values during the solution procedure is eliminated and a stable solution with good accuracy can be obtained by using the given boundary conditions in the problem. A very good agreement is observed between the obtained results by the proposed spectral collocation method and those of previously published ones.

Nonsimilar Solutions of Unsteady Mixed Convection Along a Moving Cylinder

This paper describes the nonsimilar solutions of unsteady mixed convection along a moving cylinder. The problem is formulated in a more general fashion by using the parameter \(\lambda \). When \(\lambda =0\), we have the case of stationary cylinder in a moving stream, whereas when \(\lambda =1\) we have the case of moving cylinder with no external motion imposed on the free stream. For the intermediate values of \(\lambda \) we have the case of simultaneous motion of the cylinder and the free stream. The governing unsteady boundary layer equations (nonlinear coupled PDEs) are solved by the quasilinearization technique. Fluid flow and heat transfer characteristics for a variety of accelerating and decelerating situations are presented graphically and discussed.

Flow and heat transfer in a Maxwell liquid film over an unsteady stretching sheet in a porous medium with radiation.

A problem of flow and heat transfer in a non-Newtonian Maxwell liquid film over an unsteady stretching sheet embedded in a porous medium in the presence of a thermal radiation is investigated. The unsteady boundary layer equations describing the problem are transformed to a system of non-linear ordinary differential equations which is solved numerically using the shooting method. The effects of various parameters like the Darcy parameter, the radiation parameter, the Deborah number and the Prandtl number on the flow and temperature profiles as well as on the local skin-friction coefficient and the local Nusselt number are presented and discussed. It is observed that increasing values of the Darcy parameter and the Deborah number cause an increase of the local skin-friction coefficient values and decrease in the values of the local Nusselt number. Also, it is noticed that the local Nusselt number increases as the Prandtl number increases and it decreases with increasing the radiation parameter. However, it is found that the free surface temperature increases by increasing the Darcy parameter, the radiation parameter and the Deborah number whereas it decreases by increasing the Prandtl number.

Exact solutions of unsteady boundary layer equations for power-law non-Newtonian fluids

A number of new exact solutions (with the generalized and functional separation of variables) of unsteady equations of a planar and asymmetric boundary layer of power-law non-Newtonian fluids are described. To find the solutions, the Crocco transformation reducing the order of the equations considered and simpler point transformations are used. Two theorems allowing one to generalize exact solutions of the unsteady axisymmetric boundary layer equations including additional arbitrary functions into them are proven.

Direct functional separation of variables and new exact solutions to axisymmetric unsteady boundary-layer equations

The paper deals with equations describing the unsteady axisymmetric laminar boundary layer on an extensive body of revolution as well as axisymmetric jet flows. Such equations are shown to reduce to a single nonlinear third-order PDE with variable coefficientswtz+wzwxz−wxwzz=νzwzzz+F(t,x),where w is a modified stream function. We describe a number of new generalized and functional separable solutions to this equation, which depend on two to four arbitrary functions of a single argument (a few solutions depend on an arbitrary function of two arguments). We use three methods to construct the exact solutions: (i) direct method for symmetry reductions, (ii) direct method of functional separation of variables (a special form of solutions with six undetermined functions is preset and particular solutions to an auxiliary ODE are used), and (iii) method of generalized separation of variables. Most of the solutions obtained are expressed in terms of elementary functions, provided that the arbitrary functions are also elementary. Such solutions, having relatively simple form and presenting significant arbitrariness, can be especially useful for testing numerical and approximate analytical methods for nonlinear hydrodynamic-type PDEs and solving certain model problems. The direct method of functional separation of variables used in this paper can also be effective for constructing exact solutions to other nonlinear PDEs.

Towards development of unsteady near-wall interface boundary conditions for turbulence modeling

In near-wall turbulence modeling it is required to resolve a thin layer nearby the solid boundary, which is characterized by high gradients of the solution. An accurate enough resolution of such a layer can take most computational time. The situation even becomes worse for unsteady problems. To avoid time-consuming computations, a new approach is developed, which is based on a non-overlapping domain decomposition. The boundary condition of Robin type at the interface boundary is achieved via transfer of the boundary condition from the wall. For the first time interface boundary conditions of Robin type are derived for a model nonstationary equation which simulates the key terms of the unsteady boundary layer equations. In the case of stationary solutions the approach is automatically reduced to the technique earlier developed for the steady problems. The considered test cases demonstrate that unsteady effects can be significant for near-wall domain decomposition. In particular, they can be important in the case of the wall-function-based approach.

Nanofluid flow over an unsteady stretching surface in presence of thermal radiation

This paper investigates the unsteady boundary layer flow of a nanofluid over a heated stretching sheet with thermal radiation. The transport model employed includes the effects of Brownian motion and thermophoresis. The unsteadiness in the flow field is caused by the time-dependence of the stretching velocity, free stream velocity and the surface temperature. The unsteady boundary layer equations are transformed to a system of non-linear ordinary differential equations and solved numerically using a shooting method together with Runge–Kutta–Fehlberg scheme. The clear liquid results from this study are in agreement with the results reported in the literature. It is found that the heat transfer rate at the surface increases in the presence of Brownian motion but reverse effect occurs for thermophoresis.

Effects of radiation and variable viscosity on unsteady MHD flow of a rotating fluid from stretching surface in porous medium

Abstract This work is focused on the study of unsteady magnetohydrodynamics boundary-layer flow and heat transfer for a viscous laminar incompressible electrically conducting and rotating fluid due to a stretching surface embedded in a saturated porous medium with a temperature-dependent viscosity in the presence of a magnetic field and thermal radiation effects. The fluid viscosity is assumed to vary as an inverse linear function of temperature. The Rosseland diffusion approximation is used to describe the radiative heat flux in the energy equation. With appropriate transformations, the unsteady MHD boundary layer equations are reduced to local nonsimilarity equations. Numerical solutions of these equations are obtained by using the Runge–Kutta integration scheme as well as the local nonsimilarity method with second order truncation. Comparisons with previously published work have been conducted and the results are found to be in excellent agreement. A parametric study of the physical parameters is conducted and a representative set of numerical results for the velocity in primary and secondary flows as well as the local skin-friction coefficients and the local Nusselt number are illustrated graphically to show interesting features of Darcy number, viscosity-variation, magnetic field, rotation of the fluid, and conduction radiation parameters.