System Of Boundary Layer Equations Research Articles

Flow past Rankine’s Half-Body of Revolution

An axisymmetric incompressible flow past Rankine’s half-body of revolution is investigated at high Reynolds numbers. It is established that the solution of the problem for the system of boundary layer equations does not involve the line of zero surface friction. The low under consideration is separationless, which is in complete agreement with the available experimental data.

A numerical study of hybrid nanofluids near an irregular 3D surface having slips and exothermic effects

The current article presents a comprehensive investigation of MHD viscous flow of hybrid-nanofluids (Al2O3 − Ag/ water and (Al2O3 − Cu/) over a horizontally irregular 3D plane with non-uniform thickness combined with slip effects. The foremost aim of conducting this study is to enhance thermal transportation. Based on the following novelties, the subject study holds tremendous significance: i. A comparative analysis of two hybrid nanofluids with hybrid-base fluid together with slip effects ii. An exclusive study where the Tiwari and Das nanofluid model is employed combined with Fourier's heat flux model iii. Development of finite-difference code which implements the three-stage Lobatto IIIa approach for the designed problem. We have used suitable scaling transformations to convert the three-dimensional conservation equations of mass, momentum, and energy into a dimensionless system of boundary layer equations. The numerical solution of the coupled non-linear boundary layer problem is determined using the built-in finite-difference code designed to employ the three-stage Lobatto IIIa formula. A comprehensive assessment is carried out in how the velocity components, temperature, skin friction, and heat transfer rate are affected by the physical parameters of interest. The same is presented through graphs and in tabular form to offer a pictorial overview. The fluctuating trends of skin friction coefficients (x, y-directions) and Nusselt number are investigated to explore the physical landscape of the current study. The findings of this study offer a noticeable contrast to their existing counterparts.

Velocity and thermal boundary layer equations for turbulent Rayleigh-Bénard convection

The authors derive a full system of boundary layer equations for turbulent Rayleigh-Benard convection that incorporates all the relevant physical effects, namely, buoyancy, fluctuations, and vanishing large-scale circulation velocity in the central bulk region of the convection cell. The theory predicts the time-averaged temperature and velocity profiles for fluids with a general Prandtl number.

Equations of Symmetric Boundary Layer for the Ladyzhenskaya Model of a Viscous Medium in the Crocco Variables

We consider the system of boundary layer equations governing a viscous medium subject to the nonlinear rheological law in the sense of Ladyzhenskaya. Owing to the use of the Crocco transformation for reducing the system to a single quasilinear equation, it becomes possible to study both stationary and nonstationary boundary layers. We obtain asymptotic estimates for the solution.

System of Boundary Layer Equations for a Rheologically Complicated Medium: Crocco Variables

The system of boundary-layer equations for a nonlinear generalized Newtonian viscous fluid with the Ladyzhenskaya law is studied. The correct solvability of the problem under study is proved using the Crocco transformation method, which reduces the system of boundary-layer equations to a quasilinear degenerate parabolic equation. Asymptotic estimates for the solution on the boundary of the domain are obtained.

Numerical simulation of helium injection into the accelerated turbulent xenon flow

The paper reports on numerical simulation of a weakly accelerated turbulent boundary layer over the permeable wall. Helium-xenon mixture was injected into the xenon main flow after the laminar-turbulent transition. The finite difference method was applied to solve a system of boundary layer equations with variable gas properties supplemented by the k-ω-γ turbulence model. It has been shown that the light gas injection into the accelerated turbulent flow leads to arising of velocity maximum within the boundary layer and its partial laminarization.

The effect of vertical throughflow on the boundary layer flow of a nanofluid past a stretching/shrinking sheet

PurposeThe purpose of this paper is to study the effects of vertical throughflow on the boundary layer flow and heat transfer of a nanofluid driven by a permeable stretching/shrinking surface.Design/methodology/approachSimilarity transformation is used to convert the system of boundary layer equations into a system of ordinary differential equations. The system of governing similarity equations is then reduced to a system of first-order differential equations and solved numerically using the bvp4c function in Matlab software. The generated numerical results are presented graphically and discussed based on some governing parameters.FindingsIt is found that dual solutions exist in both cases of stretching and shrinking sheet situations. Stability analysis is performed to determine which solution is stable and valid physically.Originality/valueDual solutions are found for positive and negative values of the moving parameter. A stability analysis has also been performed to show that the first (upper branch) solutions are stable and physically realizable, while the second (lower branch) solutions are not stable and, therefore, not physically possible.

On MHD radiative Jeffery nanofluid flow with convective heat and mass boundary conditions

The prime objective of present exploration is to study effects of magnetohydrodynamic, Joule heating and thermal radiation on an incompressible Jeffrey nanofluid flow over a linearly stretched surface. Simultaneous effects of convective heat and mass boundary conditions are also considered. Obtained system of boundary layer equations is converted into ordinary differential equations with high linearity using appropriate transformations. Analytical solutions via homotopy analysis method are obtained and deliberated accordingly. Discussion of graphs pertaining different prominent parameters is also added. Numerical values of skin friction coefficient, local Nusselt and Sherwood numbers are also given and well deliberated. It is noted that higher values of thermophoretic parameter boost temperature and concentration distributions. Moreover, temperature field is an increasing function of radiation parameter.

Numerical solutions of non-alignment stagnation-point flow and heat transfer over a stretching/shrinking surface in a nanofluid

Purpose – The purpose of this paper is to analyse numerically the steady stagnation-point flow of a viscous and incompressible fluid over continuously non-aligned stretching or shrinking surface in its own plane in a water-based nanofluid which contains three different types of nanoparticles, namely, Cu, Al2O3 and TiO2. Design/methodology/approach – Similarity transformation is used to convert the system of boundary layer equations which are in the form of partial differential equations into a system of ordinary differential equations. The system of similarity governing equations is then reduced to a system of first-order differential equations and solved numerically using the bvp4c function in Matlab software. Findings – Unique solution exists when the surface is stretched and dual solutions exist as the surface shrunk. For the dual solutions, stability analysis has revealed that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable. The effect of non-alignment is huge for the shrinking surface which is in contrast with the stretching surface. Practical implications – The results obtained can be used to explain the characteristics and applications of nanofluids, which are widely used as coolants, lubricants, heat exchangers and micro-channel heat sinks. This problem also applies to some situations such as materials which are manufactured by extrusion, production of glass-fibre and shrinking balloon. In this kind of circumstance, the rate of cooling and the stretching/shrinking process play an important role in moulding the final product according to preferable features. Originality/value – The present results are original and new for the study of fluid flow and heat transfer over a stretching/shrinking surface for the problem considered by Wang (2008) in a viscous fluid and extends to nanofluid by using the Tiwari and Das (2007) model.

Unsteady stagnation-point flow and heat transfer of a special third grade fluid past a permeable stretching/shrinking sheet.

In this paper, the unsteady stagnation-point boundary layer flow and heat transfer of a special third grade fluid past a permeable stretching/shrinking sheet has been studied. Similarity transformation is used to transform the system of boundary layer equations which is in the form of partial differential equations into a system of ordinary differential equations. The system of similarity equations is then reduced to a system of first order differential equations and has been solved numerically by using the bvp4c function in Matlab. The numerical solutions for the skin friction coefficient and heat transfer coefficient as well as the velocity and temperature profiles are presented in the forms of tables and graphs. Dual solutions exist for both cases of stretching and shrinking sheet. Stability analysis is performed to determine which solution is stable and valid physically. Results from the stability analysis depict that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable.

Free convective flow from a heated vertical wall immersed in a thermally stratified environment

Free convective flow along a heated vertical wall immersed in a thermally stratified environment is studied using Lie group theory. The symmetries admitted by the governing system of boundary layer equations are identified along with the admitted classes of temperature stratifications. The cases of both prescribed wall temperature and prescribed wall heat flux are considered. The resulting similarity variables, the reduced systems of equations and their corresponding solutions, are presented for each class of temperature stratification. Specific solutions illustrating the different classes are provided.

A methodology for optimal laminar flow control: Application to the damping of Tollmien–Schlichting waves in a boundary layer

A methodology for determining the optimal steady suction distribution for the delay of transition in a boundary layer is presented. The flow state is obtained from the coupled system of boundary layer equations and parabolized stability equations (PSE), to account for the spatially developing nature of the flow. The wall suction is defined by an optimal control procedure based on the iterative solution of the equations for the state and the dual state; the latter is available from the adjoint boundary layer equations and the adjoint PSE. The technique is applied to the control of two-dimensional Tollmien–Schlichting (TS) waves. Results show that the onset of the instability can be significantly postponed and/or the growth rate considerably reduced by applying an appropriate suction through the whole wall length, in a wide frequency band. Control over panels of finite length completes the study and brings useful, preliminary information on the practicality of the approach in view of implementation. Finally, a simplified methodology which does not rely on the PSE is discussed, based on the minimization of the shape factor. Satisfactory results are achieved with this simpler approach which might, thus, constitute a method of choice when results are needed rapidly, i.e., during on-line control of TS waves.

Steady-state shapes of bodies melted by aerodynamic heating

The variation of the shape of bodies exposed to aerodynamic ablation has been the subject of a considerable amount of research. One particular aspect of the general problem, namely, the problem of the steady-state shapes, i.e., those that do not vary as a result of ablation, was solved in [1, 2] for convective heat transfer on the assumption that the effective enthalpy of the material is constant. In this case a distinctive feature of the solution is the presence of corner points (breaks) on the steady-state shapes. Here, the problem is solved without assuming a constant effective enthalpy — the ablation rate is determined on the basis of a numerical solution of the equations describing the ablation of glassy materials, the flow of the molten film over the surface being calculated on the basis of the complete system of boundary-layer equations for an incompressible fluid [3]. It is shown that for a uniform heat transfer regime (laminar or turbulent) the steady-state shapes are smooth bodies without corner points. In the mixed heat transfer regime, in the general case the problem has no solution.

Combined effect of external turbulence and strong acceleration on boundary layer transition in a nozzle

The effect of external turbulence on the boundary layer flow in a convergent-divergent nozzle with a high expansion ratio has been studied numerically. The external turbulence was simulated by the turbulent viscosity ɛe, for which we used the partial differential equation that serves to close the system of boundary layer equations [1–4]. It was found that there exists a critical value ɛcr such that for all ɛe< ɛcr the flow regime in the nozzle remains perfectly laminar, whereas for ɛe⩾ɛcr a laminar-turbulent transition takes place and the boundary layer in the supersonic part of the nozzle becomes turbulent. For postcritical values of ɛe the heat fluxes and friction losses are approximately an order greater than for precritical values. With increase in the Reynolds number, determined from the parameters in the nozzle throat, the value of ɛcr decreases; as the coordinate of the onset of boundary layer formation is displaced in the direction of flow the value of ɛcr increases.

ON THE CALCULATION OF MOMENTUM, HEAT, AND MASS TRANSFER IN LAMINAR AND TURBULENT BOUNDARY LAYER FLOWS ALONG A VAPORIZING LIQUID FILM

Abstract A numerical method is presented for predicting the laminar and turbulent flow of a binary gas mixture along a plane vaporizing liquid film. The shear layer flow including the heat and mass transfer at the gas-liquid interface is described by a system of boundary-layer equations. Each of the parabolic equations is numerically solved using an implicit finite-difference method of Hermitiah type. However, the overall solution procedure handles the coupled differential equations together with the boundary conditions in an explicit manner. Polynomials in terms of temperature are used to approximate the physical properties of the gas components at 1 atm pressure. Turbulence is accounted for by the algebraic eddy viscosity approach of Cebeci and Smith. The accuracy and efficiency of the numerical procedure developed is demonstrated for various temperatures of the external flow. A comparison of the numerical results with measurements in terms of flow profiles and Stanton numbers indicates good agreement.

Direction of streamlines near the wall in a three-dimensional turbulent boundary layer

Expressions are derived that, for a large class of flows, enable one to determine the direction of the streamlines near the wall in a three-dimensional turbulent boundary layer without integrating the system of boundary-layer equations. Comparison with available experimental data reveals a completely satisfactory agreement between the calculations and the experimental data.

Character of the boundary layer near its origin on a body rotating in a fluid at rest

The evolution of the boundary layer on bodies of revolution rotating about the symmetry axis in a fluid at rest is largely determined by the position of its origin with respect to the axis of rotation. If the origin of the boundary layer coincides with a pole of the rotating body, then under fairly general assumptions as to the shape of the body the boundary layer has a nonzero thickness in the vicinity of the pole, and the flow in it is described by a particular “self-similar” solution of the boundary-layer equations [1, 2]. The applicability of existing approximate methods for calculation of the boundary layer [2, 3] is restricted to this case. The results of the present article refer to the case in which the boundary originates at the leading edge at a finite distance from the rotation axis. The behavior of the solution of the boundary-layer equations near the edge is determined. A transformation of variables that reduces the system of boundary-layer equations to a form suitable for analysis and solution is derived. This transformation is used to obtain “universal” equations determining the local behavior of the boundary layer in the vicinity of its origin in both of the cases indicated above.

Time-periodic solution of the system of boundary layer equations: PMM vol. 36, n≗3, 1972, pp. 460–470

The system of boundary layer equations of unsteady axisymmetric flow of incompressible fluid in the presence of blowing or suction through the boundary surface of a body is considered. Proof is given of the existence and uniqueness of a timeperiodic solution of such system in the neighborhood of the critical point (forced oscillations), when the external flow is periodic with respect to time and the functions defining the body shape and the blowing or suction conditions are known. This problem was considered in detail in [1] for specific initial conditions (as a whole dependent on t), where, in particular, data on the stability of such flows are presented.

Uniqueness of the self-similar solutions of three-dimensional laminar boundary-layer equations

The equations of the three-dimensional laminar boundary layer on lines of flow outflow and inflow are studied for conical outer flow under the assumption that the Prandtl number and the productρμ are constant. It is shown that in the case of a positive velocity gradient of the secondary flow (α1>0) the additional conditions which result from the physical flow pattern determine a unique solution of the system of boundary-layer equations. For a negative velocity gradient of the secondary flow (α1≤0) these conditions are satisfied by two solutions. An approximate solution is obtained for the boundary layer equations which is in rather good agreement with the numerical integration results. Compressible gas flow in a three-dimensional laminar boundary layer is described by a system of nonlinear differential equations whose solution is not unique for given boundary conditions. Therefore additional conditions resulting from the physical pattern of the gas flow are imposed on the resulting solution. In the solution of problems with a negative pressure gradient these additional conditions are sufficient for a unique selection of the solution of the boundary-layer equations. However, in the case of a positive pressure gradient the solution of the boundary-layer equations satisfying the boundary and additional conditions may not be unique. In particular, in [1] in a study of a three-dimensional laminar boundary layer in the vicinity of the stagnation point it was shown that for $$c = {{\frac{{\partial v_e }}{{\partial y}}} \mathord{\left/ {\vphantom {{\frac{{\partial v_e }}{{\partial y}}} {\frac{{\partial u_e }}{{\partial x}}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial u_e }}{{\partial x}}}} > 0$$ the solution is unique, while for c<0 there are two solutions. In the present paper we study the question of the uniqueness of the self-similar solution of the three-dimensional laminar boundary-layer equations on lines of flow outflow and inflow for a conical outer flow.

Flow in a laminar boundary layer under intensive injection and radiant heat exchange

An asymptotic solution has been obtained for the system of boundary layer equations with intense injection of a foreign absorbing substance for optically thick and thin boundary layers taking account of the influence of a magnetic field. Analytical formulas are presented which permit computation of the temperature, concentration, and heat flux profiles.