Momentum Boundary Layer Equations Research Articles

Physical Conserved Quantities for the Axisymmetric Liquid, Free and Wall Jets

A systematic way to derive the conserved quantities for the axisymmetric liquid jet, free jet and wall jet using conservation laws is presented. The flow in axisymmetric jets is governed by Prandtl’s momentum boundary layer equation and the continuity equation. The multiplier approach is used to construct a basis of conserved vectors for the system of two partial differential equations for the two velocity components. The basis consists of two conserved vectors. By integrating the corresponding conservation laws across the jet and imposing the boundary conditions, conserved quantities are derived for the axisymmetric liquid and free jet. The multiplier approach applied to the third-order partial differential equation for the stream function yields two local conserved vectors one of which is a non-local conserved vector for the system. One of the conserved vectors gives the conserved quantity for the axisymmetric free jet but the conserved quantity for the wall jet is not obtained from the second conserved vector. The conserved quantity for the axisymmetric wall jet is derived from a non-local conserved vector of the third-order partial differential equation for the stream function. This non-local conserved vector for the third-order partial differential equation for the stream function is obtained by using the stream function as multiplier. Keywords—Axisymmetric jet,liquid jet, free jet, wall jet, conservation laws, conserved quantity

Thin film flow of an unsteady shrinking sheet through porous medium with variable viscosity

In the present study, we have analyzed the effects of variable viscosity, variable thermocapillarity on the flow and heat transfer in a thin film on a horizontal porous shrinking sheet through a porous medium. The unsteady boundary layer equations for momentum and thermal energy are simplified by using similarity transformations. The resulted, coupled nonlinear differential equations are solved by homotopy analysis method. The results are presented graphically to interpret various physical parameters appearing in the problem.

The Coupling of Conduction with Free Convection Flow Along a Vertical Flat Plate in Presence of Heat Generation

The aim of this paper is to analyze the conjugate problems of heat conduction in solid walls coupled with laminar free convection flow adjacent to a vertical flat plate under boundary layer approximation. Using the similarity transformations the governing boundary layer equations for momentum and energy are reduced to a system of partial differential equations and then solved numerically using Finite Difference Method (FDM) known as the Keller-box scheme. Computed solutions to the governing equations are obtained for a wide range of non-dimensional parameters that are present in this problem, namely the coupling parameter P, the Prandtl number Pr and the heat generation parameter Q. The variations of the local heat transfer rate as well as the interface temperature and the friction along the plate and typical velocity and temperature profiles in the boundary layer are shown graphically. Numerical solutions have been consider for the Prandtl number Pr = 0.70

Heat transfer in a liquid film on an unsteady stretching sheet

The heat transfer in a liquid film driven by a horizontal sheet is examined. The stretching rate and temperature of the sheet vary with time. The boundary layer equations for momentum and thermal energy are reduced to a set of ordinary differential equations by means of an exact similarity transformation. Numerical solutions of the resulting four-parameter problem are provided. It is observed that the variation of the sheet temperature with distance and with time has analogous effects both on the free surface temperature and the heat transfer rate (Nusselt number) at the sheet.

Viscoelastic MHD flow and heat transfer over a stretching sheet with viscous and ohmic dissipations

A mathematical analysis has been carried out on momentum and heat transfer characteristics in an incompressible electrically conducting viscoelastic boundary layer fluid flow over a linear stretching sheet. Momentum boundary layer equation takes into account the effect of transverse magnetic field and electric field. Thermal boundary layer equation takes into account the viscous dissipation and Ohmic dissipation due to transverse magnetic field and electric field. Highly non-linear momentum boundary layer equation and thermal boundary layer equation are converted into similarity equations and then solved numerically by employing fifth order Runge–Kutta–Fehlberg method with shooting. The results are analysed for the situation when stretching boundary is prescribed by non-isothermal temperature, namely, prescribed surface temperature (PST) which varies quadratically with the flow directional coordinate x. The effects of various physical parameters like viscoelastic parameter, Prandtl number, local Reynolds number, Eckert number Hartmann number and electric parameter on various momentum and heat transfers characteristics are analysed. Some of the important findings of this paper are (i) The combined effect of increasing the values of local Reynolds number Rex and local electric parameter (E1) is to decrease skin-friction coefficient Cf largely. (ii) In presence of magnetic field the effect of electric field is to decrease temperature near the stretching boundary and increase the same significantly away from the stretching sheet. (iii) If electric field is present and the Prandtl number is lower then there would be a significant decrease of temperature near the boundary sheet in case of viscoelastic fluid. (iv) The presence of electric field reverses the direction of heat transfer on the boundary stretching sheet more significantly in case of viscoelastic fluid.

Analytical approximation for laminar film condensation of saturated stream on an isothermal vertical plate

In this paper, the laminar film condensation of saturated stream on an isothermal vertical plate is studied. The boundary layer equations of momentum and thermal energy are reduced to two ordinary differential equations by means of a set of similarity transformations. The problem is then solved analytically using the homotopy analysis method (HAM). The dual solutions are obtained for a range of values of the parameter η δ . However, it should be noted that the second branch solution of the considered problem has only mathematical meanings. The present work shows the validity and the great potentiality of the proposed technique for the nonlinear problems with multiple solutions.

On sheet-driven motion of power-law fluids

A rigorous analysis of non-Newtonian boundary layer flow of power-law fluids over a stretching sheet is presented. First, a systematic framework for treatment of sheet velocities of the form U ( x ) = Cx m is provided. By means of an exact similarity transformation, the non-linear boundary layer momentum equation transforms into an ordinary differential equation with m and the power-law index n as the only parameters. Earlier investigations of a continuously moving surface ( m = 0 ) and a linearly stretched sheet ( m = 1 ) are recovered as special cases. For the particular parameter value m = 1 , i.e. linear stretching, numerical solutions covering the parameter range 0.1 ⩽ n ⩽ 2.0 are presented. Particular attention is paid to the most shear-thinning fluids, which exhibit a challenging two-layer structure. Contrary to earlier observations which showed a monotonic decrease of the sheet velocity gradient - f ″ ( 0 ) with n, the present results exhibit a local minimum of - f ″ ( 0 ) close to n = 1.77 . Finally, a series expansion in ( n - 1 ) is proved to give good estimates of - f ″ ( 0 ) both for shear-thinning and shear-thickening fluids.

The effects of variable fluid properties and thermocapillarity on the flow of a thin film on an unsteady stretching sheet

The effects of variable viscosity, variable thermal conductivity and thermocapillarity on the flow and heat transfer in a laminar liquid film on a horizontal stretching sheet is analyzed. Using a similarity transformation the governing time dependent boundary layer equations for momentum and thermal energy are reduced to a set of coupled ordinary differential equations. The resulting five-parameter problem is solved numerically for some representative value of the parameters. It is shown that the film thickness increases with the increase in viscosity of the fluid. In other words viscosity resists film thinning. Further it is shown that more heat flows out of the liquid through the stretching surface when conductivity increases with temperature than that for the case when conductivity decreases with temperature.

A Noble Gas Wiping System to Prevent the Edge Overcoating in Continuous Hot-dip Galvanizing

A noble method is proposed to prevent the edge overcoating (EOC) that may develop near the edge of the steel strip in the gas wiping process of continuous hot-dip galvanizing. In our past study (ISIJ International, Vol. 27 (2003), No. 10, pp. 1495–1501), it was found that the EOC is caused by the alternating vortices which are generated by the collision of two opposing jets in the region outside the steel strip. In the present study, the flow field around the gas wiping system has been analyzed numerically and it was found that when the two opposing jets collide at an angle much less than 180°, the alternating vortices disappear and the impinging pressure on the steel strip surface becomes nearly uniform. In order to deflect both jets downward by a certain angle, a cylinder with small diameter is installed tangentially to each exit of the lower lips of the two-dimensional opposing jets. The three dimensional flow field with the proposed device is analyzed numerically by using the commercial CFD software, STAR-CD. And the coating thickness is calculated by solving the boundary layer momentum equation with an integral analysis method. In order to compare the present noble method with the conventional edge baffle plate method to prevent the EOC, the flow field with edge baffle plates is also calculated. The calculation results show that the tangentially installed cylinder at the lower lip of the jet exit is significantly more effective than the edge baffle plate.

Heat transfer in a viscoelastic fluid flow over a stretching surface with heat source/sink, suction/blowing and radiation

A boundary layer problem on heat transfer in a viscoelastic boundary layer fluid flow over a non-isothermal porous sheet, where the flow is generated due to linear stretching of the sheet and influenced by a continuous suction/blowing of the fluid through the porous boundary, has been presented. In the flow region, heat balance is maintained with a temperature dependent heat source/sink, viscous dissipation and thermal radiation. Applying suitable similarity transformations on the highly non-linear momentum boundary layer equation and thermal boundary layer equation several closed from analytical solutions have been derived for non-dimensional temperature and heat flux profiles in the from of confluent hyper geometric (Kummer’s) functions and other elementary functions as its special form. Heat transfer analysis has been carried out for two general types of boundary heating processes, namely, (i) prescribed quadratic power law surface temperature (PST) and (ii) prescribed quadratic power law surface heat flux (PHF) for various values of non-dimensional viscoelastic parameter k 1 ∗ , Prandtl number Pr, Eckert number E, radiation parameter N, suction/blowing parameter v w and source/sink parameter β. Some of the several important findings reported in this paper are (i) the combined effect of Prandtl number Pr, radiation parameter N and suction/blowing parameter v w has significant impact in controlling the rate of heat transfer to the boundary layer region through the porous stretching sheet and (ii) radiation and suction can be used as means of cooling the viscoelastic boundary layer flow region. Special cases of our results are in excellent agreement with some of the existing work.

연속식 용융아연도금 공정에서 단부 과도금 현상을 방지하기 위한 하향 대칭 분류유동 연구

In this study, a noble method is proposed to prevent the edge overcoating (EOC) that may develop near the edge of the steel strip in the gas wiping process of continuous hot-dip galvanizing. In our past study (Trans. of the KSME (B), Vol. 27, No. 8, pp. 1105~1113), it was found that EOC is caused by the alternating vortices which are generated by the collision of two opposed jets in the region outside the steel strip. When the two opposed jets collide at an angle much less than 180°, non-alternating stable vortices are established symmetrically outside the steel strip, which lead to nearly uniform pressure on the strip surface. In order to deflect both jets downward by a certain angle, a cylinder with small diameter is installed tangentially to the exit of the lower lip of the two-dimensional jet. In order to find an optimum cylinder diameter, the three dimensional flow field is analysed numerically by using the commercial code, STAR-CD., the coating thickness is calculated by using an integral analysis method to solve the boundary layer momentum equation. In order to compare the present noble method with the conventional baffle plate method to prevent the EOC, the flow field with a baffle plate is also calculated. The calculation results show that the tangentially installed cylinder at the bottom lip of the jet exit is more effective than the baffle plate to prevent EOC.

Similarity solutions of energy and momentum boundary layer equations for a power-law shear driven flow over a semi-infinite flat plate

The problem of a steady forced convection thermal boundary-layer driven by a power-law shear is investigated. The search for similarity solutions reduces the problem to a couple of ordinary differential equations containing three parameters: the exponent of the decaying exterior velocity profile, the exponent of the power-law prescribing the thermal condition on the wall and Prandtl number. The effects of these parameters on the existence and form of similarity solution are investigated and the functional dependence of the local Nusselt number on these parameters is reported and discussed. An analysis of the assumptions usually accepted to derive similarity solutions is also reported in order to show the range of values of the exterior velocity power-law exponent for which such solutions may exist.

Effects of radiation and heat source on MHD flow of a viscoelastic liquid and heat transfer over a stretching sheet

We study the MHD flow and also heat transfer in a viscoelastic liquid over a stretching sheet in the presence of radiation. The stretching of the sheet is assumed to be proportional to the distance from the slit. Two different temperature conditions are studied, namely (i) the sheet with prescribed surface temperature (PST) and (ii) the sheet with prescribed wall heat flux (PHF). The basic boundary layer equations for momentum and heat transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. The resulting non-linear momentum differential equation is solved exactly. The energy equation in the presence of viscous dissipation (or frictional heating), internal heat generation or absorption, and radiation is a differential equation with variable coefficients, which is transformed to a confluent hypergeometric differential equation using a new variable and using the Rosseland approximation for the radiation. The governing differential equations are solved analytically and the effects of various parameters on velocity profiles, skin friction coefficient, temperature profile and wall heat transfer are presented graphically. The results have possible technological applications in liquid-based systems involving stretchable materials.

Group Invariant Solution And Conservation Law for a Steady Laminar Axisymmetric Free Jet

A group invariant solution for a steady laminar axisymmetric free jet is derived. The general form of the group invariant solution is first obtained by considering a linear combination of the Lie point symmetry generators of Prandtl's momentum boundary layer equation for the stream function. The ratios of the constants in the linear combination are determined from conservation of total momentum flux in the downstream direction, from the condition of finite volume flow and finite momentum at the infinitesimal circular orifice and from specification of the arbitrary additive constant in the stream function. An elementary conservation law for the partial differential equation for the stream function is derived. A relation is established between this conservation law and conservation of total momentum flux in the downstream direction by showing that the Lie point symmetry which is associated with the conservation law is the Lie point symmetry that generates the group invariant solution.

Separation Criterion for Turbulent Boundary Layers Via Similarity Analysis

By using the RANS boundary layer equations, it will be shown that the outer part of an adverse pressure gradient turbulent boundary layer tends to remain in equilibrium similarity, even near and past separation. Such boundary layers are characterized by a single and constant pressure gradient parameter, Λ, and its value appears to be the same for all adverse pressure gradient flows, including those with eventual separation. Also it appears from the experimental data that the pressure gradient parameter, Λθ, is also approximately constant and given by Λθ=0.21±0.01. Using this and the integral momentum boundary layer equation, it is possible to show that the shape factor at separation also has to within the experimental uncertainty a single value: Hsep≅2.76±0.23. Furthermore, the conditions for equilibrium similarity and the value of Hsep are shown to be in reasonable agreement with a variety of experimental estimates, as well as the predictions from some other investigators.

Thermocapillarity in a liquid film on an unsteady stretching surface

The influence of thermocapillarity on the flow and heat transfer in a thin liquid film on a horizontal stretching sheet is analysed. The time-dependent governing boundary layer equations for momentum and thermal energy are reduced to a set of coupled ordinary differential equations by means of an exact similarity transformation. The resulting three-parameter problem is solved numerically for some representative values of an unsteadiness parameter S and a thermocapillarity number M for Prandtl numbers from 0.001 to 100. The thermocapillary surface forces drag the liquid film in the same direction as the stretching sheet and a local velocity minimum occurs inside the film. The surface velocity, the film thickness, and the Nusselt number at the sheet increase with M for Pr≲10. For higher Prandtl numbers, the thermal boundary layer is confined to the lower part of the liquid film and the temperature at the free surface remains equal to the slit temperature and the thermocapillary forces vanish.

Temperature and flow fields for the flow of a second grade fluid past a wedge

The effects of fluid viscoelasticity on the temperature and flow fields of the flow past a wedge were studied. The combined effects of the shape factor, suction/injection rates and viscoelasticity were analysed by the series expansion method, similarity transformation, fourthorder Runge-Kutta integration and the shooting method. The stream function was introduced into the momentum boundary layer equations of a second-grade fluid to eliminate the pressure terms. The energy equation is also analysed by the similarity transformation. The results indicated that the velocity profiles, temperature distributions, surface friction and heat transfer are greatly affected by the elastic parameter, the shape factor, the Prandtl number and the suction/injection rates.

Transient Convective Heat Transfer in Planar Stagnation Flows With Time-Varying Surface Heat Flux and Temperature

Impinging flows are used in a variety of applications where effective and localized heat transfer is mandated by short residence times or by space constraints, as in cooling materials moving along a conveyor or removing heat dissipated within microelectronic circuitry. A wide selection of heat transfer correlations is available for steady-state conditions. However, instantaneous heat transfer coefficients can differ significantly from steady-state values when temporal variations occur in the surface heat flux or surface temperature. Under these conditions, the temperatures of fluid layers near the surface are affected preferentially due to their proximity to the temporal variation. A theoretical model is formulated to assess the importance of a time-varying surface heat flux or temperature on convective heat transfer in a steady, planar stagnation flow. A governing equation for the transient heat transfer response is formulated analytically from the boundary layer equations for momentum and energy conservation in the fluid. Numerical solutions to the governing equation are determined for ramp and sinusoidal changes in the surface heat flux or temperature. Results indicate that the time response is chiefly governed by the velocity gradient in the free stream and to a lesser extent by the Prandtl number. Departures from steady-state Nusselt numbers are larger for more rapid transients and smaller or comparable in size to the magnitude of the imposed variation at the surface.

Boundary-layer analysis of heat and mass transfer over a circular cylinder in crossflow

Two-dimensional laminar boundary-layer equations of momentum, heat and mass transfer have been numerically solved tinder forced convection. A finite difference approximation of the governing equations in a Goertler-type variable domain has been implemented on computer. To provide rigorous initial conditions at x 0 in the boundary-layer code, differential equations governing heat and mass transfer in the stagnation region have been set up and solved numerically. The effect of outer flow condition on skin friction as well as heat and mass transfer has been demonstrated. Results obtained made an improvement over past theoretical predictions based on analytical approximation, and agreed favorably with experimental data available.

A numerical analysis of unsteady separated flow by vortex shedding model. 2nd report Flow around a circular cylinder.

A numerical method is described, which is developed to analyze an unsteady separated flow. In this method, the flow around a body is assumed to be a potential flow except in the region of free shear layers. Hence, the body is represented by a continuous vorticity sheet and the free shear layers are replaced by discrete vortices. The separation point are determined by sloving the boundary layer momentum equation. This methods is applied to calculations of the flow around a circular cylinder in which the laminar and turbulent boundary layer separation occurs. The calculated results for lift and dray coefficients, Strouhal number and separation points are compared with experiments.