Momentum Boundary Layer Equations Research Articles

Viscous Flow over Nonlinearly Stretching Sheet with Effects of Viscous Dissipation

The flow and heat transfer characteristics of incompressible viscous flow over a nonlinearly stretching sheet with the presence of viscous dissipation is investigated numerically. The similarity transformation reduces the time‐independent boundary layer equations for momentum and thermal energy into a set of coupled ordinary differential equations. The obtained equations, including nonlinear equation for the velocity field f and differential equation by variable coefficient for the temperature field θ, are solved numerically by using the fourth order of Runge‐Kutta integration scheme accompanied by shooting technique with Newton‐Raphson iteration method. The effect of various values of Prandtl number, Eckert number and nonlinear stretching parameter are studied. The results presented graphically show some behaviors such as decrease in dimensionless temperature θ due to increase in Pr number, and curve relocations are observed when heat dissipation is considered.

BOUNDARY LAYER FLOW AND DOUBLE DIFFUSION OVER AN UNSTEADY STRETCHING SURFACE WITH HALL EFFECT

The present investigation is concerned with the effect of Hall currents on boundary layer flow, and heat and mass transfer of an electrically conducting fluid over an unsteady stretching sheet in the presence of a strong magnetic field. The electron-atom collision frequency is assumed to be relatively high, so that the Hall effect is assumed to exist, while the induced magnetic field is neglected. The governing time-dependent boundary layer equations for momentum, thermal energy, and concentration are reduced using a similarity transformation to a set of coupled ordinary differential equations. The similarity ordinary differential equations are then solved numerically by the successive linearization method together with the Chebyshev pseudo-spectral collocation method. Effects of the Prandtl number, Pr, Schmidt number, Sc, magnetic field, M, Hall parameter, m, and the unsteadiness parameter, A, on the velocity, temperature, and concentration profiles as well as the local skin friction coefficient and the heat and mass transfer rates are depicted graphically and/or in tabular form. Favorable comparisons with previously published work on various special cases of the problem are also obtained.

Thermo-diffusion and diffusion-thermo effects on MHD free convection flow past a vertical porous plate embedded in a non-Darcian porous medium

An analysis is presented for the steady free convection heat and mass transfer past a vertical porous plate in a non-Darcy porous medium subjected to uniform magnetic field with Soret (thermo-diffusion) and Dufour (diffusion-thermo) effects included. The non-Darcy effects are simulated via second order Forchheimer drag force term in the momentum boundary layer equation. The governing boundary layer equations are transformed into a non-dimensional form and the resulting non-linear system of partial differential equations are solved using the efficient Keller-box implicit numerical finite difference method. Increasing magnetohydrodynamic body force parameter ( M) is found to decelerate the flow. Increasing Soret number and simultaneously decreasing Dufour number enhances the local heat transfer rate (local Nusselt number) at the plate surface with the opposite effect sustained for the mass transfer rate (local Sherwood number). Increasing Forchheimer inertial drag parameter ( Λ) retards the flow considerably but enhances temperatures. Increasing Darcy number accelerates the flow due to a corresponding rise in permeability of the regime and concomitant decrease in Darcian impedance. Applications of the model arise in metallurgical materials processing, chemical engineering flow control, geothermal hydromagnetics, etc.

Thermal radiation effects on magnetohydrodynamic free convection heat and mass transfer from a sphere in a variable porosity regime

A mathematical model is presented for multiphysical transport of an optically-dense, electrically-conducting fluid along a permeable isothermal sphere embedded in a variable-porosity medium. A constant, static, magnetic field is applied transverse to the cylinder surface. The non-Darcy effects are simulated via second order Forchheimer drag force term in the momentum boundary layer equation. The surface of the sphere is maintained at a constant temperature and concentration and is permeable, i.e. transpiration into and from the boundary layer regime is possible. The boundary layer conservation equations, which are parabolic in nature, are normalized into non-similar form and then solved numerically with the well-tested, efficient, implicit, stable Keller-box finite difference scheme. Increasing porosity ( ε) is found to elevate velocities, i.e. accelerate the flow but decrease temperatures, i.e. cool the boundary layer regime. Increasing Forchheimer inertial drag parameter ( Λ) retards the flow considerably but enhances temperatures. Increasing Darcy number accelerates the flow due to a corresponding rise in permeability of the regime and concomitant decrease in Darcian impedance. Thermal radiation is seen to reduce both velocity and temperature in the boundary layer. Local Nusselt number is also found to be enhanced with increasing both porosity and radiation parameters.

The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet

The effects of variable viscosity and thermal conductivity on the flow and heat transfer in a laminar liquid film on a horizontal shrinking/stretching sheet are analyzed. The similarity transformation reduces the time independent boundary layer equations for momentum and thermal energy into a set of coupled ordinary differential equations. The resulting five-parameter problem is solved by the homotopy perturbation method. The results are presented graphically to interpret various physical parameters appearing in the problem.

Group Invariant Solution for a Liquid Film on the Surface of a Sphere

When a circular jet of liquid strikes a sphere normal to the tangent plane at a point, a liquid film is formed on the surface of the sphere. This is a new problem. The flow in the liquid film is studied by means of boundary layer theory for laminar flow. The governing equations are Prandtl’s momentum boundary layer equation and the continuity equation. To solve the problem completely a conserved quantity is required as well as boundary conditions. The conserved quantity for the film of liquid on the sphere is derived with the help of a conserved vector. Two conservation laws for the system have been obtained by Naeem and Naz (Int. J. Nonlin. Sci. 7, 149 (2009)), and one of these is used to derive the conserved quantity for the liquid film. A stream function is introduced which reduces the system to a single third-order partial differential equation. The group invariant solution for this partial differential equation is constructed by considering a linear combination of its Lie point symmetries. The velocity profile of the liquid film on the sphere is investigated.

Combined effects of non-uniform heat source/sink and thermal radiation on heat transfer over an unsteady stretching permeable surface

The present paper is concerned with the study of flow and heat transfer characteristics in the unsteady laminar boundary layer flow of an incompressible viscous fluid over continuously stretching permeable surface in the presence of a non-uniform heat source/sink and thermal radiation. The unsteadiness in the flow and temperature fields is because of the time-dependent stretching velocity and surface temperature. Similarity transformations are used to convert the governing time-dependent nonlinear boundary layer equations for momentum and thermal energy are reduced to a system of nonlinear ordinary differential equations containing Prandtl number, non-uniform heat source/sink parameter, thermal radiation and unsteadiness parameter with appropriate boundary conditions. These equations are solved numerically by applying shooting method using Runge–Kutta–Fehlberg method. Comparison of numerical results is made with the earlier published results under limiting cases. The effects of the unsteadiness parameter, thermal radiation, suction/injection parameter, non-uniform heat source/sink parameter on flow and heat transfer characteristics as well as on the local Nusselt number are shown graphically.

Heat and mass transfer by natural convection at a stagnation point in a porous medium considering Soret and Dufour effects

Dufour and Soret effects on flow at a stagnation point in a fluid-saturated porous medium are studied in this paper. A two dimensional stagnation-point flow with suction/injection of a Darcian fluid is considered. By using an appropriate similarity transformation, the boundary layer equations of momentum, energy and concentration are reduced to a set of ordinary differential equations, which are solved numerically using the Keller-box method, which is a very efficient finite differences technique. Nusselt and Sherwood numbers are obtained, together with the velocity, temperature and concentration profiles in the boundary layer. For the large suction case, asymptotic analytical solutions of the problem are obtained, which compare favourably with the numerical solutions. A critical view of the problem is presented finally.

Mixed convection magnetohydrodynamic heat and mass transfer past a stretching surface in a micropolar fluid-saturated porous medium under the influence of Ohmic heating, Soret and Dufour effects

A numerical model is developed to examine the combined effects of Soret and Dufour on mixed convection magnetohydrodynamic heat and mass transfer in micropolar fluid-saturated Darcian porous medium in the presence of thermal radiation, non-uniform heat source/sink and Ohmic dissipation. The governing boundary layer equations for momentum, angular momentum (microrotation), energy and species transfer are transformed to a set of non-linear ordinary differential equations by using similarity solutions which are then solved numerically based on shooting algorithm with Runge–Kutta–Fehlberg integration scheme over the entire range of physical parameters with appropriate boundary conditions. The influence of Darcy number, Prandtl number, Schmidt number, Soret number and Dufour number, magnetic parameter, local thermal Grashof number and local solutal Grashof number on velocity, temperature and concentration fields are studied graphically. Finally, the effects of related physical parameters on local Skin-friction, local Nusselt number and local Sherwood number are also studied. Results showed that the fields were influenced appreciably by the Soret and Dufour effects, thermal radiation and magnetic field, etc.

Thin Film Flow of a Second Grade Fluid over a Stretching/Shrinking Sheet with Variable Temperature-Dependent Viscosity

Effects of variable viscosity on the flow and heat transfer in a thin film on a horizontal porous stretching sheet are analyzed. The steady boundary layer equations for momentum and thermal energy are simplified by using similarity transformations. The resulted and coupled nonlinear differential equations are solved by Homotopy analysis method. The results are presented graphically to interpret various physical parameters appearing in the problem.

Heat transfer in a second grade fluid through a porous medium from a permeable stretching sheet with non-uniform heat source/sink

In the present article an analysis is carried out to study the boundary layer flow and heat transfer characteristics of a second grade, non-Newtonian fluid through a porous medium. The stretching sheet is assumed to be permeable so that suction effects come into play. The effects of viscous dissipation, non-uniform heat source/sink on heat transfer are addressed. 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. Analytical solutions are obtained for the resulting boundary value problems. The effects of viscous dissipation and non-uniform heat source/sink, Prandtl number, Eckert number and suction/injection on heat transfer are shown in several plots for two different heating processes (CST and PST cases). Dimensionless surface temperature gradient is tabulated for various values of the governing the parameters.

Group invariant solution for a free jet on a hemi-spherical shell

When two circular jets impinge upon each other along the axis of a hemi-spherical shell then a free jet on a hemi-spherical shell is formed. The governing equations are Prandtl’s momentum boundary layer equation and the continuity equation. The conserved quantity is required for the free jet on a hemi-spherical shell. The conserved quantity for the free jet on a hemi-spherical shell is established with the help of a conserved vector. The group invariant solution for the third-order partial differential equation for the stream function is constructed.

Heat and mass transfer in MHD non-Darcian flow of a micropolar fluid over a stretching sheet embedded in a porous media with non-uniform heat source and thermal radiation

A mathematical analysis has been carried out to study magnetohydrodynamic boundary layer flow, heat and mass transfer characteristic on steady two-dimensional flow of a micropolar fluid over a stretching sheet embedded in a non-Darcian porous medium with uniform magnetic field. Momentum boundary layer equation takes into account of transverse magnetic field whereas energy equation takes into account of Ohmic dissipation due to transverse magnetic field, thermal radiation and non-uniform source effects. An analysis has been performed for heating process namely the prescribed wall heat flux (PHF case). The governing system of partial differential equations is first transformed into a system of non-linear ordinary differential equations using similarity transformation. The transformed equations are non-linear coupled differential equations which are then linearized by quasi-linearization method and solved very efficiently by finite-difference method. Favorable comparisons with previously published work on various special cases of the problem are obtained. The effects of various physical parameters on velocity, temperature, concentration distributions are presented graphically and in tabular form.

Heat transfer in a Walter’s liquid B fluid over an impermeable stretching sheet with non-uniform heat source/sink and elastic deformation

In this present article an analysis is carried out to study the boundary layer flow behavior and heat transfer characteristics in Walter’s liquid B fluid flow. The stretching sheet is assumed to be impermeable, the effects of viscous dissipation, non-uniform heat source/sink in the presence and in the absence of elastic deformation (which was escaped from attention of researchers while formulating the viscoelastic boundary layer flow problems)on heat transfer are addressed. 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. Analytical solutions are obtained for the resulting boundary value problems. The effects of viscous dissipation, Prandtl number, Eckert number and non-uniform heat source/sink on heat transfer (in the presence and in the absence of elastic deformation) are shown in several plots and discussed. Analytical expressions for the wall frictional drag coefficient, non-dimensional wall temperature gradient and non-dimensional wall temperature are obtained and are tabulated for various values of the governing parameters. The present study reveals that, the presence of work done by deformation in the energy equation yields an augment in the fluid’s temperature.

Flow and heat transfer over an unsteady stretching surface with Hall effect

The present investigation is concerned with the effect of Hall currents on the flow and heat transfer of an electrically conducting fluid over an unsteady stretching surface in presence of a strong magnetic field. The induced magnetic field is neglected while the electron-atom collision frequency is assumed to be relatively high, so that the Hall effect is assumed to exist. The incorrect similarity transformation of Elbashbeshy and Bazid (Heat Mass Transfer 41:1–4, 2004). is corrected and a physically realistic distribution of the velocity and temperature is obtained. 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 which are then solved numerically by the shooting method. Effects of the magnetic field, M , Hall parameter, m, and the unsteadiness parameter, S, on the velocity and temperature profiles as well as the local skin friction coefficients and the heat transfer rate are shown graphically.

Radiation effect on the flow and heat transfer over an unsteady stretching sheet

The effect of radiation on the heat and fluid flow over an unsteady stretching surface is analyzed. Using a similarity transformation the governing time dependent boundary layer equations for momentum and thermal energy are reduced to a set of ordinary differential equations. The resulting three-parameter problem is solved numerically for some representative values of the unsteadiness parameter A, the radiation parameter R and Prandtl number Pr. It is shown that the heat transfer rate is increased with increasing R, A and Pr. Also the effect of radiation parameter on the heat transfer rate is found to be more noticeable at larger values of A and Pr.

Laminar flow and heat transfer in the boundary-layer of non-Newtonian fluids over a stretching flat sheet

A theoretical analysis of the laminar boundary-layer flow and heat transfer of power-law non-Newtonian fluids over a stretching sheet with the sheet velocity distribution of the form U w = C x m and the wall temperature distribution of the form T w = T ∞ + A x γ is presented, where x denotes the distance from the slit from which the surface emerges and C and A are constants, m and γ denote, the sheet velocity exponent and the temperature exponent, respectively. Within the framework of the boundary layer approximations, the nonlinear boundary layer momentum equation and the energy equation are reduced to a set of ordinary differential equations. It is found that when the velocity exponent m = 1 / 3 or the power-law index n = 1 , the similarity solutions are in existence for both the momentum equation and the energy equation. Analytical approximations with high accuracy for the reduced velocity and temperature profiles are obtained using a new procedure based on the homotopy analysis method. Besides, the effects of the parameters m , n and the Prandlt number P r on the flow are investigated.

CONSERVATION LAWS FOR HEATED LAMINAR RADIAL LIQUID AND FREE JETS

The conserved quantities for the heated radial liquid jet and the heated radial free jet are established by using conservation laws. The flow in a heated radial jet is described by Prandtl's momentum boundary layer equation, the continuity equation and the energy equation. Viscous dissipation is neglected. The multiplier approach is used to derive the conservation laws for the system of three equations for the velocity components and the temperature and three conserved vectors are obtained. The conservation laws for the system of two partial differential equations for the stream function formulation are also computed by the multiplier approach and three conserved vectors are obtained. One of these is a non-local conserved vector for the system. The conserved quantities for the heated radial liquid jet and the heated radial free jet, emitted into a stationary fluid of uniform temperature θ∞, are derived by integrating the conservation laws across the jet.

HEAT TRANSFER IN MHD VISCOELASTIC FLUID FLOW ON STRETCHING SHEET WITH HEAT SOURCE/SINK, VISCOUS DISSIPATION, STRESS WORK, AND RADIATION FOR THE CASE OF LARGE PRANDTL NUMBER

An analysis is carried out to study the magnetohydrodynamic boundary layer flow behavior and heat transfer characteristics of a viscoelastic fluid flow over a stretching sheet with radiation and for the case of large Prandtl numbers. The basic boundary layer equations of momentum and heat transfer, which are nonlinear partial differential equations, are converted into nonlinear ordinary differential equations by means of similarity transformation. The resulting nonlinear ordinary differential equations of momentum are solved exactly. Similarly, the energy equation is transformed to a confluent hypergeometric differential equation using a new variable and Rosseland approximation for radiation. The analytic solutions for temperature profile and heat transfer characteristics are obtained in terms of Kummer's function, and their asymptotic limits for large Prandtl numbers are also obtained. The effects of magnetic field, viscoelastic parameter, viscous dissipation, heat generation/absorption, work done due to deformation, and radiation on flow and heat transfer characteristics are discussed through several graphs. To assess the validity and accuracy of the present work, heat transfer results were compared to those of previously published work of Nataraja et al. (1977). This comparison shows excellent agreement between the results.

Conservation laws and conserved quantities for laminar two-dimensional and radial jets

A systematic way to derive the conserved quantities for the liquid jet, free jet and wall jet using conservation laws is presented. Both two-dimensional and radial jets are considered. The jet flows are described by Prandtl’s momentum boundary layer equation and the continuity equation. The multiplier approach (also know as variational derivative approach) is first applied to construct a basis of conserved vectors for the system. 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 liquid jet and the free jet. The multiplier approach is then applied to construct a basis of conserved vectors for the third-order partial differential equation for the stream function. The basis consists of two local conserved vectors one of which is a non-local conserved vector for the system. The conserved quantities for the free jet and the wall jet are derived from the corresponding conservation laws and boundary conditions. The approach gives a unified treatment to the derivation of conserved quantities for jet flows and may lead to a new classification of jets through conserved vectors and their multipliers.