Momentum and heat transfer of a special case of the unsteady stagnation-point flow

This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer,which is also an exact solution to the unsteady Navier-Stokes(NS)equations.The boundary layer energy equation is solved with the closed form solutions for prescribed wall temperature and prescribed wall heat flux conditions.The wall temperature and heat flux have power dependence on both time and spatial distance.The solution domain,the velocity distribution,the flow field,and the temperature distribution in the fluids are studied for different controlling parameters.These parameters include the Prandtl number,the mass transfer parameter at the wall,the wall moving parameter,the time power index,and the spatial power index.It is found that two solution branches exist for certain combinations of the controlling parameters for the flow and heat transfer problems.The heat transfer solutions are given by the confluent hypergeometric function of the first kind,which can be simplified into the incomplete gamma functions for special conditions.The wall heat flux and temperature profiles show very complicated variation behaviors.The wall heat flux can have multiple poles under certain given controlling parameters,and the temperature can have significant oscillations with overshoot and negative values in the boundary layers.The relationship between the number of poles in the wall heat flux and the number of zero-crossing points is identified.The difference in the results of the prescribed wall temperature case and the prescribed wall heat flux case is analyzed.Results given in this paper provide a rare closed form analytical solution to the entire unsteady NS equations,which can be used as a benchmark problem for numerical code validation.     

Boundary layer flow and heat transfer of a dusty fluid flow over a stretching sheet with non-uniform heat source/sink

The present paper deals with the analysis of boundary layer flow and heat transfer of a dusty fluid over a stretching sheet with the effect of non-uniform heat source/sink. Here we consider two types of heating processes namely (i) prescribed surface temperature and (ii) prescribed surface heat flux. The momentum and thermal boundary layer equations of motion are solved numerically using Runge Kutta Fehlberg fourth–fifth order method (RKF45 Method). The effects of fluid particle interaction parameter, Eckert number, Prandtl number, Number of dust particle and non-uniform heat generation/absorption parameter on temperature distribution are analyzed and also the effect of wall temperature gradient function and wall temperature function are tabulated and discussed.     

Effects of thermal buoyancy and variable thermal conductivity on the MHD flow and heat transfer in a power-law fluid past a vertical stretching sheet in the presence of a non-uniform heat source

The paper considers the flow of a power-law fluid past a vertical stretching sheet. Effects of variable thermal conductivity and non-uniform heat source/sink on the heat transfer are addressed. The thermal conductivity is assumed to vary linearly with temperature. Similarity transformation is used to convert the governing partial differential equations into a set of coupled, non-linear ordinary differential equations. Two different types of boundary heating are considered, namely Prescribed power-law Surface Temperature (PST) and Prescribed power-law Heat Flux (PHF). Shooting method is used to obtain the numerical solution for the resulting boundary value problems. The effects of Chandrasekhar number, Grashof number, Prandtl number, non-uniform heat source/sink parameters, wall temperature parameter and variable thermal conductivity parameter on the dynamics are shown graphically in several plots. The skin friction and heat transfer coefficients are tabulated for a range of values of the parameters. Present study reveals that in a gravity affected flow buoyancy effect has a significant say in the control of flow and heat transfer.     

Effect of slip velocity on Casson thin film flow and heat transfer due to unsteady stretching sheet in presence of variable heat flux and viscous dissipation

The aim of the present paper is to study flow and heat transfer characteristics of a viscous Casson thin film flow over an unsteady stretching sheet subject to variable heat flux in the presence of slip velocity condition and viscous dissipation. The governing equations are partial differential equations. They are reduced to a set of highly nonlinear ordinary differential equations by suitable similarity transformations. The resulting similarity equations are solved numerically with a shooting method. Comparisons with previous works are made, and the results are found to be in excellent agreement. In the present work, the effects of the unsteadiness parameter, the Casson parameter, the Eckert number, the slip velocity parameter, and the Prandtl number on flow and heat transfer characteristics are discussed. Also, the local skin-friction coefficient and the local Nusselt number at the stretching sheet are computed and discussed.     

Finite element solution of heat and mass transfer flow with Hall current, heat source, and viscous dissipation

The aim of the paper is to investigate the effect of heat and mass transfer on the unsteady magnetohydrodynamic free convective flow with Hall current, heat source, and viscous dissipation. The problem is governed by the system of coupled non-linear partial differential equations whose exact solution is difficult to obtain. Therefore, the problem is solved by using the Galerkin finite element method. The effects of the various parameters like Hall current, Eckert number, heat source parameter, Prandtl number, and Schmidt number on the velocity components, the temperature, and the concentration are also examined through graphs.     

Numerical solutions of pulsating flow and heat transfer characteristics in a pipe

Numerical studies are made of flow and heat transfer characteristics of a pulsating flow in a pipe. Complete time-dependent laminar boundary-layer equations are solved numerically over broad ranges of the parameter spaces, i.e., the frequency parameter β and the amplitude of oscillation A. Recently developed numerical solution procedures for unsteady boundary-layer equations are utilized. The capabilities of the present numerical model are satisfactorily tested by comparing the instantaenous axial velocities with the existing data in various parameters. The time-mean axial velocity profiles are substantially unaffected by the changes in β and A. For high frequencies, the prominent effect of pulsations is felt principally in a thin layer near the solid wall. Skin friction is generally greateer than that of a steady flow. The influence of oscillation on skin friction is appreciable both in terms of magnitude and phase relation. Numerical results for temperature are analyzed to reveal significant heat transfer characteristics. In the downstream fully established region, the Nusselt number either increases or decreases over the steady-flow value, depending on the frequency parameter, although the deviations from the steady values are rather small in magnitude for the parameter ranges computed. The Nusselt number trend is amplified as A increases and when the Prandtl number is low below unity. These heat transfer characteristics are qualitatively consistent with previous theoretical predictions.     

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.     

Effects of thermophoresis particle deposition and of the thermal conductivity in a porous plate with dissipative heat and mass transfer

Network simulation method(NSM) is used to solve the laminar heat and mass transfer of an electricallyconducting,heat generating/absorbing fluid past a perforated horizontal surface in the presence of viscous and Joule heating problem. The governing partial differential equations are non-dimensionalized and transformed into a system of nonlinear ordinary differential similarity equations,in a single independent variable,η. The resulting coupled,nonlinear equations are solved under appropriate transformed boundary conditions. Computations are performed for a wide range of the governing flow parameters,viz Prandtl number,thermophoretic coeffcient(a function of Knudsen number),thermal conductivity parameter,wall transpiration parameter and Schmidt number. The numerical details are discussed with relevant applications. The present problem finds applications in optical fiber fabrication,aerosol filter precipitators,particle deposition on hydronautical blades,semiconductor wafer design,thermo-electronics and problems including nuclear reactor safety.     

Boundary layer flow and heat transfer over an unsteady stretching vertical surface

The solution to the unsteady mixed convection boundary layer flow and heat transfer problem due to a stretching vertical surface is presented in this paper. The unsteadiness in the flow and temperature fields is caused by the time-dependent of the stretching velocity and the surface temperature. The governing partial differential equations with three independent variables are first transformed into ordinary differential equations, before they are solved numerically by a finite-difference scheme. The effects of the unsteadiness parameter, buoyancy parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. Both assisting and opposing buoyant flows are considered. It is observed that for assisting flow, the solutions exist for all values of buoyancy parameter, whereas for opposing flow, they exist only if the magnitude of the buoyancy parameter is small. Comparison with known results for steady-state flow is excellent.     

Unsteady flow and heat transfer of porous media sandwiched between viscous fluids

The problem of unsteady oscillatory flow and heat transfer of porous medin sandwiched between viscous fluids has been considered through a horizontal channel with isothermal wall temperatures. The flow in the porous medium is modeled using the Brinkman equation. The governing partial differential equations are transformed to ordinary differential equations by collecting the non-periodic and periodic terms. Closed-form solutions for each region are found after applying the boundary and interface conditions. The influence of physical parameters, such as the porous parameter, the frequency parameter, the periodic frequency parameter, the viscosity ratios, the conductivity ratios, and the Prandtl number, on the velocity and temperature fields is computed numerically and presented graphically. In addition, the numerical values of the Nusselt number at the top and bottom walls are derived and tabulated.     

Unsteady flow with heat and mass transfer due to impingement of slot jet on an inclined plate

The unsteady laminar incompressible boundary layer flow due to a two-dimensional slot jet on a flat plate at an angle of attack has been studied. The unsteadiness in the flow field is due to the free stream velocity distribution or wall temperature (concentration) which varies with time. The governing partial differential equations in primitive variables have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The effect of the variation of the free stream velocity distribution with time is found to be more pronounced on the skin friction than on the heat or mass transfer. The Prandtl number and the variation of the wall temperature with time strongly affect the heat transfer. Similarly, the Schmidt number and the variation of the concentration at the wall with time strongly affect the mass transfer. Beyond a certain critical value of the viscous dissipation parameter, the plate gets heated instead of being cooled.     

Unsteady magnetohydrodynamic stagnation point flow — closed-form analytical solutions

This paper investigates the unsteady stagnation point flow and heat transfer of magnetohydrodynamic(MHD) fluids over a moving permeable flat surface. The unsteady Navier-Stokes(NS) equations are transformed into a similarity nonlinear ordinary differential equation, and a closed form solution is obtained for the unsteadiness parameter of 2. The boundary layer energy equation is transformed into a similarity equation,and is solved for a constant wall temperature and a time-dependent uniform wall heat flux case. The solution domain, velocity, and temperature profiles are calculated for different combinations of parameters including the Prandtl number, mass transfer parameter, wall moving parameter, and magnetic parameter. Two solution branches are obtained for certain combinations of the controlling parameters, and a stability analysis demonstrates that the lower solution branch is not stable. The present solutions provide an exact solution to the entire unsteady MHD NS equations, which can be used for validating the numerical code of computational fluid dynamics.     

Melting heat transfer effects on stagnation point flow of micropolar fluid saturated in porous medium with internal heat generation （absorption）

The effect of melting heat transfer on the two dimensional boundary layer flow of a micropolar fluid near a stagnation point embedded in a porous medium in the presence of internal heat generation/absorption is investigated. The governing non-linear partial differential equations describing the problem are reduced to a system of non-linear ordinary differential equations using similarity transformations solved numerically using the Chebyshev spectral method. Numerical results for velocity, angular velocity and temperature profiles are shown graphically and discussed for different values of the inverse Darcy number, the heat generation/absorption parameter, and the melting parameter. The effects of the pertinent parameters on the local skin-friction coefficient, the wall couple stress, and the local Nusselt number are tabulated and discussed. The results show that the inverse Darcy number has the effect of enhancing both velocity and temperature and suppressing angular velocity. It is also found that the local skin-friction coefficient decreases, while the local Nusselt number increases as the melting parameter increases.     

Steady nonlinear hydromagnetic flow and heat transfer over a stretching surface of variable temperature

The steady nonlinear hydromagnetic flow of an incompressible, viscous and electrically conducting fluid with heat transfer over a surface of variable temperature stretching with a power-law velocity in the presence of variable transverse magnetic field is analysed. Utilizing similarity transformation, governing nonlinear partial differential equations are transformed to nonlinear ordinary differential equations and they are numerically solved using fourth-order Runge–Kutta shooting method. Numerical solutions are illustrated graphically by means of graphs. The effects of magnetic field, stretching parameter and Prandtl number on velocity, skin friction, temperature distribution and rate of heat transfer are discussed.     

Unsteady natural convection Couette flow of heat generating/absorbing fluid between vertical parallel plates filled with porous material

The extended Brinkman Darcy model for momentum equations and an energy equation is used to calculate the unsteady natural convection Couette flow of a viscous incompressible heat generating/absorbing fluid in a vertical channel(formed by two infinite vertical and parallel plates) filled with the fluid-saturated porous medium.The flow is triggered by the asymmetric heating and the accelerated motion of one of the bounding plates.The governing equations are simplified by the reasonable dimensionless parameters and solved analytically by the Laplace transform techniques to obtain the closed form solutions of the velocity and temperature profiles.Then,the skin friction and the rate of heat transfer are consequently derived.It is noticed that,at different sections within the vertical channel,the fluid flow and the temperature profiles increase with time,which are both higher near the moving plate.In particular,increasing the gap between the plates increases the velocity and the temperature of the fluid,however,reduces the skin friction and the rate of heat transfer.     

Computational modeling of heat transfer over an unsteady stretching surface embedded in a porous medium

The present paper deals with the study of heat transfer characteristics in the laminar boundary layer flow of an incompressible viscous fluid over an unsteady stretching sheet which is placed in a porous medium in the presence of viscous dissipation and internal absorption or generation. Similarity transformations are used to convert the governing time dependent nonlinear boundary layer equations into a system of non-linear ordinary differential equations containing Prandtl number, Eckert number, heat source/sink parameter, porous parameter 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 parameters which determine the velocity and temperature fields are discussed in detail.     

Lie group analysis for thermophoretic and radiative augmentation of heat and mass transfer in a Brinkman-Darcy flow over a flat surface with heat generation

A boundary layer analysis is presented to investigate numerically the effects of radiation,thermophoresis and the dimensionless heat generation or absorption on hydromagnetic flow with heat and mass transfer over a flat surface in a porous medium.The boundary layer equations are transformed to non-linear ordinary differential equations using scaling group of transformations and they are solved numerically by using the fourth order Runge-Kutta method with shooting technique for some values of physical parameters.Comparisons with previously published work are performed and the results are found to be in very good agreement.Many results are obtained and a representative set is displayed graphically to illustrate the influence of the various parameters on the dimensionless velocity,temperature and concentration profiles as well as the local skin-friction coefficient,wall heat transfer,particle deposition rate and wall thermophoretic deposition velocity.The results show that the magnetic field induces acceleration of the flow,rather than deceleration(as in classical magnetohydrodynamics(MHD) boundary layer flow) but to reduce temperature and increase concentration of particles in boundary layer.Also,there is a strong dependency of the concentration in the boundary layer on both the Schmidt number and mass transfer parameter.     

MHD flow and heat transfer over a stretching surface with variable thermal conductivity and partial slip

In this paper we analyze the flow and heat transfer of an MHD fluid over an impermeable stretching surface with variable thermal conductivity and non-uniform heat source/sink in the presence of partial slip. The governing partial differential equations of the problem are reduced to nonlinear ordinary differential equations by using a similarity transformation. The temperature boundary conditions are assumed to be linear functions of the distance from the origin. Analytical solutions of the energy equations for Prescribed Surface Temperature (PST) and Prescribed Heat Flux (PHF) cases are obtained in terms of a hypergeometric function, without applying the boundary-layer approximation. The effects of the governing parameters on the flow and heat transfer fields are presented through tables and graphs, and they are discussed. Furthermore, the obtained numerical results for the skin friction, wall-temperature gradient and wall temperature are analyzed and compared with the available results in the literature for special cases.     

Generalised Couette flow of two immiscible viscous fluids with heat transfer using Brinkman model

The present study deals with generalised Couette flow of two viscous, incompressible, immiscible fluids with heat transfer in presence of heat source through two straight parallel horizontal walls. The lower wall is bounded below, by a naturally permeable material of high porosity and the flow inside the porous medium is assumed to be of moderate permeability, modelled by Brinkman equation. The flow domain is divided into three zones to obtain analytical solutions of the momentum and energy equations. To link various flow regions, appropriate matching conditions have been used. The effects of permeability parameter, Reynolds number and viscous parameter on velocity field and the effects of Reynolds number, viscous parameter, permeability parameter, constant heat source and Brinkman number on temperature distribution in different zones are discussed graphically. The mass flow rate, skin-friction factor and rates of heat transfer at the upper boundary and porous interface are discussed with the help of tables.     

Unsteady conjugate problem of a dissipative fluid in a horizontal channel with a periodic variation temperature

The unsteady two-dimensional transient heat transfer problem referring to a fully laminar flow developing in a parallel-plane channel exposed to a periodic variation surface temperature with distance is numerically studied. The effects of channel thickness, Péclet number, wall-to-fluid conductivity ratio, thermal diffusivity ratio, angular frequency and the viscous dissipation parameter are determined in the solutions. The non-linear equations are discretized by means an implicit finite difference scheme and the electric analogy to the resulting system is applied to convert these equations into a network-electrical model that was solved using a computer code (electric circuits simulator). In this scheme, only spatial discretization is necessary, while time remains as a real continuous variable, and its programming does not require manipulation of the sophisticated mathematical software that is inherent in other numerical methods. The network simulation method, which satisfies the conservation law for the heat flux variable and the uniqueness law for temperature, also permits the direct visualization of the local and/or integrated transport variables at any point or section of the medium.