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1.
The melting phenomenon in two-dimensional (2D) flow of fourth-grade material over a stretching surface is explored. The flow is created via a stretching surface. A Darcy-Forchheimer (D-F) porous medium is considered in the flow field. The heat transport is examined with the existence of the Cattaneo-Christov (C-C) heat flux. The fourth-grade material is electrically conducting subject to an applied magnetic field. The governing partial differential equations (PDEs) are reduced into ordinary differential equations (ODEs) by appropriate transformations. The solutions are constructed analytically through the optimal homotopy analysis method (OHAM). The fluid velocity, temperature, and skin friction are examined under the effects of various involved parameters. The fluid velocity increases with higher material parameters and velocity ratio parameter while decreases with higher magnetic parameter, porosity parameter, and Forchheimer number. The fluid temperature is reduced with higher melting parameter while boosts against higher Prandtl number, magnetic parameter, and thermal relaxation parameter. Furthermore, the skin friction coefficient decreases against higher melting and velocity ratio parameters while increases against higher material parameters, thermal relaxation parameter, and Forchheimer number. 相似文献
2.
This paper studied on magnetohydrodynamics flow and heat transfer outside a stretching cylinder. Momentum and energy equations are reduced using similarity transformation and converted into a system of ordinary differential equations which are solved analytically by the homotopy analysis method. The effects of the parameters involved, namely the magnetic parameter ( M), Prandtl number ( Pr) and Reynolds number ( Re) on the velocity and temperature fields are investigated. The obtained results are valid for the whole solutions' domain with high accuracy. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
3.
In this paper, the unsteady three‐dimensional boundary layer flow due to a stretching surface in a viscous and incompressible micropolar fluid is considered. The partial differential equations governing the unsteady laminar boundary layer flow are solved numerically using an implicit finite‐difference scheme. The numerical solutions are obtained which are uniformly valid for all dimensionless time from initial unsteady‐state flow to final steady‐state flow in the whole spatial region. The equations for the initial unsteady‐state flow are also solved analytically. It is found that there is a smooth transition from the small‐time solution to the large‐time solution. The features of the flow for different values of the governing parameters are analyzed and discussed. The solutions of interest for the skin friction coefficient with various values of the stretching parameter c and material parameter K are presented. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
4.
An analysis is made for the steady two-dimensional magneto-hydrodynamic flow of an incompressible viscous and electrically
conducting fluid over a stretching vertical sheet in its own plane. The stretching velocity, the surface temperature and the
transverse magnetic field are assumed to vary in a power-law with the distance from the origin. The transformed boundary layer
equations are solved numerically for some values of the involved parameters, namely the magnetic parameter M, the velocity exponent parameter m, the temperature exponent parameter n and the buoyancy parameter λ, while the Prandtl number Pr is fixed, namely Pr = 1, using a finite difference scheme known as the Keller-box method. Similarity solutions are obtained in the presence of
the buoyancy force if n = 2 m−1. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed
and discussed. It is found that both the skin friction coefficient and the local Nusselt number decrease as the magnetic parameter
M increases for fixed λ and m. For m = 0.2 (i.e. n = −0.6), although the sheet and the fluid are at different temperatures, there is no local heat transfer at the surface of
the sheet except at the singular point of the origin (fixed point). 相似文献
5.
An numerical study of the flow and heat transfer characteristics associated with a heated, continuously stretching surface
being cooled by a mixed convection flow has been carried out. The relevant heat transfer mechanisms are of interest in a wide
variety of practical applications, such as hot rolling, continuous casting, extrusion, and drawing. The surface velocity of
the continuously stretching sheet was assumed to vary according to a power-law form, that is, u
w
( x)= Cx
p
. Two conditions of surface heating were considered, a variable wall temperature (VWT) in the form T
w
( x)− T
∞= Ax
n
and a variable surface heat flux (VHF) in the form q
w
( x)= Bx
m
. The governing differential equations are transformed by introducing proper nonsimilarity variables and solved numerically
using a procedure based on finite difference approximations. Results for the local Nusselt number and the local friction coefficient
are obtained for a wide range of governing parameters, such as the surface velocity parameter p, the wall temperature exponent n, the surface heat flux exponent m, the buoyancy force parameters (ξ for the VWT case and χ for the VHF case), and Prandtl number of the fluid. It is found
that the local Nusselt number is increased with increasing the velocity exponent parameter p for the VWT case, while the opposite trend is observed for the VHF case. The local friction coefficient is increased for
a decelerated stretching surface, while it is decreased for an accelerated stretching surface. Also, appreciable effects of
the buoyancy force on the local Nusselt number and the local friction coefficient are observed for both VWT and VHF cases,
as expected.
Received on 11 January 1999 相似文献
6.
Numerical simulations have been performed for flow past two equal‐sized square cylinders in tandem arrangement subjected to incoming planar shear flow. Effect of L/ d ratio and the shear parameter has been studied. The range of L/ d ratio (ratio of center‐to‐center distance ( L) to cylinder width ( d)) is varied from 2 to 7 and the non‐dimensional shear parameter ( K) is varied from 0.0 to 0.4 in steps of 0.1. For all the cases the Reynolds number ( Re) based on centerline velocity and cylinder width is fixed at 100. The results are compared with that of isolated square cylinder with uniform flow. Strouhal number decreases with increasing shear parameter. There are more than one shedding frequency at high shear parameters and L/ d ratios. The mean drag coefficient is decreased with shear parameter and lesser than that of the single cylinder. The root mean square (RMS) value of both lift and drag coefficients is higher for the downstream cylinder for all values of shear parameter. With increasing L/ d ratio, for both lift and drag, the RMS value increases and then decreases for upstream cylinder, whereas it continuously increases for the downstream cylinder. The stagnation point is moved towards the top leading edge with increasing shear. The critical L/ d ratio, which is defined as the distance between two cylinders, beyond which the vortex shedding from the upstream cylinder occurs, decreases with increasing shear parameter. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
7.
A numerical study of pulsatile flow and mass transfer of an electrically conducting Newtonian biofluid via a channel containing
porous medium is considered. The conservation equations are transformed and solved under boundary conditions prescribed at
both walls of the channel, using a finite element method with two-noded line elements. The influence of magnetic field on
the flow is studied using the dimensionless hydromagnetic number, Nm, which defines the ratio of magnetic (Lorentz) retarding
force to the viscous hydrodynamic force. A Darcian linear impedance for low Reynolds numbers is incorporated in the transformed
momentum equation and a second order drag force term for inertial (Forchheimer) effects. Velocity and concentration profiles
across the channel width are plotted for various values of the Reynolds number ( Re), Darcy parameter (λ), Forchheimer parameter ( Nf), hydro-magnetic number ( Nm), Schmidt number ( Sc) and also with dimensionless time ( T). Profiles of velocity varying in space and time are also provided. The conduit considered is rigid with a pulsatile pressure
applied via an appropriate pressure gradient term. Increasing the hydromagnetic number ( Nm) from 1 to 15 considerably depresses biofluid velocity ( U) indicating that a magnetic field can be used as a flow control mechanism in, for example, medical applications. A rise in
Nf from 1 to 20 strongly retards the flow development and decreases the velocity, U, across the width of the channel. The effects of other parameters on the flowfield are also discussed at length. The flow
model also has applications in the analysis of electrically conducting haemotological fluids flowing through filtration media,
diffusion of drug species in pharmaceutical hydromechanics, and also in general fluid dynamics of pulsatile systems. 相似文献
8.
The flow structure and isotherms of hard disk drives (HDD) are investigated using a finite element method (FEM). The governing equations are based on the three‐dimensional axisymmetric Navier–Stokes partial differential equations (PDEs) with Galerkin FE formulation. Co‐rotating models are selected that include the non‐ventilated configuration within an enclosure. With various operating conditions for the disk system, the following important parameters are considered: disk number ( n), rotational speed ( Ro), and wall temperature. The flow structure changes rapidly when the rotational Reynolds number ( Reϕ) is increased. The flow has a greater tendency to flow radially outwards and the swirling velocity tends to be more vertically orientated, especially for high Reϕ values. The isotherms only have small varying regions near the rotating axis, forming outward arcs near the wall and inward arcs near the end gap of the disk. Different from the case without the enclosure, the vorticities exist along the outer disk ends. Both the swirling velocity and isotherms indicate nearly symmetrical characteristics, as expected. A higher temperature gradient occurs near the right outer disk ends, which implies the characteristic of higher heat flux. A commercial computational fluid dynamic (CFD) code, CFX‐5, was chosen to validate the results. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
9.
A low‐Reynolds number k– ε turbulence model is proposed that incorporates diffusion terms and modified Cε(1,2) coefficients to amplify the level of dissipation in non‐equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. Unlike the conventional k– ε model, it requires no wall function/distance parameter that bridges the near‐wall integration. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
10.
The paper presents a study of the laminar mixed convection adjacent to vertical continuously stretching sheets, taking into account the effects of variable viscosity and variable thermal diffusivity. The similarity solutions are reported for isothermal sheet moving with a velocity of the form uw= Bx0.5 and a continuous linearly stretching sheet with a linear surface temperature distribution. The equations of conservation of mass, momentum and energy, which govern the flow and heat transfer, are solved numerically by using the shooting method. The numerical results obtained for the flow and heat transfer characteristics reveal many interesting behaviors. The numerical results show that, variable viscosity, variable thermal diffusivity, the velocity exponent parameter, the temperature exponent parameter and the buoyancy force parameter have significant influences on the velocity and temperature profiles, shear stress and Nusselt number in two cases air and water. 相似文献
11.
The effects of pulsatile amplitude on sinusoidal transitional turbulent flows through a rigid pipe in the vicinity of a sharp‐edged mechanical ring‐type constriction have been studied numerically. Pulsatile flows were studied for transitional turbulent flow with Reynolds number ( Re) of the order of 10 4, Womersley number ( Nw) of the order of 50 with a corresponding Strouhal number ( St) of the order of 0.04. The pulsatile flow considered is a sinusoidal flow with dimensionless amplitudes varying from 0.0 to 1.0. Transitional laminar and turbulent flow characteristics in an alternative manner within the pulsatile flow fields were observed and studied numerically. The flow characteristics were studied through the pulsatile contours of streamlines, vorticity, shear stress and isobars. It was observed that fluid accelerations tend to suppress the development of flow disturbances. All the instantaneous maximum values of turbulent kinetic energy, turbulent viscosity, turbulent shear stress are smaller during the acceleration phase when compared with those during deceleration period. Various parametric equations within a pulsatile cycle have also been formulated through numerical experimentations with different pulsatile amplitudes. In the vicinity of constrictions, the empirical relationships were obtained for the instantaneous flow rate ( Q), the pressure gradient (d p/d z), the pressure loss ( Ploss), the maximum velocity ( Vmax), the maximum vorticity (ζ max), the maximum wall vorticity (ζ w,max), the maximum shear stress (τ max) and the maximum wall shear stress (τ w,max). Elliptic relation was observed between flow rate and pressure gradient. Quadratic relations were observed between flow rate and the pressure loss, the maximum values of shear stress, wall shear stress, turbulent kinematic energy and the turbulent viscosity. Linear relationships exist between the instantaneous flow rate and the maximum values of vorticity, wall vorticity and velocity. The time‐average axial pressure gradient and the time average pressure loss across the constriction were observed to increase linearly with the pulsatile amplitude. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
12.
This paper presents a numerical analysis of the steady boundary-layer flow of a Reiner–Philippoff fluid induced by a 90° stretching wedge in a variable free stream. The governing partial differential equations are converted into a set of two ordinary differential equations by the use of a similarity transformation. The flow is therefore governed by a stretching velocity parameter λ and two non-Newtonian fluid parameters γ and μ0. The variation of the skin friction, as well as other flow characteristics, as a function of the governing parameters is presented graphically and tabulated. A stability analysis has also been performed for this self-similar flow based on linear disturbances to the steady similarity solutions. The results presented in this paper reveal that there are no multiple (dual) solutions for the present problem and the unique solution is stable. 相似文献
13.
An analysis is presented for the steady non-linear viscous flow of an incompressible viscous fluid over a horizontal surface
of variable temperature with a power-law velocity under the influences of suction/blowing, viscous dissipation and thermal
radiation. Numerical results are illustrated by means of tables and graphs. The governing partial differential equations are
converted into nonlinear ordinary differential equations by a similarity transformation. The effects of the stretching parameter
n, suction/blowing parameter b, Prandtl number σ, Eckert number Ec( Ec * )E_{c}(E_{c}^{ *} ) and radiation parameter N
R
are discussed. Two cases are studied, namely, (i) Prescribed surface temperature (PST case) and, (ii) Prescribed heat flux
at the sheet (PHF case). 相似文献
14.
A theoretical solution is presented for the convective heat transfer of Giesekus viscoelastic fluid in pipes and channels,
under fully developed thermal and hydrodynamic flow conditions, for an imposed constant heat flux at the wall. The fluid properties
are taken as constant and axial conduction is negligible. The effect of Weissenberg number ( We), mobility parameter ( α) and Brinkman number ( Br) on the temperature profile and Nusselt number are investigated. The results emphasize the significant effect of viscous
dissipation and fluid elasticity on the Nusselt number in all circumstances. For wall cooling and the Brinkman number exceeds
a critical value ( Br
1), the heat generated by viscous dissipation overcomes the heat removed at the wall and fluid heats up longitudinally. Fluid
elasticity shifts this critical Brinkman number to higher values. 相似文献
15.
In this article, a similarity solution of the steady boundary layer flow near the stagnation-point flow on a permeable stretching
sheet in a porous medium saturated with a nanofluid and in the presence of internal heat generation/absorption is theoretically
studied. The governing partial differential equations with the corresponding boundary conditions are reduced to a set of ordinary
differential equations with the appropriate boundary conditions via Lie-group analysis. Copper (Cu) with water as its base
fluid has been considered and representative results have been obtained for the nanoparticle volume fraction parameter f{\phi} in the range 0 £ f £ 0.2{0\leq \phi \leq 0.2} with the Prandtl number of Pr = 6.8 for the water working fluid. Velocity and temperature profiles as well as the skin friction coefficient and the local
Nusselt number are determined numerically. The influence of pertinent parameters such as nanofluid volume fraction parameter,
the ratio of free stream velocity and stretching velocity parameter, the permeability parameter, suction/blowing parameter,
and heat source/sink parameter on the flow and heat transfer characteristics is discussed. Comparisons with published results
are also presented. It is shown that the inclusion of a nanoparticle into the base fluid of this problem is capable to change
the flow pattern. 相似文献
16.
A moment‐of‐fluid method is presented for computing solutions to incompressible multiphase flows in which the number of materials can be greater than two. In this work, the multimaterial moment‐of‐fluid interface representation technique is applied to simulating surface tension effects at points where three materials meet. The advection terms are solved using a directionally split cell integrated semi‐Lagrangian algorithm, and the projection method is used to evaluate the pressure gradient force term. The underlying computational grid is a dynamic block‐structured adaptive grid. The new method is applied to multiphase problems illustrating contact‐line dynamics, triple junctions, and encapsulation in order to demonstrate its capabilities. Examples are given in two‐dimensional, three‐dimensional axisymmetric ( R– Z), and three‐dimensional ( X– Y– Z) coordinate systems. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
17.
In the present study we have explored the effects of thermal buoyancy on flow of a viscoelastic second grade fluid past a
vertical, continuous stretching sheet of which the velocity and temperature distributions are assumed to vary according to
a power-law form. The governing differential equations are transformed into dimensionless form using appropriate transformations
and then solved numerically. The methods here employed are (1) the perturbation method together with the Shanks transformation,
(2) the local non-similarity method with second level of truncation and (3) the implicit finite difference method for values
of ξ ( = Gr
x
/ Re
x
2, defined as local mixed convection parameter) ranging in [0, 10]. The comparison between the solutions obtained by the aforementioned
methods found in excellent agreement. Effects of the elasticity parameter λ on the skin-friction and heat transfer coefficients
have been shown graphically for the fluids having the values of the Prandtl number equal to 0.72, 7.03 and 15.0. Effects of
the viscoelastic parameter and the mixed convection parameter, ξ, on the temperature and velocity fields have also been studied.
We notice that with the increase in visco-elastic parameter λ, velocity decreases whereas temperature increases and that velocity
gradient is higher than that of temperature.
On leave of absence from the Department of Mathematics, University of Dhaka, Bangladesh. 相似文献
18.
The analysis of laminar mixed convection in boundary layers adjacent to a vertical, continuously stretching sheet has been
presented. The velocity and temperature of the sheet were assumed to vary in a power-law form, that is, u
w
( x)= Bx
m
and T
w
( x)− T
∞= Ax
n
. In the presence of buoyancy force effects, similarity solutions were reported for the following two cases: (a) n=0 and m=0.5, which corresponds to an isothermal sheet moving with a velocity of the form u
w
= Bx
0.5 and (b) n=1 and m=1, which corresponds to a continuous, linearly stretching sheet with a linear surface temperature distribution, i.e. T
w
− T
∞= Ax. Formulation of the present problem shows that the heat transfer characteristics depends on four governing parameters, namely,
the velocity exponent parameter m, the temperature exponent parameter n, the buoyancy force parameter G
*, and Prandtl number of the fluid. Numerical solutions were generated from a finite difference method. Results for the local
Nusselt number, the local friction coefficient, and temperature profiles are presented for different governing parameters.
Effects of buoyancy force and Prandtl number on the flow and heat transfer characteristics are thoroughly examined.
Received on 17 July 1997 相似文献
19.
The present paper deals with the flow and heat transfer of a viscous fluid saturated in a porous medium past a permeable and
non-isothermal stretching sheet with internal heat generation or absorption and radiation. Closed-form solutions to steady,
two dimensional momentum equations with neglecting quadratic inertia terms and heat transfer equation are found using a similarity
transformation. Asymptotic expressions of the temperature functions are also presented valid for both very large and very
small modified Prandtl numbers. Attention is focused on the effects of porous parameter K, suction parameter R, radiation parameter Nr, viscosity ratio Λ, internal heat parameter α and Prandtl number P to the characteristics of flow and heat transfer. 相似文献
20.
The r‐ratio is a parameter that measures the local monotonicity, by which a number of high‐resolution and TVD schemes can be formed. A number of r‐ratio formulations for TVD schemes have been presented over the last few decades to solve the transport equation in shallow waters based on the finite volume method (FVM). However, unlike structured meshes, the coordinate directions are not clearly defined on an unstructured mesh; therefore, some r‐ratio formulations have been established by approximating the solute concentration at virtual nodes, which may be estimated from different assumptions. However, some formulations may introduce either oscillation or diffusion behavior within the vertex‐centered (VC) framework. In this paper, a new r‐ratio formulation, applied to an unstructured grid in the VC framework, is proposed and compared with the traditional r‐ratio formulations. Through seven commonly used benchmark tests, it is shown that the newly proposed r‐ratio formulation obtains better results than the traditional ones with less numerical diffusion and spurious oscillation. Moreover, three commonly used TVD schemes—SUPERBEE, MINMOD, and MUSCL—and two high‐order schemes—SOU and QUICK—are implemented and compared using the new r‐ratio formulation. The new r‐ratio formulation is shown to be sufficiently comprehensive to permit the general implementation of a high‐resolution scheme within the VC framework. Finally, the sensitivity test for different grid types demonstrates the good adaptability of this new r‐ratio formulation. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
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