Dimensionless Stream Function Research Articles

Effects of viscous dissipation and work done by deformation on the MHD flow and heat transfer of a viscoelastic fluid over a stretching sheet

An analysis is performed for flow and heat transfer of a steady laminar boundary-layer flow of an electrically conducting fluid of second grade subject to suction and to a transverse uniform magnetic field past a semi-infinite stretching sheet. The influence on the skin-friction parameter b of various non-dimensional parameters like viscoelastic parameter λ 1 , magnetic parameter M and suction parameter R is discussed and the variations of the dimensionless stream function f and its derivatives with these parameters are graphed and tabulated. Therewith, the effects of viscous dissipation and work due to deformation are considered in the energy equation and the variations of dimensionless surface temperature and dimensionless surface temperature gradient with the above-mentioned non-dimensional parameters and other ones like Prandtl number σ and Eckert number E ck are also plotted and tabulated. Two cases are studied, namely, (i) the sheet with constant surface temperature (CST case) and (ii) the sheet with prescribed surface temperature (PST case). A novel result of the analysis is that for a given viscoelastic fluid subjected to a constant suction the work due to deformation's effect increases with increase in the magnetic parameter M.

A direct method to calculate the heat transfer coefficient of steady similar boundary layer flows induced by continuous moving surfaces

The heat transfer coefficient mentioned in the title is calculated in this paper without to know the solution of the corresponding boundary value problem (neither in an explicit nor in an implicit form). This “direct method” makes use of a slightly modified form of the Merkin transformation which reverses the role of the dimensionless stream function f from that of an “old” dependent variable to one of a “new” independent variable and then gives the formal solution for the velocity and temperature profiles as a power series of 1 − f / f ∞ (where f ∞ denotes the dimensionless entrainment velocity). The series solutions yield for f = 0 explicit analytical expressions for the skin friction and the heat transfer coefficient of the flow. Owing to a formal mathematical analogy, the results also apply to the free convection boundary layer flows from vertical surfaces adjacent to fluid saturated porous media.

Oscillatory eddy structure in a container

We consider the periodic, two-dimensional motion of a viscous, incompressible liquid which fills a rectangular container. The motion is due to the periodic motion of the lid which moves in its own plane. If the velocities are sufficiently small the motion will be governed by the linearized Navier–Stokes equations and consequently the dimensionless stream function -plane. This fact has an important bearing on the damping of gravity waves at high Reynolds numbers. The eigenfunctions derived above are used to write down a formal expansion for the stream function and the coefficients are determined from the boundary conditions using a least-squares procedure. An examination of the resulting streamline patterns reveals interesting inertial effects even at low Reynolds numbers. In particular we examine the mechanism by which the flow field reverses its direction when the lid stops and reverses its direction of motion. If inertial effects are completely negelected, as has been done till now, one would infer an immediate response of the fluid to the changes in the lid motion; for example, one would conclude, wrongly, that when the lid is at rest so is the fluid. Our analysis shows, in fact, a very intricate and beautiful mechanism, involving among other things an apparent engulfing of the corner eddy by the new primary eddy, by which the direction of the circulation is reversed in the fluid. These results should be of importance in the analysis of mixing, where such effects appear to have been ignored till now.

The Effect Of Aspect Ratio On The Development OfConvective Motion In A Bottom Heated EnclosureContaining Ice And Water

A numerical study of the steady state flow in a rectangular enclosure with two vertical walls which are adiabatic and with two horizontal isothermal walls has been undertaken. The enclosure contains water and the upper wall is maintained at a uniform temperature that is below the freezing point of water while the lower wall is maintained at a uniform temperature that is above the freezing point of water. The upper portion of the enclosure is thus filled with ice and the lower portion is filled with water. The conditions considered in the present study are such there can be significant natural convection in the water and the effect of the density maximum that exists in the water at approximately 4°C can have a significant effect on this flow. The main aim of the study was to determine how far above 4°C the hot wall temperature can be before significant convective motion develops in the water. The governing equations have been expressed in dimensionless form and solved using a finite element procedure. The effect of the various governing parameters, and in particular the effect of the enclosure width-toheight ratio, on the value of the hot wall temperature at which convective motion commences has been considered. Nomenclature A = aspect ratio of enclosure, i.e. W/H' Advanced Computational Methods in Heat Transfer VI, C.A. Brebbia & B. Sunden (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-818-X 278 Advanced Computational Methods in Heat Transfer VI a' = coefficient in density-temperature relationship H' = height of enclosure H't — thickness of liquid layer in pure conduction case H's = thickness of solid layer in pure conduction case k' = thermal conductivity of water kr = ratio of solid to liquid thermal conductivities n = n'/H' n' — coordinate measured normal to a surface Nu = Nusselt number based on H' Pr = Prandtl number q' = heat transfer rate per unit wall area. Bo* = modified Rayleigh number based on H' T' — temperature T = dimensionless temperature T'c = temperature of top surface T'p = solidification temperature T'H = temperature of bottom surface T'rnax = temperature of maximum density TC — dimensionless temperature of top surface TH = dimensionless temperature of bottom surface THP — dimensionless temperature of bottom wall at which liquid motion starts Tmax — dimensionless temperature of maximum density t' = time t — dimensionless time x' = horizontal coordinate x = dimensionless horizontal coordinate %/ = vertical coordinate y — dimensionless vertical coordinate W' = width of enclosure x' = horizontal coordinate x = dimensionless horizontal coordinate %/ — vertical coordinate y = dimensionless vertical coordinate a' = thermal diffusivity ?// = stream function ijj — dimensionless stream function u/ = vorticity (jj = dimensionless vorticity p' — density P'max — maximum density

Thermal and solutal Marangoni convection in In–Ga–Sb system

In order to clarify the velocity feature of surface-tension-driven-flow caused by temperature differences (thermal Marangoni convection) and concentration differences (solutal Marangoni convection), the rectangular double-crucible system was developed. Measurements of surface velocities of thermal and solutal Marangoni convection in In–Ga–Sb melt were performed using the system. For solutal Marangoni convection, we solved the model in which the additional material spreads on the initial material, using a similarity parameter and a dimensionless stream function. As a result, the analytical result was in agreement with the experimental one. Moreover, we found that the typical surface velocity of solutal Marangoni convection is about 3–5 times higher than that of thermal Marangoni convection, and the results of both thermal and solutal convection could be discussed using dimensionless Reynolds (Re), Marangoni (Ma) and Prandtl (Pr) numbers. The relationship between the difference in surface tension and the surface velocity is that Re is proportional to Ma Pr −1/2 to the power of 2/3.

Full and boundary-layer solutions of the steady air sparging problem

The one previous analytical solution to the steady sparging problem in a homogeneous soil was in the boundary-layer approximation that neglects the vertical capillary pressure gradient. The present study gives an analytical solution to the full air-flow equation, with the corresponding boundary-layer solution for comparison. The full solution predicts sparging below the air-injection point, whereas the boundary-layer solution does not. Solutions are expressed both in terms of the dimensionless Kirchhoff potential θ (increasing monotonically with capillary pressure and air saturation also) and the dimensionless Stokes stream function F (mapping the pattern of air flow). Both θ and F show the full solution predicting more effective sparging near the injection point than does the boundary-layer solution. The overall boundary-layer relative error, however, decreases as dimensionless air-injection depth below the undisturbed water-table increases.

Radiation-Conduction Interaction on Mixed Convection Flow Along a Slender Vertical Cylinder

Nomenclature = Rosseland mean absorption coefficient specific heat at constant pressure dimensionless stream function Grashof number acceleration because of gravity Nusselt number Prandtl number surface heat flux Planck number radial and axial coordinates temperature of the fluid temperature of the heated surface temperature of the ambient fluid velocities in the x and r directions coefficient of thermal diffusivity coefficient of cubical expansion scaled streamwise coordinate similarity variable scaled temperature surface temperature ratio to the ambient fluid coefficient of thermal conductivity curvature parameter v = kinematic viscosity £ = nondimensional axial coordinate p = density of the fluid cr = Stefan—Boltzmann constant crs = scattering coefficient

Numerical precision and dissipation errors in rotating flows

Presents three non‐isothermal, time dependent, three dimensional examples having cylindrical geometries to show the significant effort of numerical precision and dissipation on rotating flow predictions. The examples are relevant to turbomachinery design and geophysical studies. Discusses the relationship between numerical precision, numerical dissipation and co‐ordinate system angular velocity. Compares predictions made in stationary and rotating co‐ordinate systems, using contour plots of dimensionless stream function and temperature. Shows that wrong, axisymmetric solutions are predicted if the co‐ordinate system is not selected to minimize relative tangential velocities/Peclet numbers, thereby increasing numerical precision and reducing dissipation.

Effects of a porous medium on local heat transfer and fluid flow in a forced convection field

Abstract A numerical estimation was made for the laminar flow and heat transfer through a porous medium placed in a flow passage with a miter bend. Heat transfer between solid material and fluid was considered in the numerical analysis. The calculations were performed for the Darcy number Da = 1– 2.5× 10−6, the ratio of thermal conductivity of solid material to fluid RRM = 1–1000, the parameter for heat transfer between solid material and fluid Hg = 10–1000 and the Reynolds number Re = 300. The results of local heat transfer and dimensionless stream function were compared for the cases with and without a porous medium. The behavior of the flow, the solid and gas temperatures, the Nusselt number with these parameters and the thermal performance were illustrated. It was found that a porous medium with high RRM and Hg should be selected with an optimum Darcy number to enhance the local heat transfer without a significant flow resistance.

Hiemenz flow in hydromagnetics

The laminar flow of an incompressible, viscous, electrically conducting fluid impinging normal to a plane in the presence of a transverse magnetic field is investigated. Using finite-differences and quasilinearization, an exact numerical solution is presented which takes into account the asymptotic boundary condition. It is demonstrated that iff denotes the dimensionless stream function, the value off″(0) increases monotonically withM, the Hartmann number, where a prime denotes the derivative normal to the plane.

Effects of a Porous-Medium in a Forced Convection Field on Local Heat Transfer and Flow.

A numerical estimation was performed for the laminar flow and heat transfer by a porous medium in a flow passage with a miter bend. Heat transfer between solid material and fluid was considered in the numerical analysis. The calculations were performed for the Darcy number Da = 1∼2.5×10-6, the ratio of thermal conductivity of solid material and fluid RRM = 1∼1000, the parameter of heat transfer between solid material and fluid Hg = 10∼1000 and the Reynolds number Re = 300. Local heat transfer and dimensionless stream function were compared between the flow with and without porous medium. The behavior of the flow, the solid and gas temperatures and the Nusselt number with these parameters was represented. It was found that the porous medium with low Da, high RRM and Hg should be selected to enhance the local heat transfer.

Prediction of LI-SF6 wick combustion

A theoretical and experimental investigation was conducted to study the wick combustion of lithium (Li) and sulfur hexafluoride (SF6). The wick combustion experiments were conducted in a vacuum chamber at a pressure of 0.01 MPa. At this condition, the wick combustion of Li and SF6 yields a bright luminous flame. During the combustion, the lithium bath temperature was maintained near its saturation temperature at the system pressure. A prediction model based on the conserved-scalar approach is presented along with an interface analysis. The predicted interface temperature was in agreement with the experimental liquid bath temperature. The model prediction yields a reasonable flame stand off distance as compared to the experiments reported elsewhere. The flowfleld analysis shows self-similar solutions for planar wick combustion of Li and SF6 and nonsimilar solutions for the cylindrical wick combustion. Property variations and wick curvature are found important to the flowfleld prediction of the Li-SF6 wick diffusion flame. Nomenclature A - parameter, Eq. (15) c - parameter, Eq. (8) D - binary diffusion coefficient DK - species parameter, Eq. (28) / = dimensionless stream function, Eq. (7) Gr = Grashof number, Eq. (15) g - acceleration of gravity H = plate or cylinder height h = enthalpy k - thermal conductivity L = enthalpy of gasification NK - species parameter, Eq. (28) Pr = Prandtl number T - temperature u - streamwise velocity v = cross-stream velocity x = streamwise direction y, = mass fraction of species / y = cross-stream direction y0 = radius of cylindrical wick Z = mixture fraction 17 = similarity variable, Eq. (10) jti = dynamic viscosity v — kinematic viscosity f = similarity variable, Eq. (10) p - density ^ = stream function, Eq. (9) a) = source term to account for the polymerization of gas species, Eq. (21)

A non‐iterative solution of a system of ordinary differential equations arising from boundary layer theory

Using a two parameter group transformation, the ordinary differential equations describing similarity solutions for the dimensionless stream function and temperature for flow over a flat plate held at a constant temperature have been transformed from simultaneous boundary value equations to simultaneous initial value equations. The resulting initial value equations have been solved using a standard fourth order Runge Kutta method incorporated in a teaching computer package recently developed at Napier Polytechnic. The results of the solution of the initial value problem, presented in the form of the missing initial values for the dimensionless stream function and temperature, enable the usual solution curves for these variables to be reproduced easily. Also, the results have a certain pedagogical value and may be of some use in teaching fluid dynamics. Finally, constant fluid properties are assumed throughout the analysis reported here.

Free convection boundary layers over a wedge with uniform suction or injection.

The problem of free convectional boundary layers over a wedge with uniform suction or injection is numerically studied in this paper. The ordinary equations of transformed boundary layer equation in terms of dimensionless stream function were obtained by means of the difference-differential method, and the solutions subjected to the boundary conditions can be expressed in the form of integral equations. The solutions were obtained solving the integral equations by iterative numerical quadrature. The velocity profiles and the temperature profiles were computed for various values of suction/injection parameter, Pradtl number and wedge angle.

Mixed convection wall plumes

A proper mixed convection parameter ζ = (σ Ra) 1 5 (ω Re) 1 2 , with σ = Pr (1 + Pr) and ω = Pr (1 + Pr) 1 3 , is proposed to replace the conventional parameters, Gr Re 5 2 and Re Gr 2 5 , for the analysis of mixed convection wall plumes. New coordinates, ξ = ζ (1 + ζ) and η = ( y x )[(ω Re) 1 2 + (σ Ra) 1 5 ] , and dimensionless stream function and temperature of proper scales are also introduced. The resulting non-similar equations are solved by using a very effective finite-difference scheme. The obtained solutions are uniformly valid over the entire regime of mixed convection intensity from forced convection limit to free convection limit for fluids of any Prandtl number between 0.001 and 1000.

ADI solution of free convection in a variable viscosity fluid

A numerical model of steady-state, two-dimensional free convection within an internally heated fluid of variable viscosity has been developed primarily for application to the earth's upper mantle. The dimensionless free-convection energy and stream function equations include advection, the adiabatic gradient, viscous dissipation, radiogenic heat sources, boundary heat fluxes, variable diffusivity, and variable viscosity. Both the energy equation and the fourth-order stream function equation are solved numerically by alternating direction implicit (ADI) algorithms on a special nonuniform grid first suggested by Samarskii. The Reynolds number is negligibly small, the Rayleigh number exceeds 10 6, and the Prandtl number exceeds 10 23. The numerical convergence, accuracy and reliability of the method are established by various numerical tests.

Numerical modeling of the three-dimensional flows in the ground boundary layer of a maintained axisymmetrical vortex

The three-dimensional flow in the ground boundary-layer beneath an intense Rankine vortex is studied, using a numerical technique. Equations of motion in terms of dimensionless vorticity, stream function and circulation have been derived and generalized. Upwind instead of the usual central difference scheme has been used. The method was found to be stable for the range of Reynolds numbers for the natural tornado. The results predict the known physical characteristics of the three-dimensional bound-arylayer of intense vortices, namely: The distribution of horizontal velocities in the vertical direction is oscillatory in the outer region; and the motion in the core region is upward, whilst there is weak descending motion in the outer region. The predicted pressure distribution also falls within the range observed for the natural tornado.DOI: 10.1111/j.2153-3490.1974.tb01622.x

Combined forced and free convection flow on vertical surfaces

NOMENCLATURE h dimensionless stream function ; Gr,, Grashof number, g/?A7’x3!v2 ; Gr:. modified Grashof number, gbqx4/kv2 ; 9. acceleration due to gravity ; h. local heat transfer coefficient. q/AT. k, thermal conductivity; Nu,, local Nusselt number, hxik; Prandtl number, V/K; heat flux constant ; Reynolds number, Uxiv; temperature; ambient temperature; local plate temperature : free stream velocity; streamwise velocity : transverse velocity: streamwise coordinate; transverse coordinate : T, TO: coefficient of thermal expansion ; dimensionless temperature : pseudo-similarity variable ; coefficient of thermal conductivity; kinematic viscosity ; dimensionless streamwise coordinate: stream function. criticisms made by Merkin [2] with respect to attempted series solutions about the basic free and forced convection flows as outlined by Szewczyk [3]. Essentially the error in [3] could be attributed to the choice ofa parameter expansion as opposed to the correct choice, namely a coordinate expansion. In this light it is seen that account must be taken of difficulties, which arise in the form of eigen-solutions, in the perturbed free convection solution. These difficulties are associated with the asymptotic nature of this solution at a distance far removed from the leading edge and indeed reflect the solutions’ unawareness of the precise location of the leading edge (see Stewartson [4]). ‘In [2]‘ a dimensional analysis of the governing equations yields the choice of nondimensional coordinate as

MASS TRANSFER IN A LAMINAR BOUNDARY LAYER WITH MOVING INTERFACE

Mass transfer coefficient is numerically obtained for a laminar boundary layer flow with tangentially moving interface. It is implied that the flow is similar and the dimensionless stream function is known. Dimensionless mass transfer coefficient can be approximately expressed as a function of a single parameter containing the velocity and the velocity gradient at the interface. The functional relation is analogous with that by Beek and Bakker and applicable to all similar boundary layer flows. It is also numerically shown that the general correlation gives a good estimate even in the case of small Schmidt number for uniform flow over a flat interface.

A Theoretical and Experimental Study of Confined Vortex Flow

The interaction of a vortex with a stationary surface was studied both theoretically and experimentally. The flow field examined was that produced by radially inward flow through a pair of concentric rotating porous cylinders that were perpendicular to, and in contact with, a stationary flat plane. The complete Navier-Stokes equations were solved over a range of tangential Reynolds numbers from 0–300 and a range of radial Reynolds numbers from 0 to −13, the minus sign indicating radially inward flow. In order to facilitate the solution, the original equations were recast in terms of a dimensionless stream function, vorticity, and third variable related to the tangential velocity. The general validity of the numerical technique was demonstrated by the agreement between the theoretical and experimental results. Examination of the numerical results over a wide range of parameters showed that the entire flow field is very sensitive to the amount of radial flow, especially at the transition from zero radial flow to some finite value.