Heat transfer and Helmholtz-Smoluchowski velocity in Bingham fluid flow

A mathematical study is developed for the electro-osmotic flow of a nonNewtonian fluid in a wavy microchannel in which a Bingham viscoplastic fluid model is considered. For electric potential distributions, a Poisson-Boltzmann equation is employed in the presence of an electrical double layer(EDL). The analytical solutions of dimensionless boundary value problems are obtained with the Debye-Huckel theory, the lubrication theory, and the long wavelength approximations. The effects of the Debyelen...     

Influence of a wavy boundary on turbulence.

Measurements of turbulence with laser Doppler velocimetry (LDV) are compared for turbulent flows over a flat surface and a surface with sinusoidal waves of small wavelength. The wavy boundary was highly rough in that the flow separated. The Reynolds number based on the half-height of the channel and the bulk velocity was 46,000. The wavelength was 5 mm and the height to wavelength ratio was 0.1. The root-mean-squares of the velocity fluctuations are approximately equal if normalized with the friction velocity. This can be explained as a consequence of the approximate equality of the correlation coefficients of the Reynolds shear stress. Calculations with a direct numerical simulation (DNS) are used to show that the fluid interacts with the wall in quite different ways for flat and wavy surfaces. They show similarity in that large quadrant 2 events in the outer flow, for both cases, are associated with plumes that emerge from the wall region and extend over large distances. Measurements of skewness of the streamwise and wall-normal velocity fluctuations and quadrant analyses of the Reynolds shear stresses are qualitatively similar for flat and wavy surfaces. However, the skewness magnitudes and the ratio of the quadrant 2 to quadrant 4 contributions are larger for the wavy surface. Thus, there is evidence that turbulent structures are universal in the outer flow and for quantitative differences in the statistics that reflect differences in the way in which the fluid interacts with the wall.     

Free-convection boundary-layer flow of a non-Newtonian fluid along a vertical wavy surface

A theoretical analysis of laminar free-convection flow over a vertical isothermal wavy surface in a non-Nevvtonian power-law fluid is considered. The governing equations are first cast into a nondimensional form by using suitable boundary-layer variables that substract out the effect of the wavy surface from the boundary conditions. The boundary-layer equations are then solved numerically by a very efficient implicit finite-difference method known as the Keller-Box method. A sinusoidal surface is used to elucidate the effects of the power-law index, amplitude wavelength, and Prandtl number on the velocity and temperature fields, as well as on the local Nusselt number. It is shown that the local Nusselt number varies periodically along the wavy surface. The wave-length of the local Nusselt number variation is half that of the wavy surface, irrespective of whether the fluid is a Newtonian fluid or a non-Newtonian fluid. Comparisons with earlier works are also made, and the agreement is found to be very good.     

TRPIV investigation of space-time correlation in turbulent flows over flat and wavy walls

The present experimental work is devoted to in- vestigate a new space-time correlation model for the turbulent boundary layer over a flat and a wavy walls. A turbulent boundary layer flow at Reo = 2460 is measured by tomographic time-resolved particle image velocimetry （Tomo-TRPIV）. The space-time correlations of instantaneous streamwise fluctuation velocity are calculated at 3 different wall-normal locations in logarithmic layer. It is found that the scales of coherent structure increase with moving far away from the wall. The growth of scales is a manifestation of the growth of prevalent coherent structures in the turbulent boundary layer like hairpin vortex or hairpin packets when they lift up. The resulting contours of the space-time correlation exhibit elliptic-like shapes rather than straight lines. It is suggested that, instead of Taylor hypothesis, the elliptic model of the space-time correlation is valid for the wallbounded turbulent flow over either a flat wall or a wavy wall. The elliptic iso-correlation curves have a uniform preferred orientation whose slope is determined by the convection velocity. The convection velocity derived from the space-time correlation represents the velocity at which the large-scale eddies carry small-scale eddies. The sweep velocity rep- resents the distortions of the small-scale eddies and is intimately associated with the fluctuation velocity in the logarithmic layer of turbulent boundary layers. The nondimensionalized correlation curves confirm that the elliptic model is more proper for approximating the space-time correlation than Taylor hypothesis, because the latter can not embody the small-scale motions which have non-negligible distortions. A second flow over a wavy wall is also recorded using TRPIV. Due to the combined effect of shear layers and the adverse pressure gradient, the space-time correlation does not show an elliptic-like shape at some specific heights over the wavy wall, but in the outer region of the wavy wallbounded flow, the elliptic model remai     

Advective flow in a rotating liquid film

This paper presents a new exact solution of the Navier–Stokes equations in the Boussinesq approximation that describes thermocapillary advective flow in a slowly rotating horizontal layer of incompressible fluid with free boundaries. Such flow occurs in the case of linear temperature distribution over horizontal coordinates or in the case of heat flux distribution at the layer boundaries. The influence of the Taylor, Marangoni, Grashof, and Biot numbers on the flow and temperature velocity profiles is studied.     

Heat transfer in stagnation-point flow towards a stretching sheet

 Steady two-dimensional stagnation-point flow of an incompressible viscous fluid over a flat deformable sheet is investigated when the sheet is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. It is shown that for a fluid of small kinematic viscosity, a boundary layer is formed when the stretching velocity is less than the free stream velocity and an inverted boundary layer is formed when the stretching velocity exceeds the free stream velocity. Temperature distribution in the boundary layer is found when the surface is held at constant temperature and surface heat flux is determined. Received on 12 July 2000 / Published online: 29 November 2001     

Stagnation-point flow of a viscoelastic fluid towards a stretching surface

An analysis is made of the steady two-dimensional stagnation-point flow of an incompressible viscoelastic fluid over a flat deformable surface when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. It is shown that for a viscoelastic fluid of short memory (obeying Walters’ B′ model), a boundary layer is formed when the stretching velocity of the surface is less than the inviscid free-stream velocity and velocity at a point increases with increase in the elasticity of the fluid. On the other hand, an inverted boundary layer is formed when the surface stretching velocity exceeds the velocity of the free stream and the velocity decreases with increase in the elasticity of the fluid. A novel result of the analysis is that the flow near the stretching surface is that corresponding to an inviscid stagnation-point flow when the surface stretching velocity is equal to the velocity of the free stream. Temperature distribution in the boundary layer is found when the surface is held at constant temperature and surface heat flux is determined. It is found that temperature at a point decreases with increase in the elasticity of the fluid.     

Effects of variable fluid viscosity on flow past a heated stretching sheet embedded in a porous medium in presence of heat source/sink

The boundary layer flow and heat transfer of a fluid through a porous medium towards a stretching sheet in presence of heat generation or absorption is considered in this analysis. Fluid viscosity is assumed to vary as a linear function of temperature. The symmetry groups admitted by the corresponding boundary value problem are obtained by using a special form of Lie group transformations viz. scaling group of transformations. These transformations are used to convert the partial differential equations corresponding to the momentum and the energy equations into highly non-linear ordinary differential equations. Numerical solutions of these equations are obtained by shooting method. It is found that the horizontal velocity decreases with increasing temperature-dependent fluid viscosity parameter up to the crossing-over point but increases after that point and the temperature decreases in this case. With the increase of permeability parameter of the porous medium the fluid velocity decreases but the temperature increases at a particular point of the sheet. Effects of Prandtl number on the velocity boundary layer and on the thermal boundary layer are studied and plotted.     

Effects of variable fluid properties on natural convection of MHD fluid from a heated vertical wavy surface

In the present paper, the influence of temperature-dependent fluid properties, density, viscosity and thermal conductivity on MHD natural convection flow from a heated vertical wavy surface is studied. It is assumed that, the fluid density and the thermal conductivity vary as exponential and linear functions of temperature, respectively. However, the fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature. The model analysis used here is more relevant to liquid flow. Using the appropriate variables, the wavy surface are transformed into a flat one. The transformed boundary layer equations are solved numerically, using implicit-Chebyshev pseudospectral method, for several sets of values of the physical parameters, namely, the temperature dependent fluid properties parameters, the magnetic parameter, the amplitude-wavelength ratio parameter, and the Prandtl number. The numerical values obtained for the velocity, temperature, shearing stress, and the Nusselt number are presented through graphs and tables for several sets of values of the parameters. The effects of the physical parameters on the flow and heat transfer characteristics are discussed. The results were compared with numerical solutions of previous works. The present results are found to be in good agreement.     

Momentum and heat transfer in the magnetohydrodynamic stagnation-point flow of a viscoelastic fluid toward a stretching surface

An analysis is made of the steady two-dimensional stagnation-point flow of an incompressible viscoelastic fluid over a flat deformable surface when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. It is shown that for a viscoelastic conducting fluid of short memory (obeying Walters’ Bʹ model), a boundary layer is formed when the stretching velocity of the surface is less than the inviscid free-stream velocity and velocity at a point increases with increase in the Hartmann number. On the other hand an inverted boundary layer is formed when the surface stretching velocity exceeds the velocity of the free stream and the velocity decreases with increase in the Hartmann number. A novel result of the analysis is that the flow near the stretching surface is that corresponding to an inviscid stagnation-point flow when the surface stretching velocity is equal to the velocity of the free stream. Temperature distribution in the boundary layer is found when the surface is held at constant temperature and surface heat flux is determined. It is found that in the absence of viscous and Ohmic dissipation and strain energy in the flow, temperature at a point decreases with increase in the Hartmann number.     

Structure of the temperature field in a flow over heated waves

Liquid crystal thermometry (LCT) was used to quantify temperature fields in a flow over resistively heated waves and assess the effect of the large-scale longitudinal structures that were previously obtained in the velocity field for an isothermal flow (A. Günther and P. Rudolf von Rohr, submitted article, 2002). The wavelength 6 was 10 times larger than the amplitude, and the considered Reynolds numbers were 725 and 3300, defined with the bulk velocity and the half-channel height. A constant heat flux was imposed at the wavy bottom wall. For the first time, LCT was used to determine the fluid temperature in a wall-bounded flow with heat transfer. The dominant spanwise scale obtained from a proper orthogonal decomposition (POD) of the fluid temperature field above an uphill location of the wavy wall was 1.56. It agrees well with the one previously obtained for a decomposition of the streamwise velocity.     

Turbulent transport dissimilarity with modulated turbulence structure in channel flow of viscoelastic fluid

This experimental study investigated the turbulent transport dissimilarity with a modulated turbulence structure in a channel flow of a viscoelastic fluid using simultaneous particle image velocimetry and planar laser-induced fluorescence measurements. An instantaneous dye concentration field with fluctuating velocity vectors showed that mass was transferred by hierarchically large-scale wavy motions with inclination. A co-spectral analysis showed that the spatial phase modulation of the streamwise velocity and dye concentration fluctuations for the wall-normal velocity fluctuation corresponded to the relaxation time. The occurrence of intense dye concentration fluctuation and small streamwise velocity fluctuation in a thin boundary layer caused dissimilar turbulent transport because of the non-zero negative correlation of the streamwise velocity and dye concentration fluctuations for the wall-normal velocity fluctuation only on large scales. This explains the turbulent transport dissimilarity which leads to the zero averaged Reynolds shear stress and non-zero wall-normal turbulent mass flux.     

Self-similar solutions of non-Newtonian fluid boundary layer in MHD

The self-similar solutions of the boundary layer for a non-Newtonian fluid in MHD were considered in [1, 2] for a power-law velocity distribution along the outer edge of the layer and constant electrical conductivity through the entire flow. However, the MHD flows of many conducting media, which are solutions or molten metals, cannot be described by the MHD equations for non-Newtonian fluids.The self-similar solutions of the boundary layer for a non-Newtonian fluid without account for interaction with the electromagnetic field were studied in [3].In the following we present the self-similar solutions for the boundary layer of pseudoplastic and dilatant fluids with account for the interaction with an electromagnetic field for the case of a power-law velocity distribution along the outer edge of the layer, when the conductivity of the fluid is constant throughout the flow and the magnetic Reynolds number is small.Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 2, No. 6, pp. 77–82, 1967The author wishes to thank S. V. Fal'kovich for his interest in this study.     

Exact solutions of MHD second Stokes flow of generalized Burgers fluid

This work concerns with the exact solutions of magnetohydrodynamic (MHD) flow of generalized Burgers fluid describing the second Stokes problem. The modified Darcy law is taken into account. The related velocity distribution and shear stress are expressed as a combination of steady-state and transient solutions computed by means of integral transformations. The effects of various parameters on the flow field are investigated. The MHD flow results in reduction of velocity distribution and associated thickness of the boundary layer.     

Local similarity solution for the flow of a “second-grade” viscoelastic fluid above a moving plate

The boundary layer flow of a viscoelastic fluid of the second-grade type over a rigid continuous plate moving through an otherwise quiescent fluid with constant velocity U is studied. Assuming the flow to be laminar and two-dimensional, local similarity solution is found with fluid's elasticity and plate's withdrawal speed as the main variables. Results are presented for velocity profiles, boundary layer thickness, wall skin friction coefficient and fluid entrainment in terms of the local Deborah number. A marked formation of boundary layer is predicted, even at low Reynolds numbers, provided the Deborah number is sufficiently large. The boundary layer thickness and the wall skin friction coefficient are found to scale with fluid's elasticity—both decreasing the higher the fluid's elasticity. The amount of fluid entrained is also predicted to decrease whenever a fluid exhibits elastic behavior.     

Unsteady hydromagnetic Couette flow through a porous medium in a rotating system

This paper investigates the unsteady hydromagnetic Couette fluid flow through a porous medium between two infinite horizontal plates induced by the non-torsional oscillations of one of the plates in a rotating system using boundary layer approximation. The fluid is assumed to be Newtonian and incompressible. Laplace transform technique is adopted to obtain a unified solution of the velocity fields. Such a flow model is of great interest, not only for its theoretical significance, but also for its wide applications to geophysics and engineering. Analytical expressions for the steady state velocity and shear stress on the plates are obtained, and the case of single oscillating plate is also discussed. The influence of pertinent parameters on the flow is delineated, and appropriate conclusions are drawn.     

Note on characteristics of homogeneous-heterogeneous reaction in flow of Jeffrey fluid

This letter describes the characteristics of homogeneous-heterogeneous reaction in the boundary layer flow of a Jeffrey fluid due to an impermeable horizontal stretching sheet. An analysis is carried out through the similar values of reactant and auto catalyst diffusion coefficients. Heat released by the reaction is not accounted. The exact solution for the flow of the Jeffrey fluid is constructed. The series solution for the concentration equation is derived. The velocity and concentration fields reflecting the impact of interesting parameters are plotted and examined.     

Stability of Flow with Viscous Dissipation in a Horizontal Porous Layer with an Open Boundary Having a Prescribed Temperature Gradient

A buoyancy-induced stationary flow with viscous dissipation in a horizontal porous layer is investigated. The lower boundary surface is impermeable and subject to a uniform heat flux. The upper open boundary has a prescribed, linearly varying, temperature distribution. The buoyancy-induced basic velocity profile is parallel and non-uniform. The linear stability of this basic solution is analysed numerically by solving the disturbance equations for oblique rolls arbitrarily oriented with respect to the basic velocity field. The onset conditions of thermal instability are governed by the Rayleigh number associated with the prescribed wall heat flux at the lower boundary, by the horizontal Rayleigh number associated with the imposed temperature gradient on the upper open boundary, and by the Gebhart number associated with the effect of viscous dissipation. The critical value of the Rayleigh number for the onset of the thermal instability is evaluated as a function of the horizontal Rayleigh number and of the Gebhart number. It is shown that the longitudinal rolls, having axis parallel to the basic velocity, are the most unstable in all the cases examined. Moreover, the imposed horizontal temperature gradient tends to stabilise the basic flow, while the viscous dissipation turns out to have a destabilising effect.     

Turbulent breaking of overturning gravity waves below a critical level

The interaction of an internal gravity wave with its evolving critical layer and the subsequent generation of turbulence by overturning waves are studied by three-dimensional numerical simulations. The simulation describes the flow of a stably stratified Boussinesq fluid between a bottom wavy surface and a top flat surface, both without friction and adiabatic. The amplitude of the surface wave amounts to about 0.03 of the layer depth. The horizontal flow velocity is negative near the lower surface, positive near the top surface with uniform shear and zero mean value. The bulk Richardson number is one. The flow over the wavy surface induces a standing gravity wave causing a critical layer at mid altitude. After a successful comparison of a two-dimensional version of the model with experimental observations (Thorpe [21]), results obtained with two different models of viscosity are discussed: a direct numerical simulation (DNS) with constant viscosity and a large-eddy simulation (LES) where the subgrid scales are modelled by a stability-dependent first-order closure. Both simulations are similar in the build-up of a primary overturning roll and show the expected early stage of the interaction between wave and critical level. Afterwards, the flows become nonlinear and evolve differently in both cases: the flow structure in the DNS consists of coherent smaller-scale secondary rolls with increasing vertical depth. On the other hand, in the LES the convectively unstable primary roll collapses into three-dimensional turbulence. The results show that convectively overturning regions are always formed but the details of breaking and the resulting structure of the mixed layer depend on the effective Reynolds number of the flow. With sufficient viscous damping, three-dimensional turbulent convective instabilities are more easily suppressed than two-dimensional laminar overturning.     

Fluctuating flow of a second-order fluid near a stagnation point

Summary The boundary layer equations for axisymmetric flow of an incompressible second-order fluid have been deduced. The flow of such a fluid near a stagnation point when the main stream outside the boundary layer fluctuates in magnitude but not in direction has been discussed. The velocity distribution is found for various values of the steady mean in two limiting cases of small and large values of the frequency of the oscillation of the main stream. The frequency for which two approximate solutions overlap has been calculated in each case.