Mixed convection boundary layer flow of a viscoelastic fluid over a horizontal circular cylinder

The steady mixed convection boundary layer flow of a viscoelastic fluid over a horizontal circular cylinder in a stream flowing vertically upwards is numerically studied for both cases of heated and cooled cylinders. The governing partial differential equations are transformed into dimensionless forms using an appropriate transformation and then solved numerically using the Keller-box method. The comparison between the solutions obtained and those for a Newtonian fluid is found to be very good. Effects of the mixed convection and elasticity parameters on the skin friction and heat transfer coefficients for a fluid having the Prandtl number equal to one are also discussed. It is found that for some values of the viscoelastic parameter and some negative values of the mixed convection parameter (opposing flow) the boundary layer separates from the cylinder. Heating the cylinder delays separation and can, if the cylinder is warm enough, suppress the separation completely. Similar to the case of a Newtonian fluid, cooling the cylinder brings the separation point nearer to the lower stagnation point. However, for a sufficiently cold cylinder there will not be a boundary layer.     

MHD mixed convection for viscoelastic fluid past a porous wedge

A magnetic hydrodynamic (MHD) mixed convective heat transfer problem of a second-grade viscoelastic fluid past a wedge with porous suction or injection has been studied. Governing equations include continuity equation, momentum equation and energy equation of the fluid. It has been analyzed by a combination of a series expansion method, the similarity transformation and a second-order accurate finite-difference method. Solutions of wedge flow on the wedge surface have been obtained by a generalized Falkner-Skan flow derivation. Some important parameters have been discussed by this study, which include the Prandtl number (Pr), the elastic number (E), the free convection parameter (G) and the magnetic parameter (M), the porous suction and injection parameter (C) and the wedge shape factor (β). Results indicated that elastic effect (E) in the flow could increase the local heat transfer coefficient and enhance the heat transfer of a wedge. In addition, similar to the results from Newtonian fluid flow and conduction analysis of a wedge, better heat transfer is obtained with a larger G and Pr.     

Mixed convection boundary layer flow along vertical moving thin needles with variable heat flux

The problem of steady laminar mixed convection boundary layer flow of an incompressible viscous fluid along vertical moving thin needles with variable heat flux for both assisting and opposing flow cases is theoretically considered in this paper. The governing boundary layer equations are first transformed into non-dimensional forms. The curvature effects are incorporated into the analysis whereas the pressure variation in the axial direction has been neglected. These equations are then transformed into similarity equations using the similarity variables, which are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The solutions are obtained for a blunt-nosed needle (m = 0). Numerical calculations are carried out for various values of the dimensionless parameters of the problem, which include the mixed convection parameter λ, the Prandtl number Pr and the parameter a representing the needle size. It is shown from the numerical results that the skin friction coefficient, the surface (wall) temperature and the velocity and temperature profiles are significantly influenced by these parameters. The results are presented in graphical form and are discussed in detail.     

The onset of double diffusive convection in a viscoelastic fluid layer

The onset of double diffusive convection in a viscoelastic fluid layer is studied using a linear and a weak nonlinear stability analyses. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. There is a competition between the processes of thermal diffusion, solute diffusion and viscoelasticity that causes the convection to set in through oscillatory mode rather than stationary. The effect of Deborah number, retardation parameter, solutal Rayleigh number, Prandtl number, Lewis number on the stability of the system is investigated. It is shown that the critical frequency increases with Deborah number and solutal Rayleigh number while it decreases with retardation parameter and Lewis number. The nonlinear theory based on the truncated representation of Fourier series method is used to find the heat and mass transfers. The transient behaviour of the Nusselt number and Sherwood number is investigated by solving the finite amplitude equations using Runge-Kutta method. The effect of viscoelastic parameters on heat and mass transfer is brought out.     

A note on the free convection boundary layer on a horizontal circular cylinder with constant heat flux

The equations governing the free convection boundary-layer flow on a horizontal circular cylinder on which there is a prescribed surface heat flux are solved using a finite-difference scheme. This numerical solution is then used to compare the accuracy of two proposed series expansions, a Blasius expansion and a Görtler-type expansion. It is shown that the former method is better at estimating temperature profiles while the latter is better at estimating velocity profiles.

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Bemerkung über die freie Konvektionsgrenzschicht an einem horizontalen Kreiszylinder mit gleichförmigem Wärmestrom

Zusammenfassung Die Grenzschichtgleichungen für die freie Konvektion an einem horizontalen Kreiszylinder mit gleichförmigem Wärmestrom durch die Oberfläche wird mit Hilfe eines finiten Differenzverfahrens gelöst. Die numerisch ermittelten Ergebnisse werden nachher für den Vergleich der Genauigkeit von zwei Reihendarstellungen der Lösung der Grenzschichtsgleichungen benützt. Diese Reihen sind vom Blasiusbzw. Görtier-Typ. Es wird bemerkt, daß die Reihendarstellung von Blasius die Temperaturprofile besser beschreibt, während die Reihenentwicklung der Görtlerschen Art für die Geschwindigkeitsprofile eine gute Übereinstimmung mit der exakten Lösung zeigt.

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Nomenclature a radius of the cylinder - g acceleration of gravity - P _r Prandtl number - Q prescribed (constant) heat flux - T temperature of the fluid - t ₀ temperature of the ambient fluid - u velocity in thex-direction - v velocity in they-direction - x co-ordinate measuring distance round the cylinder - y co-ordinate measuring distance normal to the cylinder - G _r Grashof number=g Q a ⁴/v² - coefficient of thermal expansion - x thermal conductivity - v kinematic viscosity - _w skin friction     

Forced convection heat transfer of Giesekus viscoelastic fluid in pipes and channels

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.     

Local nonsimilarity method for the two-phase boundary layer in mixed convection laminar film condensation

The two-phase boundary layer in laminar film condensation was solved by Koh for the free convection regime and forced convection regime using the similarity method. But the problem on mixed convection does not admit similarity solutions. The current work develops a local nonsimilarity method for the full spectrum of mixed convection, with a generic boundary layer formulation reduced to two specific cases mathematically identical to Koh’s formulations on the two limiting cases for either free or forced convection. Other solution methods for mixed convection in the literature are compared and critically evaluated to ensure a high level of accuracy in the current method. The current solution is used to extend Fujii’s correlation for mixed convection to the region where the energy convection effect is significant but has been hitherto neglected. The modified Fujii correlation provides a practical engineering tool for evaluating laminar film condensation with a mixed convection boundary layer.     

Oscillatory convection in a viscoelastic fluid through a porous layer heated from below

The stability of a viscoelastic fluid in a densely packed horizontal porous layer heated from below is considered using an Oldroyd model. Critical Rayleigh number, wave number, and frequency for overstability are determined by applying the linear stability theory. It is shown that the critical Rayleigh number is invariant under all relevant boundary combinations. Also, it is found that the effect of elasticity of the fluid is to destabilize the system and that of porosity is to stabilize the same. The limiting case of very high Prandtl number and the degenerate case corresponding to the Maxwell model are analyzed in some detail.     

Natural convection of a viscoelastic fluid

The influence of elasticity and shear thinning viscosity on the temperature distribution and heat transfer in natural thermal convection is discussed. The numerical investigations are based on a four-parameter Oldroyd constitutive equation, which represents the typical fluid response of dilute solutions and melts. It was found that especially the second normal-stress difference affects the heat transfer mechanism.     

Forced convection heat transfer at the boundary layer of a packed bed

Numerical investigations of the nature of the fluid flow pattern and heat transfer at the boundary layer of a packed bed are reported. A volume averaged Navier-Stokes equation is used to predict the fluid flow and a volume averaged heat balance equation the heat transfer. A variable porosity in the packing is assumed in the region near the wall. Simulations are performed using a modified penalty Galerkin finite element method. The case of fully developed hydrodynamic flow and developing thermal flow is studied. The Nusselt number is found to depend on the Reynolds number, Graetz number and ratio of thermal conductivity of the solid and fluid phases. Comparison is made to some experimental literature values.Nomenclature A constant - [A] Navier-Stokes type matrix - B constant - [B] divergence matrix - C _p constant pressure heat capacity - d characteristic length - D _p particle diameter - D _t tube diameter - {F} solicitation vector - Gz Graetz number, z D _t ^–1 Pr _f Re _p - k permeability term - k _f Thermal conductivity of the fluid phase - k _s Thermal conductivity of the solid phase - [K] coefficient matrix for temperature equation - n normal vector - P pressure - Pr _f Prandtl number for the fluid _f C _p k _f ^-1 - r radial coordinate - R _t tube radius - R residual - R _m residual - Re _p Reynolds number for particle, - t tortuosity factor - T temperature - interstitial velocity - z axial coordinate - effective thermal conductivity - penalty parameter - boundary of solution domain - porosity - viscosity - density - test function - solution domain - test function     

Two-dimensional convection in a horizontal fluid layer with spatially periodic boundary temperatures

Two-dimensional thermal convection in a fluid layer confined between two horizontal rigid walls kept at spatially periodic temperatures is investigated by direct numerical simulations. With increasing the Rayleigh number, convection evolves from a steady state to a temporally chaotic flow. It is observed that the transition to the chaos occurs via quasi-periodic states with two or three basic frequencies or via sequences of period-doubling bifurcations, according to the boundary temperature distributions.     

Free convection boundary layer flow of a micropolar fluid past slender cones

Using the theory of micropolar fluids developed by Eringen, the transverse curvature effects on axisymmetric free convection boundary layer flow of a micropolar fluid past slender vertical cones are investigated. The case of constant surface heat flux is considered in this paper. Using perturbation techniques, the governing equations for momentum, angular momentum and energy have been solved numerically. Graphical representations for the velocity, angular velocity and thermal functions are presented for various physical and fluid property parameters.     

Thermal convection of a viscoelastic fluid in an open-top porous layer heated from below

Buoyancy-driven convection of a viscoelastic fluid saturated in an open-top porous square box is studied based on a modified Darcy's law. The results are compared with those for a Newtonian fluid under the same boundary conditions and those for the viscoelastic fluid under a closed-top boundary. In particular, the critical Darcy–Rayleigh number Ra for onset of convection is determined first by using the linear stability theory. Then the effects of the relaxation time and the retardation time of the viscoelastic fluid on the heat transfer rate and the flow pattern are investigated numerically. The results reveal some interesting properties of thermal convection for the viscoelastic fluid. The relaxation time makes the fluid easier to destabilize while the retardation time tends to stabilize the fluid motion in the porous medium, and larger heat transfer rate can be achieved with larger value of the relaxation time and decreased retardation time. Furthermore, larger relaxation time facilitates earlier bifurcation of the flow pattern as Ra increases, but bifurcation can be postponed with increased retardation time. For larger ratio of relaxation time over retardation time, the flow pattern is more complicated and the frequency of flow oscillation also increases. Finally, large ratio of relaxation time over retardation time can make the open-top boundary impermeable due to the viscoelastic effect on the fluid.     

Non-Darcy mixed convection from a vertical plate in saturated porous media-variable surface heat flux

Nonsimilarity solutions for non-Darcy mixed convection from a vertical impermeable surface embedded in a saturated porous medium are presented for variable surface heat flux (VHF) of the power-law form. The entire mixed convection region is divided into two regimes. One region covers the forced convection dominated regime and the other one covers the natural convection dominated regime. The governing equations are first transformed into a dimensionless form by the nonsimilar transformation and then solved by a finite-difference scheme. Computations are based on Keller Box method and a tolerance of iteration of 10⁻⁵ as a criterion for convergence. Three physical aspects are introduced. One measures the strength of mixed convection where the dimensionless parameter Ra^* _x /Pe^3/2 _x characterizes the effect of buoyancy forces on the forced convection; while the parameter Pe _x /Ra^*2/3 _x characterizes the effect of forced flow on the natural convection. The second aspect represents the effect of the inertial resistance where the parameter K′U _∞/ν is found to characterize the effect of inertial force in the forced convection dominated regime, while the parameter (K′U _∞/ν)(Ra^*2/3 _x /Pe _x ) characterizes the effect of inertial force in the natural convection dominated regime. The third aspect is the effect of the heating condition at the wall on the mixed convection, which is presented by m, the power index of the power-law form heating condition. Numerical results for both heating conditions are carried out. Distributions of dimensionless temperature and velocity profiles for both Darcy and non-Darcy models are presented. Received on 26 May 1997     

Non-uniform slot suction/injection into mixed convection boundary layer flow over vertical cone

The aim of this work is to study the effect of non-uniform single and double slot suction/injection into a steady mixed convection boundary layer flow over a vertical cone, while the axis of the cone is inline with the flow. The governing boundary layer equations are transformed into a non-dimensional form by a group of non-similar trans- formations. The resulting coupled non-linear partial differential equations are solved nu- merically by employing the quasi-linearization technique and an implicit finite-difference scheme. Numerical computations are performed for different values of the dimensionless parameters to display the velocity and temperature profiles graphically. Also, numerical results are presented for the skin friction and heat transfer coefficients. Results indicate that the skin friction and heat transfer coefficients increase with non-uniform slot suction, but the effect of non-uniform slot injection is just opposite.