Hall effects on MHD flow in a rotating system with heat transfer characteristics

Closed-form solutions are derived for the steady magnetohydrodynamic (MHD) viscous flow in a parallel plate channel system with perfectly conducting walls in a rotating frame of reference, in the presence of Hall currents, heat transfer and a transverse uniform magnetic field. A mathematical analysis is described to evaluate the velocity, induced magnetic field and mass flow rate distributions, for a wide range of the governing parameters. Asymptotic behavior of the solution is analyzed for large M ² (Hartmann number squared) and K ² (rotation parameter). The heat transfer aspect is considered also with Joule and viscous heating effects present. Boundary layers arise close to the channel walls for large K ², i.e. strong rotation of the channel. For slowly rotating systems (small K ²), Hall current parameter (m) reduces primary mass flow rate (Q _x /R ρ v). Heat transfer rate at the upper plate (d θ/d η) _η=1 decreases, while at the lower plate (d θ/d η) _η=−1 increases, with increase in either K ² or m. For constant values of the rotation parameter, K ², heat transfer rate at both plates exhibits an oscillatory pattern with an increase in Hall current parameter, m. The response of the primary and secondary velocity components and also the primary and secondary induced magnetic field components to the control parameters is also studied graphically. Applications of the study arise in rotating MHD induction machine energy generators, planetary and solar plasma fluid dynamics systems, magnetic field control of materials processing systems, hybrid magnetic propulsion systems for space travel etc.     

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     

Nonlinear stability theory of channel flow with heat transfer – an asymptotic approach

A fully developed laminar Poiseuille flow subject to constant heat flux across the wall is analysed with respect to its stability behavior by applying a weakly nonlinear stability theory. It is based on an expansion of the disturbance control equations with respect to a perturbation parameter ε. This parameter is the small initial amplitude of the fundamental wave. This fundamental wave which is the solution of the linear (Orr-Sommerfeld) first order equation triggers all higher order effects with respect to ε. Heat transfer is accounted for asymptotically through an expansion with respect to a small heat transfer parameter ε _T . Both perturbation parameters, ε and ε _T , are linked by the assumption ε _T =O(ε²) by which a certain distinguished limit is assumed. The results for a fluid with temperature dependent viscosity show that heat transfer effects in the nonlinear range continue to act in the same way as in the initial linear range. Received on 11 August 1997     

Exact analytic solutions for free convection boundary layers on a heated vertical plate with lateral mass flux embedded in a saturated porous medium

 The problem of the self-similar boundary flow of a “Darcy-Boussinesq fluid” on a vertical plate with temperature distribution T _w(x) = T _∞+A·x ^λ and lateral mass flux v _w(x) = a·x ^(λ−1)/2, embedded in a saturated porous medium is revisited. For the parameter values λ = 1,−1/3 and −1/2 exact analytic solutions are written down and the characteristics of the corresponding boundary layers are discussed as functions of the suction/ injection parameter in detail. The results are compared with the numerical findings of previous authors. Received on 8 March 1999     

Oscillatory Couette flow in the presence of an inclined magnetic field

Oscillatory MHD Couette flow of electrically conducting fluid between two parallel plates in a rotating system in the presence of an inclined magnetic field is considered when the upper plate is held at rest and the lower plate oscillates non-torsionally . An exact solution of the governing equations has been obtained by using Laplace transform technique. Asymptotic behavior of the solution is analyzed for M ² ≪1, K ² ≪1 and ω ≪1 and for large M ², K ² and ω. Numerical results of velocities are depicted graphically and the frictional shearing stresses are presented in tables. It is found that a thin boundary layer is formed near the lower plate, for large values of rotation parameter K ², Hartman number M ² and frequency parameter ω. The thickness of this boundary layer increases with increase in inclination of the magnetic field with the axis of rotation.     

Fatigue gage utilizing slip-initiation phenomenon in electroplated copper foil

Fundamental experiments are carried out to examine the parameter that dominates the slip-band initiation in electroplated copper foil under the condition where the mean stress as well as the stress amplitude varies. In the case of constant-amplitude stressing, the relation between the critical stress for the slip-band initiation σ _p and the number of cyclesN is well represented by σ _p ^α N=C. In other words, the slip bands appear when the total hysteresis energy applied to the copper foil attains a critical value. In the case of variable stresses, the range-pair mainly dominates the occurrence of the slip bands, and Miner's linear cumulative damage rule holds for the accumulation of the fatigue damage for the slip-band initiation. Accordingly, the parameter (Σσ _i ^α n _i/Σn _i)^1/α is equivalent to the critical stress σ _p under constant amplitude stressing, where σ _i andn _i are the stress amplitude and the number of cycles counted by the range-pair method, respectively, and α is the exponent of the σ _p -N relation. Based on these results, the applicability of the copper foil to the fatigue gage that accumulates and indicates a load experience is discussed.     

Near wake properties of axisymmetric bluff body flows

Experiments have been made to measure some of the near wake properties of axisymmetric bluff body flows with fixed points of separation, including the detention or residence time of fluid borne scalar entities, base pressure coefficient, wake bubble length parameter and shape parameter. Measurements were made in smooth and turbulent air flow for Reynolds number in the range 2×10³<Re<4×10⁴. The results for a given bluff body were found to be uniquely controlled by a free-stream turbulence parameter. The data for all the shapes of bluff body in the class under consideration were found to collapse into unique inter-relationships by the introduction of the face pressure coefficient as a quantitative measure of “bluffness”. This paper was originally presented at the 14th International Congress of Theoretical and Applied Mechanics, Delft (August–September, 1976).     

Mixed convection flow of second grade fluid along a vertical stretching flat surface with variable surface temperature

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.     

The second-law analysis of mixed convection in rectangular ducts

 The rate of entropy generation, S˙ ^′ _G[W/mK], is examined both theoretically and numerically for forced and mixed convection in a rectangular duct heated at the bottom. Under fully-developed flow conditions S˙ ^′ _G is expressed in terms of relevant non-dimensional hydrodynamic and thermal parameters. Numerically, it is demonstrated that S˙ ^′ _G is a single, effective parameter to examine both thermal and hydrodynamic fields and their variations. Received on 22 November 1999     

High-order asymptotic analysis for the crack in nonlinear material

Accurate high-order asymptotic analyses were carried out for Mode II plane strain crack in power hardening materials. The second-order crack tip fields have been obtained. It is found that the amplitude coefficientk ₂ of the second term of the asymptotic field is correlated to the first order field as the hardening exponentn<n ^* (n ^*≈5), but asn≥n ^*,k ₂ turns to become an independent parameter. Our results also indicated that, the second term of the asymptotic field has little influence on the near-crack-tip field and can be neglected whenn<n ^*. In fact,k ₂ directly reflects the effects of triaxiality near the crack tip, the crack geometry and the loading mode, so that besidesJ-integral it can be used as another characteristic parameter in the two-parameter criterion. The project supported by National Natural Science Foundation of China     

Natural convection from a vertical cylinder in a thermally stratified porous medium

The problem of non-Darcy natural convection adjacent to a vertical cylinder embedded in a thermally stratified porous medium has been analyzed. Nonsimilarity solutions are obtained for the case that the ambient temperature increases linearly with height of the cylinder. A generalized flow model was used in the present study to include the effects of the macroscopic viscous term and the microscopic inertial force. Also, the thermal dispersion effect is considered in the energy equation. Thus, the main aim of this work is to examine the effects of thermal stratification and non-Darcy flow phenomena on the free convection flow and heat transfer characteristics. It was found that the present problem depends on six parameters, namely, the local thermal stratification parameter ξ, the boundary effect parameter Bp, the modified Grashof number Gr^*, wall temperature exponent m, the curvature parameter ω, and the modified Rayleigh number based on pore diameter Ra _d . The impacts of these governing parameters on the local heat transfer parameter are discussed in great detail. Also, representative velocity and temperature profiles are presented at selected values of the thermal stratification parameter. In general, the local heat transfer parameter is increased with increasing the values of m, ω, and Ra _d ; while it is decreased with increasing the values of ξ, Bp, and Gr^*. Received on 19 May 1998     

Rigorous Results in Steady Finger Selection in Viscous Fingering

This paper concerns the existence of a steadily translating finger solution in a Hele-Shaw cell for small but non-zero surface tension (ɛ²). Though there are numerous numerical and formal asymptotic results for this problem, we know of no mathematically rigorous results that address the selection problem. We rigorously conclude that for relative finger width λ in the range , with small, analytic symmetric finger solutions exist in the asymptotic limit of surface tension if and only if the Stokes constant for a relatively simple nonlinear differential equation is zero. This Stokes constant S depends on the parameter and earlier calculations by a number of authors have shown it to be zero for a discrete set of values of a. The methodology consists of proving the existence and uniqueness of analytic solutions for a weak half-strip problem for any λ in a compact subset of (0, 1). The weak problem is shown to be equivalent to the original finger problem in the function space considered, provided we invoke a symmetry condition. Next, we consider the behavior of the solution in a neighborhood of an appropriate complex turning point for the restricted case , for some . This turning point accounts for exponentially small terms in ɛ, as ɛ→0⁺ that generally violate the symmetry condition. We prove that the symmetry condition is satisfied for small ɛ when the parameter a is constrained appropriately. (Accepted July 4, 2002 Published online January 15, 2003) Communicated by F. OTTO     

Steady mixed convection boundary-layer flow over a vertical flat surface in a porous medium filled with water at 4°C: variable surface heat flux

The steady mixed convection boundary-layer flow over a vertical impermeable surface in a porous medium saturated with water at 4°C (maximum density) when the surface heat flux varies as x ^m and the velocity outside the boundary layer varies as x ^(1+2m)/2, where x measures the distance from the leading edge, is discussed. Assisting and opposing flows are considered with numerical solutions of the governing equations being obtained for general values of the flow parameters. For opposing flows, there are dual solutions when the mixed convection parameter λ is greater than some critical value λ _c (dependent on the power-law index m). For assisting flows, solutions are possible for all values of λ. A lower bound on m is found, m > −1 being required for solutions. The nature of the critical point λ _c is considered as well as various limiting forms; the forced convection limit (λ = 0), the free convection limit (λ → ∞) and the limits as m → ∞ and as m → −1.     

Nonsimilarity solutions for non-Darcy mixed convection from horizontal surfaces in a porous medium

Non-Darcy mixed convection in a porous medium from horizontal surfaces with variable surface heat flux of the power-law distribution is analyzed. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ζ _f =Ra^* _x /Pe² _x is found to characterize the effect of buoyancy forces on the forced convection with K ^′ U _∞/ν characterizing the effect of inertia resistance. The second region covers the natural convection dominated regime where the dimensionless parameter ζ _n =Pe _x /Ra^*1/2 _x is found to characterize the effect of the forced flow on the natural convection, with (K ^′ U _∞/ν)Ra^*1/2 _x /Pe _x characterizing the effect of inertia resistance. To obtain the solution that covers the entire mixed convection regime the solution of the first regime is carried out for ζ _f =0, the pure forced convection limit, to ζ _f =1 and the solution of the second is carried out for ζ _n =0, the pure natural convection limit, to ζ _n =1. The two solutions meet and match at ζ _f =ζ _n =1, and R ^* _h =G ^* _h . Also a non-Darcy model was used to analyze mixed convection in a porous medium from horizontal surfaces with variable wall temperature of the power-law form. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ξ _f =Ra _x /Pe _x ^3/2 is found to measure the buoyancy effects on mixed convection with Da _x Pe _x /ɛ as the wall effects. The second region covers the natural convection dominated region where ξ _n =Pe _x /Ra _x ^2/3 is found to measure the force effects on mixed convection with Da _x Ra _x ^2/3/ɛ as the wall effects. Numerical results for different inertia, wall, variable surface heat flux and variable wall temperature exponents are presented. Received on 8 July 1996     

Well Adapted Normal Linearization in Singular Perturbation Problems

We provide smooth local normal forms near singularities that appear in planar singular perturbation problems after application of the well-known family blow up technique. The local normal forms preserve the structure that is provided by the blow-up transformation. In a similar context, C ^k -structure-preserving normal forms were shown to exist, for any finite k. In this paper, we improve the smoothness by showing the existence of a C ^∞ normalizing transformation, or in other cases by showing the existence of a single normalizing transformation that is C ^k for each k, provided one restricts the singular parameter ε to a (k-dependent) sufficiently small neighborhood of the origin.     

Laminar mixed convection from a continuously moving vertical surface with suction or injection

The boundary layer flow over a uniformly moving vertical surface with suction or injection is studied when the buoyancy forces assist or oppose the flow. Similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity boundary conditions. The effect is of various governing parameters, such as Prandtl number Pr, temperature exponent n, injection parameter d, and the mixed convection parameter λ=Gr/Re², which determine the velocity and temperature distributions and the heat transfer coefficient, are studied. The heat transfer coefficient increases as λ assisting the flow for all d at Pr=0.72 however, for n=−1 it decreases sharply with λ. On the other hand, increasing λ has no effect on heat transfer coefficient for Pr=10 at n=0, and 1 for almost all values of d studied. However, for n=−1 it has similar effect as for Pr=0.72. It is also found that Nusselt number increases as n increases for fixed λ and d. Received on 26 March 1997     

A NOTE ON BIFURCATIONS OF u″+μ(u-uk)=0(4≤k∈Z+)

Bifurcations of one kind of reaction-diffusion equations, u″+μ(u-u^k)=0(μ is a parameter,4≤k∈Z⁺), with boundary value condition u(0)=u(π)=0 are discussed. By means of singularity theory based on the method of Liapunov-Schmidt reduction, satisfactory results can be acquired.     

Mixed convection cooling of a heated, continuously stretching surface

 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     

Heat transfer in nucleate pool boiling of aqueous SDS and triton X-100 solutions

Variation in degree of surface wettability is presented through the application of Cooper’s correlative approach (h ∝ M ^−0.5 q _w ^″0.67) for computing enhancement (ϕ) in nucleate pool boiling of aqueous solutions of SDS and Triton X-100 and its presentation with Marangoni parameter (χ) that represents the dynamic convection effects due to surface tension gradients. Dynamic spreading coefficient defined as σ _dyn N _a , which relates spreading and wetting characteristics with the active nucleation site density on the heated surface and bubble evolution process, represents cavity filling and activation process and eliminates the concentration dependence of nucleate pool boiling heat transfer in boiling of aqueous surfactant solutions. Using the dynamic spreading coefficient (σ _dyn N _a  = 0.09q _w ^″0.71), correlation predictions within ±15% for both SDS and Triton X-100 solutions for low heat flux boiling condition (q _w^″ ≤ 100 kW/m²) characterised primarily by isolated bubble regime are presented.     

Tailoring turbulence with an active grid

Using an active grid in a wind tunnel, we generate homogeneous shear turbulence and initiate turbulent boundary layers with adjustable properties. Homogeneous shear turbulence is characterized by a constant gradient of the mean velocity and a constant turbulence intensity. It is the simplest anisotropic turbulent flow thinkable, and it is generated traditionally by equipping a wind tunnel with screens which have a varying transparency and flow straighteners. This is not done easily, and the reachable turbulence levels are modest. We describe a new technique for generating homogeneous shear turbulence using an active grid only. Our active grid consists of a grid of rods with attached vanes which can be rotated by servo motors. We control the grid by prescribing the time-dependent angle of each axis. We tune the vertical transparency profile of the grid by setting appropriate angles of each rod such as to generate a uniform velocity gradient, and set the rods in flapping motion around these angles to tailor the turbulence intensity. The Taylor Reynolds number reached was R _λ = 870, the shear rate S = ∂U/∂y = 9.2 s⁻¹, the nondimensional shear parameter S ^*≡ Sq ²/ε = 12 and u = 1.4 ms⁻¹. As a further application of this idea we demonstrate the generation of a simulated atmospheric boundary layer in a wind tunnel which has tunable properties. This method offers a great advantage over the traditional one, in which vortex-generating structures need to be placed in the wind tunnel to initiate a fat boundary layer.