Heat transfer in micropolar boundary layer flow over a flat plate

Boundary layer solutions are presented to study the steady state heat transfer from a semi-infinite flat plate to a micropolar fluid. The boundary conditions of isothermal wall, constant surface heat flux and insulated wall with viscous dissipation effects have been treated in this paper. Numerical results for the temperature distribution and the missing wall values of the thermal functions have been given. The range of Prandtl numbers investigated was from 10 to 1000 while the dimensionless grouping of the material properties was allowed to vary over a wide range.     

Moving wedge and flat plate in a micropolar fluid

Similarity solutions for a moving wedge and flat plate in a micropolar fluid may be obtained when the fluid and boundary velocities are proportional to the same power-law of the downstream coordinate. The governing partial differential equations are transformed to the ordinary differential equations using similarity variables, and then solve numerically using a finite-difference scheme known as the Keller-box method. Numerical results are given for the dimensionless velocity and microrotation profiles, as well as the skin friction coefficient for several values of the Falkner–Skan power-law parameter (m), the ratio of the boundary velocity to the free stream velocity parameter (λ) and the material parameter (K). Important features of these flow characteristics are plotted and discussed. It is found that multiple solutions exist when the boundary is moving in the opposite direction to the free stream, and the micropolar fluids display a drag reduction compared to Newtonian fluids.     

Unsteady thermal boundary layer flow of a non-newtonian fluid over a flat plate

The transient thermal response of a power law type non-Newtonian, laminar boundary layer flow past a flat plate is investigated. Consideration is given to the case of a step change in surface temperature. The transient heat flux and details of the temperature field are obtained and have been illustrated graphically. The range of Prandtl numbers investigated was 5–1000 while the viscosity index was allowed to vary 0.1–5.0.     

Axisymmetric thermal boundary layer of a micropolar fluid on a cylinder

The laminar micropolar thermal boundary layer in axial flow along a long, thin circular cylinder is investigated using the theory of micropolar fluids formulated by Eringen. The governing energy equation has been solved numerically by the power series approach. Missing values of the thermal functions are tabulated for a wide range of the material parameters, the transverse curvature parameter, and Prandtl number of the fluid.     

Thermal boundary layer flow of a micropolar fluid past a wedge with constant wall temperature

Summary The steady laminar flow of micropolar fluids past a wedge has been examined with constant surface temperature. The similarity variables found by Falkner and Skan are employed to reduce the streamwise-dependence in the coupled nonlinear boundary layer equation. Numerical solutions are presented for the heat transfer characteristics with Pr=1 using the fourth-order Runge-Kutta method, and their dependence on the material parameters is discussed. The distributions of dimensionless temperature and Nusselt number across the boundary layer are compared with the corresponding flow problems for a Newtonian fluid over wedges. Numerical results show that for a constant wedge angle with a given Prandtl number Pr=1, the effect of increasing values ofK results in an increasing thermal boundary thickness for a micropolar fluid, as compared with a Newtonian fluid. For the case of the constant material parameterK, however, the heat transfer rate for a micropolar fluid is lower than that of a Newtonian fluid.Nomenclature h Dimensionless microrotation - j Micro-inertia density - K Dimensionless parameter of vortex viscosity - m Falkner-Skan power-law parameter - Re Reynolds number - T Temperature - u, v Fluid velocities in thex andy directions, respectively - U Free stream velocity - x Streamwise coordinate along the body surface - y Coordinate normal to the body surface Greek symbols Thermal diffusivity - Wedge angle parameter - Spin gradient viscosity - Pseudo-similarity variable - Vortex viscosity - Absolute viscosity of the fluid - v Kinematic viscosity - Dimensionless temperature - Density of the micropolar fluid - Angular velocity of micropolar fluid - Stream function     

Longitudinal surface curvature effects on boundary layer of a micropolar fluid

Summary The effect of longitudinal surface curvature on steady two-dimensional incompressible laminar boundary layer of a micropolar fluid has been considered. Van Dyke's first oder perturbation analysis is applied to the full equations of motion derived in curvilinear coordinate system which facilitates to carry out boundary layer approximation for flow past a curved surface. This results into two systems of partial differential equations which are called the zeroth order and the first order boundary layer equations. The zeroth order equations are the usual boundary layer equations for a micropolar fluid. The first order equations take into account the longitudinal surface curvature effect explicitly. Similar solution of the governing equations exists if (i) the inviscid flow velocity on the surface varies linearly along the surface and (ii) the longitudinal surface curvature is constant. Numerical results are presented illustrating the dependence of the important flow quantities of both zeroth order and first order boundary layers on the micropolar fluid parameters. The results have been compared with the corresponding results for a Newtonian fluid. It has been found that the skin friction decreases and the wall couple stress increases for convex side of the surface and vice versa for the concave side.

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Einfluß der longitudinalen Oberflächenkrümmung auf die Grenzschicht einer mikropolaren Flüssigkeit

Zusammenfassung Der Einfluß einer longitudinalen Oberflächenkrümmung auf die stationäre, zweidimensionale, inkompressible, laminare Grenzschicht einer mikropolaren Flüssigkeit wird untersucht. Van Dyke's Strömungsanalyse erster Ordnung wird angewendet auf die Bewegungsgleichungen in krummlinigen Koordinaten, die einfach die Grenzschichtnäherung für die Strömung hinter einer gekrümmten Oberfläche aufzeigen. Dies führt zu zwei Systemen partieller Differentialgleichungen, die die Grenzschichtgleichungen nullter und erster Ordnung genannt werden. Die Gleichungen nullter Ordnung sind die üblichen Grenzschichtgleichungen für eine mikropolare Flüssigkeit. Die Gleichungen erster Ordnung berücksichtigen den Einfluß der longitudinalen Oberflächenkrümmung explizit. Ähnliche Lösungen der Grundgleichungen existieren wenn: (i) sich die reibungsfreie Strömungsgeschwindigkeit an der Oberfläche linear entlang der Oberfläche ändert und (ii) die longitudinale Oberflächenkrümmung konstant ist. Numerische Ergebnisse werden angegeben, um die Abhängigkeit der wichtigsten Strömungsgrößen der Grenzschicht nullter und erster Ordnung von den Parametern der mikropolaren Flüssigkeit aufzuzeigen. Die Ergebnisse werden verglichen mit den entsprechenden Ergebnissen für eine Newtonsche Flüssigkeit. Es wurde festgestellt, daß für eine konvexe Fläche die Wandreibung abnimmt und die gekoppelten Spannungen an der Wand zunehmen; das Umgekehrte gilt für eine konkave Fläche.

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With 2 Figures     

Stability of flow of a structurally viscous fluid in the boundary layer on a flat plate

The method of small perturbations has been used to study the hydrodynamical stability of flow of a structurally viscous fluid in the boundary layer on a flat plate.     

Effects of fluid layer at micropolar orthotropic boundary surface

Effects of a fluid layer at a micropolar orthotropic elastic solid interface to a moving point load have been studied. After using the Fourier transform an eigen value approach has been employed to solve the problem. The displacement, microrotation and stress components for a micropolar orthotropic elastic solid so obtained in the physical domain are computed numerically by applying the numerical inversion technique. Micropolarity and anisotropy effects along with that of the depth of the fluid layer on various expressions have been depicted graphically for a particular model. Some special cases of interest have been presented     

Thermal boundary layer on a plate in a non-Newtonian fluid with nonlinear heat conduction law

The flat-plate boundary layer equation for a rheological power law and a proposed nonlinear law of heat conduction is reduced to an ordinary differential equation, which is solved in quadratures using previously calculated [2] velocity profiles. Graphs of the temperature and heat transfer coefficient profiles are presented.     

Similarity solution for boundary layer development of a swirling jet impinging on a normal plane

When a smooth swirling jet of water impinges vertically on a horizontal plane it spreads outwards forming a boundary layer on the plane. At large radial distance r, the flow is completely absorbed by the boundary layer and in this region a similarity solution is obtained.     

On the hydromagnetic boundary layer flow past a flat plate

Summary The Falkner-Skan transformation has been generalized to study the boundary layer flow of a viscous incompressible conducting fluid past a flat plate in the presence of a uniform transverse magnetic field fixed to the plate. Keeping the convergence aspect in view series solution, different from the conventional, for the velocity boundary layer has been developed. It is found that only few terms in the series expansion are enough to give the skin-friction coefficient to a good degree of accuracy.

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Über die hydromagnetische Grenzschichtströmung an der ebenen Platte

Zusammenfassung Die Falkner-Skan-Transformation wird verallgemeinert für den Fall der Grenzschichtströmung eines leitenden, zähen, inkompressiblen Mediums längs einer ebenen Platte in einem konstanten Magnetfeld senkrecht zur Platte. Es wird eine neue Reihenentwicklung mit guten Konvergenzeigenschaften für die Geschwindigkeit in der Grenzschicht hergeleitet. Hier reichen wenige Terme der Reihenentwicklung aus, um den Koeffizienten der Oberflächenreibung in guter Genauigkeit zu bestimmen.

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With 1 Figure     

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

The transverse curvature effects on axisymmetric free convection boundary layer flow of a micropolar fluid past vertical cylinders are investigated using the theory of micropolar fluids formulated by Eringen. The governing equations for momentum, angular momentum and energy have been solved numerically. Missing values of the velocity, angular velocity and thermal functions are tabulated for a wide range of the material parameters, transverse curvature parameter and Prandtl number of the fluid. A comparison has been made with the corresponding results for Newtonian fluids. Micropolar fluids display drag reduction and reduced surface heat transfer rate as compared with Newtonian fluids.     

The asymptotic boundary layer on a circular cylinder in axisymmetric micropolar fluid flow

An asymptotic boundary layer analysis is presented using the theory of micropolar fluids due to Eringen. The laminar boundary layer induced on the outside of a long, slender cylinder due to the flow of an incompressible micropolar fluid parallel to the axis of the cylinder is investigated. For reasons of both analytical and practical interests the boundary layer characteristics far down stream from the leading edge are analyzed on the basis of their asymptotic nature. Asymptotic series solutions for the velocity and micro-rotation fields are obtained. An expression for the new micropolar boundary layer thickness is derived. Central to the present investigation is the result that while calculating the skin friction one should take into account the total surface stress effects, not only due to the usual shear stresses but also due to the couple stresses. As a result, it is shown that the micropolar theory does predict a reduction in skin friction as is observed in experiments thus confirming Eringen's well known conjecture.     

Effect of pressure gradient on MHD boundary layer over a flat plate

Summary The magnetohydrodynamic (MHD) boundary layer flow over a flat plate is examined here for two cases, viz. a uniform free-stream velocity and a uniform hydrostatic pressure. The nonlinear boundary layer equations are solved using a reliable finite-difference method. The boundary layer physical parameters such as skin-friction coefficient, displacement, momentum and energy thicknesses of the boundary layer are determined. It is found that the normal surface velocity gradient decreases with the local magnetic interaction parameter for the cases of a uniform hydrostatic pressure, whereas in the case of a uniform free-stream volocity it increases with the interaction parameter.     

Deposition of solid particles in laminar boundary layer on a flat plate

Results are given of experimental and theoretical investigation of deposition of small solid particles on the surface of a flat plate under conditions of vertical laminar boundary layer. The present investigation is aimed at qualitatively and quantitatively estimating the effect made by the parameters of two-phase flow of the “gas—solid particles” type and by the adhesive properties of particles and surface on the deposition of particles on the plate surface. The flow velocity is 1.5 and 3 m/s. In so doing, the value of Reynolds number along the plate does not exceed 10⁵. Synthetic corundum powders with average sizes of 12, 23, and 32 μm are used as the dispersed phase of two-phase flow. The mass concentration of particles in the flow is 0.01 kg/m³. A flat plate of stainless steel is used as the object of investigation. The distributions of gas velocity and concentration of particles within the boundary layer are measured using laser optical diagnostics. The number of particles deposited along the plate surface is measured by the gravimetric method. The adhesive properties of the “particle-surface” pair are studied using the centrifugal method of detachment of particles from the surface. Logarithmic-normal dependences of the number of adhesion of particles on the force of detachment are obtained. The hydrodynamic parameters of two-phase flow in the vicinity of the plate surface are calculated using the model of two-phase laminar boundary layer. The mathematical expression is suggested for the calculation of the magnitude of deposition of solid particles along the surface of a flat plate, which includes the special features of hydrodynamics of flow, the adhesive properties of the particles and surface, and the probabilistic pattern of the process of entrapment of particles by the surface.     

Unsteady incompressible boundary layer flow of a micropolar fluid at a stagnation point

The flow, heat and mass transfer on the unsteady laminar incompressible boundary layer in micropolar fluid at the stagnation point of a 2-dimensional and an axisymmetric body have been studied when the free stream velocity and the wall temperature vary arbitrarily with time. The partial defferential equations governing the flow have been solved numerically using a quasilinear finite-difference scheme. The skin friction, microrotation gradient and heat transfer parameters are found to be strongly dependent on the coupling parameter, mass transfer and time, whereas the effect of the microrotation parameter on the skin friction and heat transfer is rather weak, but microrotation gradient is strongly affected by it. The Prandtl number and the variation of the wall temperature with time affect the heat-transfer very significantly but the skin friction and micrortation gradient are unaffected by them.     

Thermal instability in a micropolar fluid layer subject to a magnetic field

In the present paper we have considered thermal instability in a heat conducting micropolar fluid layer under the influence of a transverse magnetic field. Assuming the bounding surfaces to be rigid the eigenvalue problem is solved using finite-difference and Wilkinson's iteration techniques. Here it is seen that the instability sets in not only for adverse temperature gradient but also for positive temperature gradient. Both the microtation and the magnetic field are seen to stabilize the fluid layer. However, the stabilizing effect of microrotation becomes less significant when the strength of the magnetic field is large. In the case of heating from below, the critical wave number is seen to be insensitive to increase in the strength of the magnetic field, while it increases significantly when the fluid is heated from above.