Hall effects on the unsteady hydromagnetic oscillatory flow of a second-grade fluid

An exact solution of an oscillatory flow is constructed in a rotating fluid under the influence of an uniform transverse magnetic field. The fluid is considered as second-grade (non-Newtonian). The influence of Hall currents and material parameters of the second-grade fluid is investigated. The hydromagnetic flow is generated in the uniformly rotating fluid bounded between two rigid non-conducting parallel plates by small amplitude oscillations of the upper plate. The exact solutions of the steady and unsteady velocity fields are constructed. It is found that the steady solution depends on the Hall parameter but is independent of the material parameter of the fluid. The unsteady part of the solution depends upon both (Hall and material) parameters. Attention is focused upon the physical nature of the solution, and the structure of the various kinds of boundary layers is examined. Several results of physical interest have been deduced in limiting cases.     

Moving boundary in a non-Newtonian fluid

Moving boundary value problem in non-Newtonian fluid is considered. Exact analytical solution for the flow of second-grade fluid for a rigid moving plate oscillating in its own plane, is obtained. The Doppler effect has been observed due to the motion of the plate. The shearing stress on the plate is also calculated. It is concluded that the solutions for stationary porous boundaries can be obtained from the solutions of moving rigid boundaries.     

Effects of the side walls on the unsteady flow of a second-grade fluid in a duct of uniform cross-section

In this paper, the effects of the side walls on the unsteady flow of a second-grade fluid in a duct of rectangular cross-section are considered. Two types of unsteady flows are investigated. One of them is the unsteady flow in a duct of rectangular cross-section moving parallel to its length and the other is the unsteady flow due to an applied pressure gradient in a duct of rectangular cross-section whose sides are at rest. It is shown that a Newtonian fluid reaches steady-state earlier than a second-grade fluid and the effect of the side walls on a second-grade fluid is more effective than that on a Newtonian fluid.     

Application of differential constraint method to exact solution of second-grade fluid

A differential constraint method is used to obtain analytical solutions of a second-grade fluid flow. By using the first-order differential constraint condition, exact solutions of Poiseuille flows, jet flows and Couette flows subjected to suction or blowing forces, and planar elongational flows are derived. In addition, two new classes of exact solutions for a second-grade fluid flow are found. The obtained exact solutions show that the non-Newtonian second-grade flow behavior depends not only on the material viscosity but also on the material elasticity. Finally, some boundary value problems are discussed.     

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.     

Stokes’ first problem for some non-Newtonian fluids: Results and mistakes

The well-known problem of unidirectional plane flow of a fluid in a half-space due to the impulsive motion of the plate it rests upon is discussed in the context of the second-grade and the Oldroyd-B non-Newtonian fluids. The governing equations are derived from the conservation laws of mass and momentum and three correct known representations of their exact solutions given. Common mistakes made in the literature are identified. Simple numerical schemes that corroborate the analytical solutions are constructed.     

Heat and Mass Transfer in MHD Micropolar Flow Over a Vertical Moving Porous Plate in a Porous Medium

An analysis is presented for the problem of free convection with mass transfer flow for a micropolar fluid via a porous medium bounded by a semi-infinite vertical porous plate in the presence of a transverse magnetic field. The plate moves with constant velocity in the longitudinal direction, and the free stream velocity follows an exponentially small perturbation law. A uniform magnetic field acts perpendicularly to the porous surface in which absorbs the micropolar fluid with a suction velocity varying with time. Numerical results of velocity distribution of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. Also, the results of the skin-friction coefficient, the couple stress coefficient, the rate of the heat and mass transfers at the wall are prepared with various values of fluid properties and flow conditions.     

Transient oscillatory and constantly accelerated non-Newtonian flow in a porous medium

This paper looks at the magnetohydrodynamic (MHD) analysis for transient flow of an Oldroyd-B fluid in a porous medium. The presented analysis takes into account the modified Darcy's law. The flow is induced due to constantly accelerated and oscillating plate. Expressions for the corresponding velocity field and the adequate tangential stress are determined by means of the Fourier sine transform. The influence of various parameters of interest on the velocity and tangential stress has been shown and discussed. A comparison for different kinds of fluids is also provided.     

Stagnation-point flow of a second-grade fluid with slip

The steady two-dimensional stagnation-point flow of a second-grade fluid with slip is examined. The fluid impinges on the wall either orthogonally or obliquely. Numerical solutions are obtained using a quasi-linearization technique.     

Resistance of rough plates in a rotating fluid

Generalizing Navier’s partial slip condition, the flow due to a rough or striated plate moving in a rotating fluid is studied. It is found that the motion of the plate, the fluid surface velocity, and the shear stress are in general not in the same direction. The solution is extended to the case of finite depth, or Couette slip flow in a rotating system. In this case an optimum depth for minimum drag is found. The solutions are also closed form exact solutions of the Navier–Stokes equations. The results are fundamental to flows with Coriolis effects.     

Flow of a biomagnetic viscoelastic fluid: application to estimation of blood flow in arteries during electromagnetic hyperthermia,a therapeutic procedure for cancer treatment

The paper deals with the theoretical investigation of a fundamental problem of biomaguetic fluid flow through a porous medium subject to a magnetic field by using the principles of biomagnetic fluid dynamics （BFD）. The study pertains to a situation where magnetization of the fluid varies with temperature. The fluid is considered to be non-Newtonian, whose flow is governed by the equation of a second-grade viscoelastic fluid. The walls of the channel are assumed to be stretchable, where the surface velocity is proportional to the longitudinal distance from the origin of coordinates. The problem is first reduced to solving a system of coupled nonlinear differential equations involving seven parameters. Considering blood as a biomagnetic fluid and using the present analysis, an attempt is made to compute some parameters of the blood flow by developing a suitable numerical method and by devising an appropriate finite difference scheme. The computational results are presented in graphical form, and thereby some theoretical predictions are made with respect to the hemodynamical flow of the blood in a hyperthermal state under the action of a magnetic field. The results clearly indicate that the presence of a magnetic dipole bears the potential so as to affect the characteristics of the blood flow in arteries to a significant extent during the therapeutic procedure of electromagnetic hyperthermia. The study will attract the attention of clinicians, to whom the results would be useful in the treatment of cancer patients by the method of electromagnetic hyperthermia.     

二阶非牛顿流体蠕流精确解

二阶流体是工业界常见的非牛顿流体,因为其本构关系简单而被广泛采用和研究.逆方法预先假定流场满足某类特定的物理的或几何的特性,从而求出流体运动方程的精确解.本文通过假定平面定常二阶非牛顿流体的涡量场与受到扰动的流函数相等这一特定形式,采用求解非线性微分方程常用的逆方法,推导并获得了平面二阶蠕流流场的精确解,由此容易进一步获得流场的压力.所获得的精确解包含了Poiseuille,简单Couette平行流动以及两相向流体的相互作用等流动.这些精确解为实验,数值以及渐进解的检验提供了借鉴和参考.     

Impulsive motion of a non-Newtonian fluid between two oscillating parallel plates

An exact solution for the flow of an incompressible viscoelastic fluid between two infinitely extended parallel plates, due to the harmonic oscillations of the upper plate and the impulsively started harmonic oscillations of the lower plate from rest, in the respective planes of the plates, has been obtained. The momentum transfer towards the central region and the skin friction of the lower plate are found to be greater for the viscoelastic fluid than that for viscous fluid. The effect of out-of-phase oscillations of the plates with different amplitudes on the flow characteristics has also been investigated.     

Sakiadis flow of an upper-convected Maxwell fluid

The flow of an upper-convected Maxwell (UCM) fluid is studied theoretically above a rigid plate moving steadily in an otherwise quiescent fluid. It is assumed that the Reynolds number of the flow is high enough for boundary layer approximation to be valid. Assuming a laminar, two-dimensional flow above the plate, the concept of stream function coupled with the concept of similarity solution is utilized to reduce the governing equations into a single third-order ODE. It is concluded that the fluid's elasticity destroys similarity between velocity profiles; thus an attempt was made to find local similarity solutions. Three different methods will be used to solve the governing equation: (i) the perturbation method, (ii) the fourth-order Runge-Kutta method, and (iii) the finite-difference method. The velocity profiles obtained using the latter two methods are shown to be virtually the same at corresponding Deborah number. The velocity profiles obtained using perturbation method, in addition to being different from those of the other two methods, are dubious in that they imply some degree of reverse flow. The wall skin friction coefficient is predicted to decrease with an increase in the Deborah number for Sakiadis flow of a UCM fluid. This prediction is in direct contradiction with that reported in the literature for a second-grade fluid.     

Stagnation-point flow of upper-convected Maxwell fluids

Two-dimensional stagnation-point flow of viscoelastic fluids is studied theoretically assuming that the fluid obeys the upper-convected Maxwell (UCM) model. Boundary-layer theory is used to simplify the equations of motion which are further reduced to a single non-linear third-order ODE using the concept of stream function coupled with the technique of the similarity solution. The equation so obtained was solved using Chebyshev pseudo-spectral collocation-point method. Based on the results obtained in the present work, it is concluded that the well-established but controversial prediction that in stagnation-point flows of viscoelastic fluids the velocity inside the boundary layer may exceed that outside the layer may just be an artifact of the rheological model used in previous studies (namely, the second-grade model). No such peculiarity is predicted to exist for the Maxwell model. For a UCM fluid, a thickening of the boundary layer and a drop in wall skin friction coefficient is predicted to occur the higher the elasticity number. These predictions are in direct contradiction with those reported in the literature for a second-grade fluid.     

On Solutions of Some Non-Linear Differential Equations Arising in Newtonian and Non-Newtonian Fluids

Some steady as well as unsteady solutions of the equations of motion for an incompressible Newtonian and non-Newtonian (second-grade) fluids are obtained by applying different methods including the Lie symmetry group method. The flows considered are axially symmetric with the swirling motion, and the governing equations for second-grade fluid flow have been modeled. Expressions for streamlines, velocity and vorticity components are constructed explicitly in each case. Exact analytical solutions in second-grade fluid are obtained and compared with the corresponding viscous solutions.     

Transient mixed convection flow of a second grade viscoelastic fluid past an inclined backward facing step

The transient mixed convection of a second-grade viscoelastic fluid past an inclined backward facing step was studied numerically. The combined effects of the Reynolds number, the elastic effect, the inclined angle of the flow channel on the reattachment length, and the phenomena of heat transfer are examined during the development of the flow field. The Gauss-Seidal method with successive over relaxation was implemented to solve the stream-vorticity and energy equations. The results indicate that the reattachment length increases to the maximum as the inclined angle increases up to 150° or 180°. At these cases, the point of reattachment is close to the point of the local maximum value of Nu_x or is overshooting it. It is observed that the reattachment length increases as the Reynolds number increases or the elastic coefficient decreases. In the meantime, the contact point of the isotherm on the upper plate moves upward and is close to upstream flow as the inclined angle is around 150°.     

Flow of a fluid-solid mixture: Normal stress differences and slip boundary condition

In this paper, using mixture theory we study the flow of a dense suspension, composed of solid particles and a fluid; the emphasis is on the influence of the slip boundary condition and the effect of normal stress differences. Very little work has been done considering both the slip at the walls and the normal stress effects in the frame of a two-component flow. In this paper, the stress tensor for the solid component is modeled as a nonlinear fluid which not only includes the viscous effects but also the normal stress effects; the fluid constituent is modeled as a viscous fluid. We look at the flow between two flat plates.     

A domain perturbation study of steady flow in a cone-and-plate rheometer of non-ideal geometry

The method of domain perturbation developed by Joseph is used to calculate velocity and stress profiles in a slightly misaligned cone-and-plate rheometer where the cone is spinning and the plate is stationary. Results for a Newtonian fluid, a Criminale-Ericksen-Filbey fluid, an upper-convected Maxwell fluid, and a White-Metzner fluid are presented and compared with earlier results in which the cone is stationary and the plate is spinning (Dudgeon and Wedgewood, 1993). Streamlines calculated for the Newtonian fluid show a very small recirculation region near the stationary plate. Velocity and stress contours are symmetric around the plane of largest gap width. For the elastic fluids studied, streamlines are asymmetric. The fluid response lags where the fluid is dominated by memory effects. Much larger recirculation regions are calculated for fluids dominated by shear thinning. These recirculation regions contain a large fraction of the fluid in the apparatus and have the effect of changing the shape of the flow domain for the remaining fluid that rotates around the cone's axis. Elasticity also has a pronounced effect on the stress profile, indicating that the accuracy of the cone and plate may be compromised even for small mis-alignments.     

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.