Influence of temperature dependent viscosity on peristaltic transport of a Newtonian fluid: Application of an endoscope

The effects of variable temperature dependent viscosity on peristaltic flow of Newtonian fluid in an annulus has been investigated with long wave length approximations. A regular perturbation method has been used to obtain explicit form for the velocity, temperature and relation between flow rate and pressure gradient. The expression for the pressure rise, friction force, velocity and temperature were plotted for different values of variable viscosity parameter β, radius ratio, amplitude ratio ?, heat absorption parameter β₁, and force convection parameter G_r. It is found that the pressure rise decrease as the viscosity parameter β increases and increases as the radius ratio as ? increases and β decreases.     

Effects of mass transfer on free convective flow of a dissipative,incompressible fluid past an infinite vertical porous plate with suction

An analysis of the mass transfer effects on the free convective flow of an incompressible, dissipative, viscous fluid past an infinite, vertical porous plate with constant suction, has been carried out. Approximate solutions to coupled non-linear equations governing the flow are derived. The velocity and the temperature profiles are shown graphically for air (P=0·71). The effects of Gr (Grashof number), Gc (the modified Grashof number), Sc (Schmidt number), E (Eckert number) are discussed qualitatively during the course of discussion. It is observed that due to the addition of the foreign mass, there is a rise in the velocity and a fall in the temperature. But the skin-friction increases when Sc?1 and it decreases when Sc ～1. The rate of heat transfer increases for Sc<1 and decreases for Sc>1.     

Unsteady MHD Couette flow in a rotating system

The unsteady hydromagnetic Couette flow of a viscous incompressible electrically conducting fluid in a rotating system has been considered. An exact solution of the governing equations has been obtained by using a Laplace transform. Solutions for the velocity distributions as well as shear stresses have been obtained for small times as well as for large times. It is found that for large times the primary velocity decreases with increase in the rotation parameter K² while it increases with increase in the magnetic parameter M². It is also found that with increase in K², the secondary velocity v₁ decreases near the stationary plate while it increases near the moving plate. On the other hand, the secondary velocity decreases with increase in the magnetic parameter.     

Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition

In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux.     

Nano boundary layer equation with nonlinear Navier boundary condition

At the micro and nano scale the standard no slip boundary condition of classical fluid mechanics does not apply and must be replaced by a boundary condition that allows some degree of tangential slip. In this study the classical laminar boundary layer equations are studied using Lie symmetries with the no-slip boundary condition replaced by a nonlinear Navier boundary condition. This boundary condition contains an arbitrary index parameter, denoted by n>0, which appears in the coefficients of the ordinary differential equation to be solved. The case of a boundary layer formed in a convergent channel with a sink, which corresponds to n=1/2, is solved analytically. Another analytical but non-unique solution is found corresponding to the value n=1/3, while other values of n for n>1/2 correspond to the boundary layer formed in the flow past a wedge and are solved numerically. It is found that for fixed slip length the velocity components are reduced in magnitude as n increases, while for fixed n the velocity components are increased in magnitude as the slip length is increased.     

Nano boundary layers over stretching surfaces

In this paper, we present similarity solutions for the nano boundary layer flows with Navier boundary condition. We consider viscous flows over a two-dimensional stretching surface and an axisymmetric stretching surface. The resulting nonlinear ordinary differential equations are solved analytically by the Homotopy Analysis Method. Numerical solutions are obtained by using a boundary value problem solver, and are shown to agree well with the analytical solutions. The effects of the slip parameter K and the suction parameter s on the fluid velocity and on the tangential stress are investigated and discussed. As expected, we find that for such fluid flows at nano scales, the shear stress at the wall decreases (in an absolute sense) with an increase in the slip parameter K.     

Unsteady flow of fluids with microstructure between parallel plates—II

Two types of unsteady flow between two parallel plates of a fluid with microstructure have been considered. (i) Initially the fluid is at rest and the motion is caused by a sudden change of pressure gradient from zero to a constant value. It is shown that for a given time asα* decreases from infinity to zero, the velocity in the flow decreases. (ii) One plate is fixed and the other is accelerated with a velocity ν = Ut ⁿ (U andn being positive constants). The flow pattern is discussed in both cases in detail.     

The effect of cyclone inlet dimensions on the flow pattern and performance

The effect of the cyclone inlet dimensions on the performance and flow field pattern has been investigated computationally using the Reynolds stress turbulence model (RSM) for five cyclone separators. The results show that, the maximum tangential velocity in the cyclone decreases with increasing the cyclone inlet dimensions. No acceleration occurs in the cyclone space (the maximum tangential velocity is nearly constant throughout the cyclone). Increasing the cyclone inlet dimensions decreases the pressure drop. The cyclone cut-off diameter increases with increasing cyclone inlet dimension (consequently, the cyclone overall efficiency decreases due to weakness of the vortex strength). The effect of changing the inlet width is more significant than the inlet height especially for the cut-off diameter. The optimum ratio of inlet width to inlet height b/a is from 0.5 to 0.7.     

Weak solutions for immiscible compressible multifluid flows in porous media

We consider a multifluid flow model describing evolution of pressures and relative saturations of non-miscible and compressible phases in porous media. This model is obtained by a macroscopic representation of the flow. It takes into account capillary effects and velocity fields are described by Darcy laws. Global weak solutions for such a model is introduced. To cite this article: C. Galusinski, M. Saad, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Analysis of viscoelastic fluid flow with temperature dependent properties in plane Couette flow and thin annuli

The steady state flow in very thin annuli has been studied analytically for the case where the annular gap is much smaller than the radius of the inner cylinder and for the outer cylinder rotating at constant angular speed and the inner cylinder at rest. The cylinders were subjected to two different thermal boundary conditions. The exponential effect of temperature on the relaxation time and the viscosity coefficient was accounted into the governing differential equations using Nahme’s law. Effects of viscous dissipation as well as εDe² (viscoelastic index for SPTT constitutive equation) on the dimensionless velocity and temperature profiles have been investigated. Results show that while the properties of the fluid depend on temperature, the velocity and temperature profiles are different compared to those obtained with constant physical properties. The Nahme–Griffith number increases whereas εDe² as a viscoelastic index decreases when temperature dependent physical properties are considered. In addition, the results indicate that the viscous dissipation has a sensible effect on heat transfer and the Nusselt number decreases with an increase in the Nahme–Griffith number.     

Computational modelling of fluid structure interaction at nano-scale boundaries with modified Maxwellian velocity distribution

The soft collisions among fluid–fluid and fluid-wall molecules are modeled from first principles. In particular, the assumption of Maxwellian distribution of velocities for thermalized molecules, in both parallel and perpendicular directions to the wall, has been re-evaluated with supporting experimental and/or numerical evidence.It is proposed that the normal component of molecular velocity post collision is conserved for all fluid molecules. The slip effect at the wall boundary, introduced by the surface roughness, is accounted by an accommodation coefficient f. A moving least square method is used to calculate macroscopic velocity values. The influence of molecular interaction on the macroscopic velocity distribution is investigated at 40 MPa and 300 K for slit pore, inclined and stepped wall configurations. The accommodation coefficient values f = 0, 0.07, 0.257, 0.45, 0.681 and 1; and acceleration values ranging from zero to 1 × 10¹¹ m/s² and 250 × 10¹¹ m/s² are used for comparison.The distribution of macroscopic velocity parallel to the wall is studied to observe the effect of the slip behaviour. The detailed study of average of velocity values at various magnitudes of acceleration has shown an evidence of characteristic low and high speed of molecular flows that is considered as significant and a comparison is sought with an equivalent laminar and turbulent flow style behaviour. The two dimensional vector and contour plots of macroscopic velocity provide further insights in understanding Continuum velocity distributions resulting from molecular fluid-wall interaction at nanoscale. The research has highlighted the need to develop molecular dynamics simulation techniques for non-periodic boundary conditions.     

存在滑移时两圆球间的幂律流体挤压流动

基于Reynolds润滑理论分析了壁面滑移对任意圆球颗粒间幂律流体的挤压流动的影响。研究表明有壁面滑移时挤压流动的粘性力可通过引进本文定义的滑移修正系数分离出无滑移解。推导出的挤压力滑移修正系数是一积分表达式，依赖于滑移参数、幂律指数、球间隙和积分上限。一般地壁面滑移导致粘性力减小，粘性力的减小量随幂律指数的增大而增大，表明壁面滑移对剪切增稠流变材料有更大的影响；粘性力的减小量还随着滑移参数的增大而增加，而这恰与假设相符合；粘性力的减小量又随球间隙减小或积分上限的增大（从液桥情况到完全浸渍）而减小直到趋于常数，这一特性在离散元模拟时可以有效地减少计算量。     

A non-Newtonian fluid flow model for blood flow through a catheterized artery—Steady flow

In this paper the effects catheterization and non-Newtonian nature of blood in small arteries of diameter less than 100 μm, on velocity, flow resistance and wall shear stress are analyzed mathematically by modeling blood as a Herschel–Bulkley fluid with parameters n and θ and the artery and catheter by coaxial rigid circular cylinders. The influence of the catheter radius and the yield stress of the fluid on the yield plane locations, velocity distributions, flow rate, wall shear stress and frictional resistance are investigated assuming the flow to be steady. It is shown that the velocity decreases as the yield stress increases for given values of other parameters. The frictional resistance as well as the wall shear stress increases with increasing yield stress, whereas the frictional resistance increases and the wall shear stress decreases with increasing catheter radius ratio k (catheter radius to vessel radius). For the range of catheter radius ratio 0.3–0.6, in smaller arteries where blood is modeled by Herschel–Bulkley fluid with yield stress θ = 0.1, the resistance increases by a factor 3.98–21.12 for n = 0.95 and by a factor 4.35–25.09 for n = 1.05. When θ = 0.3, these factors are 7.47–124.6 when n = 0.95 and 8.97–247.76 when n = 1.05.     

Pulsatile flow of particle-fluid suspension model of blood under periodic body acceleration

A particle-fluid suspension model is applied to the problem of pulsatile blood flow through a circular tube under the influence of body acceleration. With the help of finite Hankel and Laplace transforms, analytic expressions for axial velocity for both fluid and particle phase, fluid acceleration, wall shear stress and instantaneous flow rate have been obtained. It is observed that the solutions can be used for all feasible values of pulsatile and body acceleration Reynolds numbers R_p and R_b. Using physiological data, the following qualitative and quantitative results have been obtained. The amplitude Q_b of instantaneous flow rate due to body acceleration decreases as the tube radius decreases. The effect of the volume fraction of particle C on Q_b is to increase it with increase of C in arteriole and to decrease Q_b as C increases in coronary and femoral arteries. The maximum of the axial velocity and fluid acceleration shifts from the axis of the tube to the vicinity of the tube wall as the tube diameter increases. The effect of C on the velocity and acceleration are nonuniform. The wall shear amplitude t_b\tau_b due to body acceleration increases as the tube diameter decreases from femoral to coronary and a further decrease in the tube diameter leads to a decrease in t_b\tau_b. The effects of C on t_b\tau_b are again nonuniform.     

Pulsatile flow of Herschel–Bulkley fluid through catheterized arteries – A mathematical model

The pulsatile flow of blood through catheterized artery has been studied in this paper by modeling blood as Herschel–Bulkley fluid and the catheter and artery as rigid coaxial circular cylinders. The Herschel–Bulkley fluid has two parameters, the yield stress θ and the power index n. Perturbation method is used to solve the resulting quasi-steady nonlinear coupled implicit system of differential equations. The effects of catheterization and non-Newtonian nature of blood on yield plane locations, velocity, flow rate, wall shear stress and longitudinal impedance of the artery are discussed. The existence of two yield plane locations is investigated and their dependence on yield stress θ, amplitude A, and time t are analyzed. The width of the plug core region increases with increasing value of yield stress at any time. The velocity and flow rate decrease, whereas wall shear stress and longitudinal impedance increase for increasing value of yield stress with other parameters held fixed. On the other hand, the velocity, flow rate and wall shear stress decrease but resistance to flow increases as the catheter radius ratio (ratio of catheter radius to vessel radius) increases with other parameters fixed. The results for power law fluid, Newtonian fluid and Bingham fluid are obtained as special cases from this model.     

Hydromagnetic flow near an accelerated plate with suction

The two-dimensional unsteady flow of a conducting viscous incompressible fluid past, an infinite flat plate with uniform suction, is considered in the presence of a uniform magnetic field. For a constant time, it is shown that for a given Hartmann numbera, as the cross Reynolds number β (corresponding to the suction velocity of the plate) increases, the velocity at any point of the fluid decreases and the skin friction at the plate increases. The results also hold good for a given β, asa increases if the magnetic lines of force are fixed relative to the fluid and are just opposite for the magnetic lines of force fixed relative to the plate.     

Falkner-Skan equation for flow past a stretching surface with suction or blowing: Analytical solutions

The simultaneous effects of suction and injection on tangential movement of a nonlinear power-law stretching surface governed by laminar boundary layer flow of a viscous and incompressible fluid beneath a non-uniform free with stream pressure gradient is considered. The self-similar flow is governed by Falkner-Skan equation, with transpiration parameter γ, wall slip velocity λ and stretching sheet (or pressure gradient) parameter β. The exact solution for β = −1 and three closed form asymptotic solutions for β large, large suction γ, and λ → 1 have also been presented. Dual solutions are found for β = −1 for each value of the transpiration parameter, including the non-permeable surface, for each prescribed value of the wall slip velocity λ. The large β asymptotic solution also dual with respect to wall slip velocity λ, but do not depend on suction and blowing. The critical values of γ, β and λ are obtained and their significance on the skin friction and velocity profiles is discussed. An approximate solution by integral method for a trial velocity profile is presented and results are compared with the exact solutions.     

Flow and heat transfer of a non-Newtonian fluid past a stretching sheet with partial slip

The entrained flow and heat transfer of a non-Newtonian third grade fluid due to a linearly stretching surface with partial slip is considered. The partial slip is controlled by a dimensionless slip factor, which varies between zero (total adhesion) and infinity (full slip). Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. The issue of paucity of boundary conditions is addressed and an effective second order numerical scheme has been adopted to solve the obtained differential equations even without augmenting any extra boundary conditions. The important finding in this communication is the combined effects of the partial slip and the third grade fluid parameter on the velocity, skin-friction coefficient and the temperature field. It is interesting to find that the slip and the third grade fluid parameter have opposite effects on the velocity and the thermal boundary layers.     

Some remarks on permutation designs

A definition of isomorphism of two permutation designs is proposed, which differs from the definition in Bandt [J. Combinatorial Theory Ser. A21 (1976), 384–392]. The proposed definition has the (generally required) property that the allowed permutations always transform a permutation design into a permutation design. It is shown that the n permutation designs coming from the partitioning of S_n into permutation designs, as constructed in Bandt [J. Combinatorial Theory Ser. A21 (1976), 384–392] are all isomorphic. Further we find that this modified definition does not increase the number of nonisomorphic (6, 4) permutation designs. The same investigation showed that one of the designs, claimed to be a (6, 4) permutation design in [J. Combinatorial Theory Ser. A21 (1976), 384–392], is actually not a (6, 4) permutation design.     

Simultaneous effects of partial slip and thermal-diffusion and diffusion-thermo on steady MHD convective flow due to a rotating disk

The purpose of present research is to derive analytical expressions for the solution of steady MHD convective and slip flow due to a rotating disk. Viscous dissipation and Ohmic heating are taken into account. The nonlinear partial differential equations for MHD laminar flow of the homogeneous fluid are reduced to a system of five coupled ordinary differential equations by using similarity transformation. The derived solution is expressed in series of exponentially-decaying functions using homotopy analysis method (HAM). The convergence of the obtained series solutions is examined. Finally some figures are sketched to show the accuracy of the applied method and assessment of various slip parameter γ, magnetic field parameter M, Eckert Ec, Schmidt Sc and Soret Sr numbers on the profiles of the dimensionless velocity, temperature and concentration distributions. Validity of the obtained results are verified by the numerical results.