Non-orthogonal stagnation point flow towards a stretching sheet

The steady flow of a viscous and incompressible fluid impinging at some angle of incidence on a stretching sheet is studied. It is shown that the stream function splits into a Hiemenz and a tangential component. Numerical solutions of the relevant functions as well as the structure of the flow field are presented and discussed. It is found that the free stream obliqueness is the shift of the stagnation point toward the incoming flow and it depends on the inclination angle.     

Influence of the velocity gradient on the stagnation point heating in hypersonic flow

In a number of experimental and numerical publications a deviation has been found between the measured or computed stagnation point heat flux and that given by the theory of Fay and Riddell. Since the formula of Fay and Riddell is used in many applications to yield a reference heat flux for experiments performed in wind tunnels, for flight testing and numerical simulations, it is important that this reference heat flux is as accurate as possible. There are some shortcomings in experiments and numerical simulations which are responsible in some part for the deviations observed. But, as will be shown in the present paper, there is also a shortcoming on the theoretical side which plays a major role in the deviation between the theoretical and experimental/numerical stagnation point heat fluxes. This is caused by the method used so far to determine the tangential velocity gradient at the stagnation point. This value is important for the stagnation point heat flux, which so far has been determined by a simple Newtonian flow model. In the present paper a new expression for the tangential velocity gradient is derived, which is based on a more realistic flow model. An integral method is used to solve the conservation equations and, for the stagnation point, yields an explicit solution of the tangential velocity gradient. The solution achieved is also valid for high temperature flows with real gas effects. A comparison of numerical and experimental results shows good agreement with the stagnation point heat flux according to the theory of Fay and Riddell, if the tangential velocity gradient is determined by the new theory presented in this paper.This article was processed by the author using theLAT_EX style filepljour2 from Springer-Verlag.     

Microfluidic-based single cell trapping using a combination of stagnation point flow and physical barrier

Single cell trapping in vitro by microfluidic device is an emerging approach for the study of the rela-tionship between single cells and their dynamic biochemical microenvironments. In this paper, a hydrodynamic-based microfluidic device for single cell trapping is designed using a combination of stagnation point flow and physical barrier. The microfluidic device overcomes the weakness of the tra-ditional ones, which have been only based upon either stag-nation point flows or physical barriers, and can conveniently load dynamic biochemical signals to the trapped cell. In addi-tion, it can connect with a programmable syringe pump and a microscope to constitute an integrated experimental system. It is experimentally verified that the microfluidic system can trap single cells in vitro even under flow disturbance and con-veniently load biochemical signals to the trapped cell. The designed micro-device would provide a simple yet effective experimental platform for further study of the interactions between single cells and their microenvironments.     

Mixed convection stagnation point flow of a micropolar fluid towards a stretching sheet

The mixed convection two-dimensional boundary layer flow of a micropolar fluid near the stagnation point on a stretching vertical sheet is investigated. The stretching velocity and the surface temperature are assumed to vary linearly with the distance from the stagnation point. The transformed ordinary differential equations are solved numerically for some values of the parameters involved using a finite-difference scheme known as the Keller-box method. The features of the flow and heat transfer characteristics are analyzed and discussed. Both assisting and opposing flows are considered. Results are presented in terms of the skin friction coefficient and the local Nusselt number with selections of velocity, microrotation and temperature profiles. Dual solutions are found to exist for the opposing flow.     

Stagnation point flow of a micropolar fluid towards a stretching sheet

The steady two-dimensional stagnation point flow of an incompressible micropolar fluid over a stretching sheet when the sheet is stretched in its own plane with a velocity proportional to the distance from the stagnation point, has been studied in this paper. The resulting equations of non-linear ordinary coupled differential equations are solved numerically using the Keller-box method. The results obtained for velocity, microrotation and skin friction are shown in tables and graphs. Comparison with the recent results of Mahapatra and Gupta {Heat Mass Transfer 38 (2002) 517} for the corresponding problem of a viscous fluid (K=0) has been done and it has been shown that the results are in excellent agreement.     

Viscoelastic stresses in the stagnation flow of a dilute polymer solution

In this paper, we consider viscoelastic stresses T₁₁, T₁₂ and T₂₂ arising in the stagnation flow of a dilute polymer solution; in particular, we consider an upper convected Maxwell (UCM) fluid. We present exact solutions to the coupled partial differential equations describing the viscoelastic stresses and deduce the results for the stress T₂₂ of Becherer et al. [P. Becherer, A.N. Morozov, W. van Saarloos, Scaling of singular structures in extensional flow of dilute polymer solutions, J. Non-Newtonian Fluid Mech. 153 (2008) 183–190]. As we considered the viscoelastic stresses over two spatial variables, we are able to study the effect of variable boundary data at the inflow. As such, our results are applicable to a wider range of fluid flow problems.     

MHD flow in the vicinity of the stagnation point in a purely azimuthal magnetic field

Axisymmetric MHD flow in the vicinity of the stagnation point in the presence of a purely azimuthal nonhomogeneous magnetic field B {0, B, 0} is studied. This problem belongs to the class of MHD problems whose solutions are known as solutions of the layer type [1]. This class also includes, in particular, the classical exact solutions of the Navier-Stokes equations.The approximate solutions of the analogous MHD problems for the limiting cases of large and small values of the diffusion number ==/ have been considered in [2–5]. In this case it is possible to divide the flow into the so-called viscous and current layers, for each of which the approximate equations, simpler than the exact equations, are solved numerically or in quadratures. Using this technique it is possible to avoid the basic mathematical difficulty, which is that the sought solution of the boundary-value problem must be selected from a family of two-parameter solutions. The approximate method permits dividing the problem into two stages (corresponding to the two boundary layers) in each of which one unknown parameter is determined (in place of their simultaneous determination by direct integration of the basic equations).The drawback of the approximate methods [2–5] is their nonapplicability in the most interesting case, when the thicknesses of the current and viscous layers are of comparable magnitude, i. e., when the kinematic and magnetic viscosities ( and ) are quantities of the same order. We should also note the poor accuracy of the methods in the framework of the considered approximations for a comparatively large volume of the calculations required, which, in turn, prevents obtaining more exact solutions.The present paper presents a numerical integration of the equations describing MHD flow in the vicinity of the stagnation point over a wide range of S and numbers (Alfvén and diffusion numbers), without the assumption of their smallness, with preliminary determination of the unknowns at the zero of the derivatives of the sought functions with the aid of the method of asymptotic integration.A critical value of the Alfvén number is found, for which the retardation of the fluid by the magnetic field (for the first considered configuration of the magnetic field) at the wall is so intense that the friction vanishes everywhere on the surface of the solid body. It is also found that with further increase of the number S a region of reverse flow appears near the wall, which is separated from the remaining flow by a plane on which the z-component of the velocity is equal to zero.     

Mixed convection unsteady stagnation point flow over a stretching sheet with heat transfer in the presence of variable free stream

An analysis is performed for the unsteady mixed convection flow of an incompressible viscous fluid about a stagnation point on a stretching sheet in the presence of a variable free stream. The equations of motion and energy are transformed into the ordinary differential equations by using similarity transformations. Homotopy analysis method is used for the solution of the governing problem. The results have been discussed by plots. The present values of the function are shown very close to the previous limiting solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

An application of interacting shear flows theory: exact solution for unsteady oblique stagnation point flow

An analytical solution of the governing equations of the interacting shear flows for unsteady oblique stagnation point flow is obtained. It has the same form as that of the exact solution obtained from the complete NS equations and physical analysis and relevant discussions are then presented.The English text was polished by Yunming Chen.     

Slip effects on unsteady mixed convection of hybrid nanofluid flow near the stagnation point

The unsteady mixed convection of the Al2O3-Cu/H2O hybrid nanofluid flow near the stagnation point past a vertical plate is analyzed. The bvp4c technique is used to solve the resulting ordinary differential equations. The combined effects of the velocity and thermal slip are addressed. The effects of different relevant physical parameters are studied numerically. The results show that the heat transfer rate is reduced when the volume fraction of the nanoparticles increases, while the unsteadiness...     

Nonequilibrium viscous flow of a multicomponent gas in the vicinity of the stagnation point of a blunt body

This paper gives the results of numerical calculations characterizing the effect of variation of the shock layer parameters on the heat transfer in the case of a multicomponent nonequilibrium-dissociating air on a wall with finite catalycity in the vicinity of the stagnation point of a spherical blunt body. Similar results for the case of a binary mixture can be found in [1–3]. It is shown that a consideration of the variation of the parameters in the nonequilibrium shock layer leads to a significant increase in heat flux to the noncatalytic wall in comparison with the theory of an asymptotically thin nonequilibrium boundary layer with equilibrium parameters on its outer boundary.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 144–147, March–April, 1971.The author thanks V. V. Lunev for useful comments in the discussion of this work.     

Boundary layer in the vicinity of the stagnation point of a body of axial symmetry in a turbulent stream

The flow in the boundary layer in the vicinity of the stagnation point of a flat plate is examined. The outer stream consists of turbulent flow of the jet type, directed normally to the plate. Assumptions concerning the connection between the pulsations in velocity and temperature in the boundary layer and the average parameters chosen on the basis of experimental data made it possible to obtain an isomorphic solution of the boundary layer equations. Equations are obtained for the friction and heat transfer at the wall in the region of gradient flow taking into account the effect of the turbulence of the impinging stream. It is shown that the friction at the wall is insensitive to the turbulence of the impinging stream, while the heat transfer is significantly increased with an increase in the pulsations of the outer flow. These properties are confirmed by the results of experimental studies [1–4].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 83–87, September–October, 1973.     

局部狭窄血管中血液振荡流的速度和切应力

本文建立一种分析局部缓慢狭窄血管中血液振荡流的数学模型，给出了血液的轴向流速，径向流速和切应力的包含压力梯度项的解析表达式，并讨论了血管内由局部狭窄引起的压力梯度沿轴向变化的规律。文章以局部余弦狭窄为例进行数值计算，详细讨论上游均匀管段压力梯度的定常部分和不同次谐波对狭窄管段内流速和切应力的影响。数值结果表明，与均匀管情况相比，在狭窄段内，血液振荡流轴向流速无论平均值还是脉动幅值均明显增大，且径向流速不再为零。但径向流速仍远小于轴向流速。同时，切应力也不再仅由轴向流速梯度提供，径向流速梯度也将产生切应力，但是在计算管壁切向上的切应力时，径向流速梯度的贡献仍相当大。与均匀管管壁切应力沿流运方向保持恒定不同。狭窄管管壁切应力（平均值和脉动值）将随着狭窄高度的增大而增大，在狭窄最大高度处达到最大，因而沿流动方向产生了较大的切应力梯度。     

滑滚表面间的流体膜壁面滑移和流体动力学

经典雷诺润滑理论建立在无壁面滑移的假设基础之上。近年来许多试验报告了发生在流体膜流动的壁面滑移证据。本文研究了两固体表面间的流体膜流动特性和流体动力学,发现壁面滑移显著影响膜的流体动力学问题,流体动压力不仅受黏度和几何间隙的影响,而且还由壁面滑移和表面运动强力控制,通过控制表面的吸附性质,甚至可以得到零摩擦表面。另一方面,如果两个表面具有相同的滑移特性,存在一个临界滑动速度使得流体动压效应完全消失;但是在纯滚动条件下,即使界面极限剪应力很小,仍然有相当可观的流体动压效应。     

水平旋转空腔环流的壁面应力

通过对典型的水平旋转内消能泄洪洞空腔环流的试验观测，研究了其壁面应力的变化规律。空腔环流的壁面压强在水平洞的起始段壁面压强急剧减小然后回升，具有过渡段的性质，沿程波状减小，符合对数变化规律，但不同的流态，对数律的参数的变化是不同的：内界面相对压强Po/Pwz。在淹沿流流态时，随(H-h)/h的变化率显著不同，在吸允流流态时却基本相同；壁面切应力沿z的变化规律为先急剧减小，随后缓慢减小至零，主要与环流特性有密切的关系。     

Mixed Convection Boundary-Layer Flow Near the Stagnation Point on a Vertical Surface in a Porous Medium: Brinkman Model with Slip

The steady boundary-layer flow near the stagnation point on an impermeable vertical surface with slip that is embedded in a fluid-saturated porous medium is investigated. Using appropriate similarity variables, the governing system of partial differential equations is transformed into a system of ordinary differential equations. This system is then solved numerically. The features of the flow and the heat transfer characteristics for different values of the governing parameters, namely, the Darcy–Brinkman, Γ, mixed convection, λ, and slip, γ, parameters, are analysed and discussed in detail for the cases of assisting and opposing flows. It is found that dual solutions exist for assisting flows, as well as those usually reported in the literature for opposing flows. A stability analysis of the steady flow solutions encountered for different values of the mixed convection parameter λ is performed using a linear temporal stability analysis. This analysis reveals that for γ  =  0 (slip absent) and Γ  =  1 the lower solution branch is unstable while the upper solution branch is stable.     

Turbulence enhancement of stagnation point heat transfer on a body of revolution

Enhancement, by free stream turbulence, of convective heat transfer to the stagnation region of a hemispherical-nosed cylinder has been studied. Increases in heat transfer were found to depend primarily on the Reynolds number and turbulence intensity of the free stream, experimental results being most successfully correlated on a NuRe^?0.5 versus TuRe^0.5 basis. Flow visualization studies have demonstrated the validity of a phenomenological model of the enhancement process, predictions of this theory showing good agreement with experimental results. The effect of free stream turbulence on the stagnation point velocity gradient has also been evaluated.     

Calculation of convective heat flux in the vicinity of the stagnation point for partially ionized gas flow past a body

From numerical solutions of the boundary layer equations for a four-component gas mixture (E, N⁺, N₂, and N) with gas injection, approximate formulas for the heat flux as a function of the variation of λρ/c_p and h^* across the boundary layer and the magnitude of the objection are obtained (λ is the thermal conductivity of the mixture,ρ is density, c_p is the specific heat, and h^* is the enthalpy of the ideal gas state of the mixture). An effective ambipolar diffusion coefficient D^(a)(i) is introduced, making possible finite formulas for the convective heat fluxes in the “frozen” boundary layer. We study the behavior of these coefficients within the boundary layer. A formula is obtained for convective heat flux to the wall from partially ionized air for a nine-component mixture (E, O⁺, N⁺, NO⁺, O, N, NO, O₂ N₂). Even for simpler four-component gas model three effective ambipolar diffusion coefficients are necessary: $$\begin{gathered} D^{(a)} (A) = D (A, M) D^{(a)} (I) = 2D (A, M), \hfill \\ D^{(a)} (M) = [ 1 + c_e (I)] D(A, M). \hfill \\ \end{gathered} $$ Here D(A, M) is the binary diffusion coefficient of the atoms into molecules, and c_e(I) is the ion concentration at the outer edge of the boundary layer. The assumption of an infinitely large charge-exchange cross section and the other simplifying assumptions used in [1] lead to overestimation of the magnitude of the dimensionless heat flux by 7–15% for the “frozen” boundary layer case.     

Boundary layer flow at a three-dimensional stagnation point in power-law non-Newtonian fluids

Similarity solutions for the three-dimensional flow and heat transfer of a power-law fluid near a stagnation point of an isothermal surface are presented. The results of the numerical integrations are given in tables and shown on graphs for some different values of the power-law index n, geometric parameter c, and the Prandtl number Pr. Whenever possible, these results are compared with available analytical solutions and found to be highly accurate.