Similarity solutions in free convection boundary-layer flows adjacent to vertical permeable surfaces in porous media: II prescribed surface heat flux

The equation which governs the similarity solution for free convection boundary-layer flow along a vertical permeable surface with prescribed surface heating and mass transfer rate is discussed. The solution is seen to depend on two non-dimensional parameters;m, the power-law exponent, and γ, the mass transfer parameter. It is shown that solutions exist for allm>?1 for γ>0 (fluid injection) whereas for γ<0 (fluid withdrawal), solution exist form>m ₀(γ), wherem ₀ is determined as a function of γ. Solutions for large mass transfer rates are obtained, for both γ>0 and γ<0. For γ>0 the form of the asymptotic solution for γ large is seen to depend on the value ofm. Solutions form large are derived, these are seen to be different depending on whether γ is positive or negative.     

Characterization of micellar structure dynamics for a drag-reducing surfactant solution under shear: normal stress studies and flow geometry effects

Some surfactant solutions have been observed to exhibit a strong drag reduction behavior in turbulent flow. This effect is generally believed to result from the formation of large cylindrical micelles or micellar structures. To characterize and understand better these fluids, we have studied the transient rheological properties of an efficient drag-reducing aqueous solution: tris (2-hydroxyethyl) tallowalkyl ammonium acetate (TTAA) with added sodium salicylate (NaSal) as counter ion. For a 5/5 mM equimolar TTAA/NaSal solution, there is no measurable first normal stress difference (N ₁) immediately after the inception of shear, but N ₁ begins to increase after a well-defined induction time — presumably as shear-induced structures (SIS) are formed — and it finally reaches a fluctuating plateau region where its average value is two orders of magnitude larger than that of the shear stress. The SIS buildup times obtained by first normal stress measurements were approximately inversely proportional to the shear rate, which is consistent with a kinetic process during which individual micelles are incorporated through shear into large micellar structures. The SIS buildup after a strong preshear and the relaxation processes after flow cessation were also studied and quantified with first normal stress difference measurements. The SIS buildup times and final state were also found to be highly dependent on flow geometry. With an increase in gap between parallel plates, for example, the SIS buildup times decreased, whereas the plateau viscosity increased.     

Flow of viscoelastic fluids through a porous channel—I

The flow of viscoelastic fluids through a porous channel with one impermeable wall is computed. The flow is characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. Three solutions are developed: (i) an exact numerical solution, (ii) a perturbation solution for small R, the cross-flow Reynold's number and (iii) an asymptotic solution for large R. The results from exact numerical integration reveal that the solutions for a non-Newtonian fluid are possible only up to a critical value of the viscoelastic fluid parameter, which decreases with an increase in R. It is further demonstrated that the perturbation solution gives acceptable results only if the viscoelastic fluid parameter is also small. Two more related problems are considered: fluid dynamics of a long porous slider, and injection of fluid through one side of a long vertical porous channel. For both the problems, exact numerical and other solutions are derived and appropriate conclusions drawn.     

Modes of bubble growth in the slow-formation regime of nucleate pool boiling

The shape and size of a bubble formed slowly on a sharp- or round-edged orifice are derived with the help of a new analytical solution for the bubble profile. Two modes of formation are distinguished, depending on the natural contact angle, ?₀: bubble confined to the orifice (?₀ small); bubble spreading beyond the orifice (?₀ large: Fritz mode). The limits of the slow-formation regime in mucleate pool boiling are estimated, involving an assessment of the influences of liquid inertia, viscosity and surface-tension gradients.“Slow” formation is predicted for large cavities or high pressures and this is borne out by data for water. The Fritz mode of growth, however, is seen to be suppressed.     

TTS in LAOS: validation of time-temperature superposition under large amplitude oscillatory shear

The validation of time-temperature superposition of non-linear parameters obtained from large amplitude oscillatory shear is investigated for a model viscoelastic fluid. Oscillatory time sweeps were performed on a 11?wt.% solution of high molecular weight polyisobutylene in pristane as a function of temperature and frequency and for a broad range of strain amplitudes varying from the linear to the highly non-linear regime. Lissajous curves show that this reference material displays strong non-linear behaviour when the strain amplitude is exceeding a critical value. Elastic and viscous Chebyshev coefficients and alternative non-linear parameters were obtained based on the framework of Ewoldt et al. (J Rheol 52(6):1427?C1458, 2008) as a function of temperature, frequency and strain amplitude. For each strain amplitude, temperature shift factors a _T (T) were calculated for the first order elastic and viscous Chebyshev coefficients simultaneously, so that master curves at a certain reference temperature T _ref were obtained. It is shown that the expected independency of these shift factors on strain amplitude holds even in the non-linear regime. The shift factors a _T (T) can be used to also superpose the higher order elastic and viscous Chebyshev coefficients and the alternative moduli and viscosities onto master curves. It was shown that the Rutgers-Delaware rule also holds for a viscoelastic solution at large strain amplitudes.     

Singular perturbation of nonlinear vector boundary value problem

In this paper we study the perturbed boundary value problem of the form dx/dt=f(x,y,t;ε), εdy/dt=g(x,y,t;ε), a_1(ε)x(0,ε)+a_2(ε)y(0,ε)=a(ε) b_1(ε)x(1,ε)+εb_2(ε)y(1,ε)=β(ε)in whichx,f,β∈E~m, y,g,a∈E~n, 0<ε(?)1and a_1(ε), a_2(ε), b_2(ε)and b_2(ε) are matrices of the appropriate size. Under the condition that g_y(t) is nonsingular and other suitable restrictions, the existence of the solution is proved, the asymptotic expansion of solution of order n is constructed, and the remainder term is estimated.     

基于边界值方法的微分动力系统数值计算方法

对高维非线性初值问题,微分求积法在每一步的积分过程中需要求解一个更高维的非线性方程组,因而计算量巨大。基于微分求积法与边界值方法两者之间的关系,可以将广义向后差分方法和扩展的隐式梯形积分方法看作是经典微分求积法的稀疏表达形式。将广义向后差分方法以及扩展的隐式梯形积分方法这两类边界值方法应用于微分动力系统的数值计算,提出了一类新的数值计算方法。理论分析及算例结果表明,对高维非线性微分初值问题的数值计算,本文方法相对于经典的微分求积法具有更高的计算效率。     

SINGULAR PERTURBATION OF NONLINEAR BOUNDARY VALUE PROBLEMS

In this paper we consider the boundary value problem where ε.μ are two positive parameters. Under f_y≤-k<0 and other suitable restrictions, there exists a solution and it satisfied where y_(0,0)(x) is solution of reduced problem while y_i-j,j(x)(j=0,1,...,i;i=1,2,...,m) can be obtained successively from certain linear equations.     

The unsteady heat transfer due to a heat source in an MHD stretching sheet flow

The time evolution in the temperature field resulting from the sudden introduction of a heat source into the already fully established steady MHD flow of an electrically conducting fluid past a linearly stretching isothermal surface is considered. The problem is shown to be fully described by two dimensionless parameters, a modified magnetic field strength ?? and a heat source strength Q. Numerical solutions of the initial-value problem show that there is a critical value Q _c of the parameter Q, dependent on ??, such that, for Q<Q _c , the solution approaches a steady state at large times and, for Q>Q _c , the solutions grows exponentially large as time increases. This growth rate is determined through an eigenvalue problem which also determines the critical value Q _c . The limits of Q _c for both small and large values of ?? are discussed.     

An exact solution for a model of pressure-dependent plasticity in an un-steady plane strain process

An incompressible material obeying a pressure-dependent yield condition is confined between two planar plates which are inclined at an angle 2α. The plates intersect in a hinged line and the angle α slowly decreases from an initial value. An initial/boundary value problem for the flow of the material is formulated and solved for the stress and the velocity fields, the solution being in closed form. The material is assumed to obey a special case of the double slip and rotation model, which generalises the classical plastic potential model and is also a variant of the double shearing model. The solution for the velocity field may exhibit sliding or sticking at the plates. Solutions which exhibit sticking may have a rigidly rotating zone in the region adjacent to the plates. It is shown that sliding occurs when the value of α is less than a certain critical value α_c ; that sticking occurs without a rigid zone if α exceeds or equals α_c but is less than a second critical value α₀; and that sticking with a rigid zone adjacent to the plates occurs if α exceeds α₀. The values of α_c and α₀ coincide for a certain range of model parameters. Solutions which exhibit sliding are singular. Qualitative features of the solution found are compared with those of the solution for the classical plastic potential model.     

Matched asymptotic expansions in nonlinear fracture mechanics—III. In-plane loading of an elastic perfectly-plastic symmetric specimen

The matched expansion method, introduced by the authors in two earlier papers (1976) devoted to mode III loading, is applied to the practically important case of mode I loading of a symmetric specimen. The method allows the linear elastic far-field to be considered separately from the elasto-plastic near-tip field, except for coupling through a set of parameters that are determined explicitly in the matching. The effects of the plasticity are thus found, once and for all, from the solution of a set of standard elasto-plastic problems for a semi-infinite crack in an infinite body, whose properties may be tabulated. The solution for any particular specimen geometry and loading then follows from a small set of linear elastic solutions for the specimen, which define, through coefficients γ_ij appearing in their near-tip expansions, all the parameters in the “inner” and “outer” solutions. The effects of plasticity appear in these parameters only through a set of constants C_ti that define the far-field expansions of the “inner” (near-tip) solutions: they are material constants, depending upon the constitutive relation for the material, but not upon specimen geometry and loading. The J-integral, being obtainable from the far-field, is expressed as an explicit asymptotic series in the loading parameter ε, whose coefficients are given as functions of the “elastic” parameters γ_ij and the material constants C_i. It is demonstrated that a plastic-zone correction term, r_y, can be chosen to yield a two-term asymptotic expansion for J; the value of r_y depends upon the yielding model only through the constant C₁.The Dugdale (1960) model of yielding is treated, as a simple example for which all calculations can be performed analytically, and for which exact solutions are available for comparison.Finally, the near-tip solutions are constructed for a material obeying the Mises yield criterion and associated flow-rule, using a specially developed finite element program. The first eight of the constants C_i are tabulated, which suffice to define the J-integral up to terms of order ε⁶ (where ε is a loading parameter) and some representative near-tip features are displayed graphically. The computed value of C₁ shows that the conventionally adopted value for the plastic-zone correction r_y is too large by a factor of roughly 2.8, if it is to yield a genuine asymptotic estimate for J. As an example, the “elastic” parameters γ_ij are found, from a boundary collocation program, for a centre-cracked square plate subjected to tensile loading; and a plot of J versus load, and the plastic-zone shape at a particular load level, are displayed.     

Axisymmetric membrane in adhesive contact with rigid substrates: Analytical solutions under large deformation

The large deformation of an elastic axisymmetric membrane in adhesive contact with a rigid flat punch is studied. Detachment of membrane is analyzed using a critical energy release rate criterion. Two types of incompressible hyperelastic material models are considered: neo-Hookean and a class of materials whose elastic energy density functions are independent of the trace of the Cauchy–Green tensor (I₂-based material). We also include pre-stretch in our formulation and study the stability of detachment process. Closed form analytical solutions for the membrane stresses, deformed profiles and energy release rate are obtained in the regime of large longitudinal stretch. For the I₂-based material, we discover an interesting “pinching” instability where the contact angle suddenly increases in a displacement controlled test. The region of validity of our analytical solutions is determined by comparing them with numerical solutions of the governing equations. We found that the accuracy of our solution improves with pre-stretch; for pre-stretch ratios greater than 1.3, our analytical solution also works well in the small deformation regime.     

Steady-state solutions to Bentsen's equation

Based on experimentally observed phenomena and the physical requirement of a unique value of saturation at any location within a porous medium, a restrictive condition for a valid solution to Bentsen's equation is derived: ?² f/?S ²≤0. The steady-state solution to Bentsen's equation is shown to be identical to the Buckley-Leverett solution to the displacement equation, and the steady-state solution for the fractional flow is shown to be independent of the capillary number. It is proved that under steady-state conditions, the capillary term of the fractional flow equation in the frontal region does not depend on the capillary number. Therefore, the unrealistic triple-valued saturation profile of the original Buckley-Leverett solution resulted because the capillary term was in-appropriately neglected. The break-through recovery efficiency,Τ _bt , is shown to be a function of the capillary number. As the capillary number decreases, the break-through recovery efficiency increases and the maximum value ofΤ _bt can be obtained asN _c → 0. The Buckley-Leverett solution is the limiting solution asN _c → 0.     

Arbitrarily oriented crack near interface in piezoelectric bimaterials

Arbitrarily oriented crack near interface in piezoelectric bimaterials is considered. After deriving the fundamental solution for an edge dislocation near the interface, the present problem can be expressed as a system of singular integral equations by modeling the crack as continuously distributed edge dislocations. In the paper, the dislocations are described by a density function defined on the crack line. By solving the singular integral equations numerically, the dislocation density function is determined. Then, the stress intensity factors (SIFs) and the electric displacement intensity factor (EDIF) at the crack tips are evaluated. Subsequently, the influences of the interface on crack tip SIFs, EDIF, and the mechanical strain energy release rate (MSERR) are investigated. The J-integral analysis in piezoelectric bimaterals is also performed. It is found that the path-independent of J₁-integral and the path-dependent of J₂-integral found in no-piezoelectric bimaterials are still valid in piezoelectric bimaterials.     

Near-field of coaxial jets with large density differences

 The paper describes an experimental investigation of coaxial jets with large density differences. Measurements by various techniques show that density effects on the flow dynamics are taken into account to first order by considering the specific outer to inner jet momentum flux ratio M and not separately the density and velocity ratios. A regime of recirculation occurs for M higher than a critical value (M _c ≈50). For a given value of M, however, the position of the recirculation bubble is slightly shifted in the upstream direction for density ratios much smaller than one. An unexpected result is obtained for an extremely low density ratio: the onset of recirculation occurs for a significantly higher value of M (100<M _c <140) for helium/SF₆ jets (density ratio : 0.028). Received: 1 October 1997 / Accepted: 8 March 1998     

Stress in elastic plates reinforced by fibres lying in concentric circles

We Consider fibre-reinforced elastic plates with the reinforcement continuously distributed in concentric circles ; such a material is locally transversely isotropic, with the circumferential direction as the preferred direction. For an annulus bounded by concentric circles, the exact solution of the traction boundary value problem is obtained. When the extension modulus in the fibre direction is large compared to other extension and shear moduli, the material is strongly anisotropic. For this case a simpler approximate solution is obtained by treating the material as inextensible in the fibre direction. It is shown that the exact solution reduces to the inextensible solution in the appropriate limit. The inextensible theory predicts the occurrence of stress concentration layers in which the direct stress is infinite ; for materials with small but finite extensibility these layers correspond to thin regions of high stress and high stress gradient. A boundary layer theory is developed for these regions. For a typical carbon fibre-resin composite, the combined boundary layer and inextensible solutions give an excellent approximation to the exact solution. The theory is applied to the problem of an isotropic plate, under uniform stress at infinity, containing a circular hole which is strengthened by the addition of an annulus of fibre-reinforced material.     

Exact and asymptotic solutions of the Navier-Stokes equations in a fluid layer between plates approaching and moving apart from one another

Exact solutions of the Navier-Stokes equations are investigated in the layer between parallel plates the distance between which changes proportionally to the square root of time. At the boundaries of the plates the no-slip condition is assigned. For approaching plates a countable family of exact solutions each of which continuously depends on the Reynolds number is obtained. At a sufficiently large Reynolds number, near the boundary a counterflow is formed: the velocity is directed oppositely to the average velocity. On the basis of the exact solution obtained, relative errors are calculated for the asymptotic theories of Reynolds lubricating layer and Prandtl boundary layer.     

Singular perturbation of boundary value problem for a vector fourth order nonlinear differential equation

We study the vector boundary value problem with boundary perturbations: ε~2y~((4))=f(x,y,y″,ε, μ) ( μ<χ<1-μ) y(χ,ε,μ)l_(χ-μ)= A_1(ε,μ), y(χ,ε,μ)l_(χ-1-μ)=B_1(ε,μ) y″(χ,ε,μ)l_(χ-μ)=A_2(ε,μ),y″(χ,ε,μ)l_(χ-1-μ)=B_2(ε,μ)where yf, A_j and B_j (j=1,2) are n-dimensional vector functions and ε,μ are two small positive parameters. This vector boundary value problem does not appear to have been studied, although the scalar boundary value problem has been treated. Under appropriate assumptions, using the method of differential inequalities we find a solution of the vector boundary value problem and obtain the uniformly valid asymptotic expansions.     

Some asymptotic results concerning the buckling of a spherical shell of arbitrary thickness

For a spherical shell of arbitrary thickness which is subjected to an external hydrostatic pressure, symmetrical buckling takes place at a value of μ₁ which depends on and the mode number, where A₁ and A₂ are the undeformed inner and outer radii, and μ₁ is the ratio of the deformed inner radius to the undeformed inner radius. In the large mode number limit, we find that the dependence of μ₁ on has a boundary layer structure: it is a constant over almost the entire region of and decreases sharply from this constant value to unity as tends to unity (the thin-shell limit). Simple asymptotic expressions for the bifurcation condition are obtained. The classical result for thin shells is recovered directly from the equations of finite elasticity, and an asymptotic critical neutral curve (which envelops the neutral curves corresponding to different mode numbers) is obtained.     

Large free vibrations of a beam carrying a moving mass

The focus of this work is to develop a technique to obtain numerical solution over a long range of time for non-linear multi-body dynamic systems undergoing large amplitude motion. The system considered is an idealization of an important class of problems characterized by non-linear interaction between continuously distributed mass and stiffness and lumped mass and stiffness. This characteristic results in some distinctive features in the system response and also poses significant challenges in obtaining a solution.

In this paper, equations of motion are developed for large amplitude motion of a beam carrying a moving spring–mass. The equations of motion are solved using a new approach that uses average acceleration method to reduce non-linear ordinary differential equations to non-linear algebraic equations. The resulting non-linear algebraic equations are solved using an iterative method developed in this paper. Dynamics of the system is investigated using a time-frequency analysis technique.