Heat transfer in oblique stagnation-point flow of an incompressible viscous fluid towards a stretching surface

Steady two-dimensional oblique stagnation-point flow of an incompressible viscous fluid over a flat deformable sheet is investigated when the sheet is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. It is shown that the flow has a boundary layer structure for values of a/c (> 1), where ax+2by and cx are the x-component of the free stream velocity and the stretching velocity of the plate respectively, x being the distance from the stagnation-point. On the other hand when a/c < 1, the flow has an inverted boundary layer structure. It is also observed that the velocity at a point increases with increase in the free stream shear. For a fixed value of a/c, the streamlines becomes more and more oblique towards the left of the stagnation-point with increase in b/c where b > 0. On the other hand the streamlines become increasingly oblique to the right of the stagnation-point with increase in |b/c| when b < 0. For a fixed value of the Prandtl number Pr, temperature at a point decreases with increase in a/c. Further for a given value of a/c, the surface heat flux increases with increase in Pr.     

Effect of material nonhomogeneities on the HRR dominance

This work studies the asymptotic stress and displacement fields near the tip of a stationary crack in an elastic–plastic nonhomogeneous material with the emphasis on the effect of material nonhomogeneities on the dominance of the crack tip field. While the HRR singular field still prevails near the crack tip if the material properties are continuous and piecewise continuously differentiable, a simple asymptotic analysis shows that the size of the HRR dominance zone decreases with increasing magnitude of material property gradients. The HRR field dominates at points that satisfy |α⁻¹ ∂α/∂x_δ|1/r, |α⁻¹ ∂²α/(∂x_δ ∂x_γ)|1/r², |n⁻¹ ∂n/∂x_δ|1/[r|ln(r/A)|] and |n⁻¹ ∂²n/(∂x_δ ∂x_γ)|1/[r²|ln(r/A)|], in addition to other general requirements for asymptotic solutions, where α is a material property in the Ramberg–Osgood model, n is the strain hardening exponent, r is the distance from the crack tip, x_δ are Cartesian coordinates, and A is a length parameter. For linear hardening materials, the crack tip field dominates at points that satisfy |E_tan⁻¹ ∂E_tan/∂x_δ|1/r, |E_tan⁻¹ ∂²E_tan/(∂x_δ ∂x_γ)|1/r², |E⁻¹ ∂E/∂x_δ|1/r, and |E⁻¹ ∂²E/(∂x_δ ∂x_γ)|1/r², where E_tan is the tangent modulus and E is Young’s modulus.     

Crack size and speed interaction characteristics at micro-, meso- and macro-scale

The motivation to examine physical events at even smaller size scale arises from the development of use-specific materials where information transfer from one micro- or macro-element to another could be pre-assigned. There is the growing belief that the cumulated macroscopic experiences could be related to those at the lower size scales. Otherwise, there serves little purpose to examine material behavior at the different scale levels. Size scale, however, is intimately associated with time, not to mention temperature. As the size and time scales are shifted, different physical events may be identified. Dislocations with the movements of atoms, shear and rotation of clusters of molecules with inhomogeneity of polycrystals; and yielding/fracture with bulk properties of continuum specimens. Piecemeal results at the different scale levels are vulnerable to the possibility that they may be incompatible. The attention should therefore be focused on a single formulation that has the characteristics of multiscaling in size and time. The fact that the task may be overwhelmingly difficult cannot be used as an excuse for ignoring the fundamental aspects of the problem.Local nonlinearity is smeared into a small zone ahead of the crack. A “restrain stress” is introduced to also account for cracking at the meso-scale.The major emphasis is placed on developing a model that could exhibit the evolution characteristics of change in cracking behavior due to size and speed. Material inhomogeneity is assumed to favor self-similar crack growth although this may not always be the case. For relatively high restrain stress, the possible nucleation of micro-, meso- and macro-crack can be distinguished near the crack tip region. This distinction quickly disappears after a small distance after which scaling is no longer possible. This character prevails for Mode I and II cracking at different speeds. Special efforts are made to confine discussions within the framework of assumed conditions. To be kept in mind are the words of Isaac Newton in the Fourth Regula Philosophandi:

“Men are often led into error by the love of simplicity which disposes us to reduce things to few principles, and to conceive a greater simplicity in nature than there really is … We may learn something of the way in which nature operates from fact and observation; but if we conclude that it operates in such a manner, only because to our understanding that operates to be the best and simplest manner, we shall always go wrong.”––Isaac Newton

Mixed convection flow of second grade fluid along a vertical stretching flat surface with variable surface temperature

In the present study we have explored the effects of thermal buoyancy on flow of a viscoelastic second grade fluid past a vertical, continuous stretching sheet of which the velocity and temperature distributions are assumed to vary according to a power-law form. The governing differential equations are transformed into dimensionless form using appropriate transformations and then solved numerically. The methods here employed are (1) the perturbation method together with the Shanks transformation, (2) the local non-similarity method with second level of truncation and (3) the implicit finite difference method for values of ξ ( = Gr _x /Re _x ², defined as local mixed convection parameter) ranging in [0, 10]. The comparison between the solutions obtained by the aforementioned methods found in excellent agreement. Effects of the elasticity parameter λ on the skin-friction and heat transfer coefficients have been shown graphically for the fluids having the values of the Prandtl number equal to 0.72, 7.03 and 15.0. Effects of the viscoelastic parameter and the mixed convection parameter, ξ, on the temperature and velocity fields have also been studied. We notice that with the increase in visco-elastic parameter λ, velocity decreases whereas temperature increases and that velocity gradient is higher than that of temperature. On leave of absence from the Department of Mathematics, University of Dhaka, Bangladesh.     

Natural Convection Boundary-Layer Flow in a Porous Medium with Temperature-Dependent Boundary Conditions

The natural convection boundary-layer flow on a surface embedded in a fluid-saturated porous medium is discussed in the case when the wall heat flux is related to the wall temperature through a power-law variation. The flow within the porous medium is assumed to be described by Darcy’s law and the Boussinesq approximation is assumed for the density variations. Two cases are discussed, (i) stagnation-point flow and (ii) flow along a vertical surface. The possible steady states are considered first with the governing partial equations reduced to ordinary differential equations by similarity transformations and these latter equations further transformed to previously studied free-convection problems. This identifies values of the exponent N in the power-law wall temperature variation, N = 3/2 for stagnation-point flows and 3/2 ≤ N ≤ 3 for the vertical surface, where similarity solutions do not exist. Time development for stagnation-point flows is seen to depend on N, for N <  3/2 the steady state is approached at large times, for N ≥ 3/2 a singularity develops at finite time leading to thermal runaway. Numerical solutions for the vertical surface, where the temperature-dependent boundary condition becomes more significant as the solution develops, show that, for N < 3/2, the corresponding similarity solution is approached, whereas for N >  3/2 the solution breaks down at a finite distance along the surface.     

Double diffusive natural convection in a partially heated enclosure with Soret and Dufour effects

The effect of double-diffusive natural convection of water in a partially heated enclosure with Soret and Dufour coefficients around the density maximum is studied numerically. The right vertical wall has constant temperature θ_c, while left vertical wall is partially heated θ_h, with θ_h > θ_c. The concentration in right wall is maintained higher than left wall (C_c < C_h) for case I, and concentration is lower in right wall than left wall (C_h > C_c) for case II. The remaining left vertical wall and the two horizontal walls are considered adiabatic. Water is considered as the working fluid. The governing equations are solved by control volume method using SIMPLE algorithm with QUICK scheme. The effect of the various parameters (thermal Rayleigh number, center of the heating location, density inversion parameter, Buoyancy ratio number, Schmidt number, and Soret and Dufour coefficients) on the flow pattern and heat and mass transfer has been depicted. Comprehensive Nusselt and Sherwood numbers data are presented as functions of the governing parameters mentioned above.     

On some nearly viscometric flows of viscoelastic liquids

Superposition of oscillatory shear imposed from the boundary and through pressure gradient oscillations and simple shear is investigated. The integral fluid with fading memory shows flow enhancement effects due to the nonlinear structure. Closed-form expressions for the change in the mass transport rate are given at the lowest significant order in the perturbation algorithm. The elasticity of the liquid plays as important a role in determining the enhancement as does the shear dependent viscosity. Coupling of shear thinning and elasticity may produce sharp increases in the flow rate. The interaction of oscillatory shear components may generate a steady flow, either longitudinal or orthogonal, resulting in increases in flow rates akin to resonance, and due to frequency cancellation, even in the absence of a mean gradient. An algorithm to determine the constitutive functions of the integral fluid of order three is outlined.Nomenclature A _n Rivlin-Ericksen tensor of order . - A _k Non-oscillatory component of the first order linear viscoelastic oscillatory velocity field induced by the kth wave in the pressure gradient - d Half the gap between the plates - e _x, e _z Unit vectors in the longitudinal and orthogonal directions, respectively - G(s) Relaxation modulus - G History of the deformation - Stress response functional - I() Enhancement defined as the ratio of the frequency dependent part of the discharge to the frequencyindependent part of it at the third order - I ^*() Enhancement defined as the ratio of the increase in discharge due to oscillations to the total discharge without the oscillations - k Power index in the relaxation modulus G(s) - k _i ^–1 Relaxation times in the Maxwell representation of the quadratic shear relaxation modulus (s ₁, s ₂) - m _i ^–1, n _i ^–1 Relaxation times in the Maxwell representations of the constitutive functions ₁(s ₁,s ₂,s ₃) and ₄ (s ₁, s ₂,s ₃), respectively - P Constant longitudinal pressure gradient - p Pressure field - _mx ,⁽³⁾ _nz ,⁽³⁾ Mean volume transport rates at the third order in the longitudinal and orthogonal directions, respectively - ₀,⁽³⁾, ₁,⁽³⁾ Frequency independent and dependent volume transport rates, respectively, at the third order - s = t- Difference between present and past times t and      

Phase portraits and bifurcations of the non-linear oscillator:

Experimental data of two low-density polyethylene (LDPE) melts at 200°C for both shear flow (transient and steady shear viscosity as well as transient and steady first normal stress coefficient) and elongational flow (transient and steady-state elongational viscosity) as published by Pivokonsky et al. (J Non-Newtonian Fluid Mech 135:58–67, 2006) were analysed using the molecular stress function model for broadly distributed, randomly branched molecular structures. For quantitative modelling of melt rheology in both types of flow and in a very wide range of deformation rates, only three nonlinear viscoelastic material parameters are needed: Whilst the rotational parameter, a ₂, and the structural parameter, β, are found to be equal for the two melts considered, the melts differ in the parameter describing maximum stretch of the polymer chains.     

Modelling elongational and shear rheology of two LDPE melts

Experimental data of two low-density polyethylene (LDPE) melts at 200°C for both shear flow (transient and steady shear viscosity as well as transient and steady first normal stress coefficient) and elongational flow (transient and steady-state elongational viscosity) as published by Pivokonsky et al. (J Non-Newtonian Fluid Mech 135:58–67, 2006) were analysed using the molecular stress function model for broadly distributed, randomly branched molecular structures. For quantitative modelling of melt rheology in both types of flow and in a very wide range of deformation rates, only three nonlinear viscoelastic material parameters are needed: Whilst the rotational parameter, a ₂, and the structural parameter, β, are found to be equal for the two melts considered, the melts differ in the parameter describing maximum stretch of the polymer chains.     

Hydromagnetic free convection flow with induced magnetic field effects

An exact solution is presented for the hydromagnetic natural convection boundary layer flow past an infinite vertical flat plate under the influence of a transverse magnetic field with magnetic induction effects included. The transformed ordinary differential equations are solved exactly, under physically appropriate boundary conditions. Closed-form expressions are obtained for the non-dimensional velocity (u), non-dimensional induced magnetic field component (B _x ) and wall frictional shearing stress i.e. skin friction function (τ _x ) as functions of dimensionless transverse coordinate (η), Grashof free convection number (G _r ) and the Hartmann number (M). The bulk temperature in the boundary layer (Θ) is also evaluated and shown to be purely a function of M. The Rayleigh flow distribution (R) is derived and found to be a function of both Hartmann number (M) and the buoyant diffusivity parameter (ϑ ^*). The influence of Grashof number on velocity, induced magnetic field and wall shear stress profiles is computed. The response of Rayleigh flow distribution to Grashof numbers ranging from 2 to 200 is also discussed as is the influence of Hartmann number on the bulk temperature. Rayleigh flow is demonstrated to become stable with respect to the width of the boundary layer region and intensifies with greater magnetic field i.e. larger Hartman number M, for constant buoyant diffusivity parameter ϑ ^*. The induced magnetic field (B _x ), is elevated in the vicinity of the plate surface with a rise in free convection (buoyancy) parameter G _r , but is reduced over the central zone of the boundary layer regime. Applications of the study include laminar magneto-aerodynamics, materials processing and MHD propulsion thermo-fluid dynamics.     

Effects of deformed volume, volume fraction and particle size on the deformation behaviour of W/Cu composites

Tungsten/copper (W/Cu) particle reinforced composites were used to investigate the scaling effects on the deformation and fracture behaviour. The effects of the volume fraction and the particle size of the reinforcement (tungsten particles) were studied. W/Cu-80/20, 70/30 and 60/40 wt.% each with tungsten particle size of 10 μm and 30 μm were tested under compression and shear loading. Cylindrical compression specimens with different volumes (D_S = H) were investigated with strain rates between 0.001 s⁻¹ and about 5750 s⁻¹ at temperatures from 20 °C to 800 °C. Axis-symmetric hat-shaped shear specimens with different shear zone widths were examined at different strain rates as well. A clear dependence of the flow stress on the deformed volume and the particle size was found under compression and shear loading. Metallographic investigation was carried out to show a relation between the deformation of the tungsten particles and the global deformation of the specimens. The size of the deformed zone under either compression or shear loading has shown a clear size effect on the fracture of the hat-shaped specimens.The quasi-static flow curves were described with the material law from Swift. The parameters of the material law were presented as a function of the temperature and the specimen size. The mechanical behaviour of the composite materials were numerically computed for an idealized axis-symmetric hat-shaped specimen to verify the determined material law.     

Oblique stagnation-point flow of an incompressible visco-elastic fluid towards a stretching surface

An analysis is made of the steady two-dimensional stagnation-point flow of an incompressible viscoelastic fluid over a flat deformable surface when the surface is stretched in its own plane with a velocity cx, where x is the distance from the stagnation-point and c is a positive constant. It is shown that for a viscoelastic fluid of short memory (obeying Walters’ B model), a boundary layer is formed when the stretching velocity of the surface is less than ax, where ax+2by is the inviscid free-stream velocity and y is the distance normal to the plate, a and b being constants and the velocity at a point increases with increase in the elasticity of the fluid. On the other hand an inverted boundary layer is formed when the surface stretching velocity exceeds ax and the velocity decreases with increase in the elasticity of the fluid. A novel result of the analysis is that the flow near the stretching surface is that corresponding to an inviscid stagnation-point flow when a=c. Temperature distribution in the boundary layer is found in three cases, namely: (i) the sheet with constant surface temperature (CST); (ii) the sheet with variable surface temperature (VST) and (iii) the sheet with prescribed quadratic power law surface heat flux (PHF) for various values of non-dimensional parameters. It is found that in all the three cases when a/c>1, temperature at a point decreases with increase in the elasticity of the fluid and when a/c<1, temperature at a point increases with increase in the elasticity of the fluid. Further temperature at a point decreases with increase in the radiation parameter and wall temperature parameter.     

Uniform Local Existence for Inhomogeneous Rotating Fluid Equations

We investigate the equations of anisotropic incompressible viscous fluids in , rotating around an inhomogeneous vector B(t, x ₁, x ₂). We prove the global existence of strong solutions in suitable anisotropic Sobolev spaces for small initial data, as well as uniform local existence result with respect to the Rossby number in the same functional spaces under the additional assumption that B = B(t, x ₁) or B = B(t, x ₂). We also obtain the propagation of the isotropic Sobolev regularity using a new refined product law.     

Symmetry and secondary bifurcation of elastic structures

We consider non-linear bifurcation problems for elastic structures modeled by the operator equation F[w;α]=0 where F:X×R^k→Y,X,Y are Banach spaces and XY. We focus attention on problems whose bifurcation equations are of the form

f_i(α₁,α₂;λ,μ)=(a_iμ+b_iλ)α_i+p_iα_i³+q_iα_i∑_j=1,j≠i^kα_j+1²+α_ih_i(λ,μ;α₁,α₂,…α_k) i=1,2,…k

which emanates from bifurcation problems for which the linearization of F is Fredholm operators of index 0. Under the assumption of F being odd we prove an important theorem of existence of secondary bifurcation. Under this same assumption we prove a symmetry condition for the reduced equations and consequently we got an existence result for secondary bifurcation. We also include a stability analysis of the bifurcating solutions.     

A study of the upper limit of solid scatters density for gray Lattice Boltzmann Method

The upper limit of the solid scatters density ns （x）, a key parameter for the simulation of flows in porous media with a gray Lattice Boltzmann Method, is studied by an analytical way for the infiltration Poiseuille flow between two infinite parallel plates. Analyses of three different gray Lattice Boltzmann schemes, separately proposed by Gao and Sharma et al., Dardis and McCloskey, and Thorne and Sukop, indicate that the effective domain of Gao and Sharma＇s scheme is restricted to ns 〈 1/2√3≈0.289, Dardis and McCloskey＇s scheme is restricted to ns 〈 （√57-1）/28≈0.234, and that there is no extra restriction on ns（x） with Thorne and Sukop＇s scheme. These results are obtained for the dimensionless relaxation time τ= 1. The above analytical results are verified by our numerical simulations. The use of a gray LBM is further illustrated by simulating the flow at the interface of a porous medium. Simulation results yield velocity profiles which agree very well with Brinkman＇s prediction.     

Inviscid Dynamical Structures Near Couette Flow

Consider inviscid fluids in a channel {-1\leqq y\leqq1}{\{-1\leqq y\leqq1\}} . For the Couette flow u ₀ = (y, 0), the vertical velocity of solutions to the linearized Euler equation at u ₀ decays in time. Whether the same happens at the non-linear level is an open question. Here we study issues related to this problem. First, we show that in any (vorticity) H^s(s < \frac32){H^{s}\left(s<\frac{3}{2}\right)} neighborhood of Couette flow, there exist non-parallel steady flows with arbitrary minimal horizontal periods. This implies that nonlinear inviscid damping is not true in any (vorticity) H^s(s < \frac32){H^{s}\left(s<\frac{3}{2}\right)} neighborhood of Couette flow for any horizontal period. Indeed, the long time behaviors in such neighborhoods are very rich, including nontrivial steady flows and stable and unstable manifolds of nearby unstable shears. Second, in the (vorticity) ${H^{s}\left(s>\frac{3}{2}\right)}${H^{s}\left(s>\frac{3}{2}\right)} neighborhoods of Couette flow, we show that there exist no non-parallel steadily travelling flows \varvecv(x-ct,y){\varvec{v}\left(x-ct,y\right)} , and no unstable shears. This suggests that the long time dynamics in ${H^{s}\left(s>\frac{3}{2}\right)}${H^{s}\left(s>\frac{3}{2}\right)} neighborhoods of Couette flow might be much simpler. Such contrasting dynamics in H ^s spaces with the critical power s=\frac32{s=\frac{3}{2}} is a truly nonlinear phenomena, since the linear inviscid damping near Couette flow is true for any initial vorticity in L ².     

Viscoelastic properties of ultra-high viscosity alginates

Ultra-high viscosity alginates were extracted from the brown seaweeds Lessonia nigrescens (UHV_N, containing 61% mannuronate (M) and 2% guluronate (G)) and Lessonia trabeculata (UHV_T, containing 22% M and 78% G). The viscoelastic behavior of the aqueous solutions of these alginates was determined in shear flow in terms of the shear stress σ ₂₁, the first normal stress difference N ₁, and the shear viscosity η in isotonic NaCl solutions (0.154 mol/L) at T = 298 K in dependence of the shear rate [(g)\dot]\dot{\gamma} for solutions of varying concentrations and molar masses (3–10 × 10⁵ g/mol, homologous series was prepared by ultrasonic degradation). Data obtained in small-amplitude oscillatory shear (SAOS) experiments obey the Cox–Merz rule. For comparison, a commercial alginate with intermediate chemical composition was additionally characterized. Particulate substances which are omnipresent in most alginates influenced the determination of the material functions at low shear rates. We have calculated structure–property relationships for the prediction of the viscosity yield, e.g., η–M _w–c–[(g)\dot]\dot{\gamma} for the Newtonian and non-Newtonian region. For the highest molar masses and concentrations, the elasticity yield in terms of N ₁ could be determined. In addition, the extensional flow behavior of the alginates was measured using capillary breakup extensional rheometry. The results demonstrate that even samples with the same average molar mass but different molar mass distributions can be differentiated in contrast to shear flow or SAOS experiments.     

Tailoring turbulence with an active grid

Using an active grid in a wind tunnel, we generate homogeneous shear turbulence and initiate turbulent boundary layers with adjustable properties. Homogeneous shear turbulence is characterized by a constant gradient of the mean velocity and a constant turbulence intensity. It is the simplest anisotropic turbulent flow thinkable, and it is generated traditionally by equipping a wind tunnel with screens which have a varying transparency and flow straighteners. This is not done easily, and the reachable turbulence levels are modest. We describe a new technique for generating homogeneous shear turbulence using an active grid only. Our active grid consists of a grid of rods with attached vanes which can be rotated by servo motors. We control the grid by prescribing the time-dependent angle of each axis. We tune the vertical transparency profile of the grid by setting appropriate angles of each rod such as to generate a uniform velocity gradient, and set the rods in flapping motion around these angles to tailor the turbulence intensity. The Taylor Reynolds number reached was R _λ = 870, the shear rate S = ∂U/∂y = 9.2 s⁻¹, the nondimensional shear parameter S ^*≡ Sq ²/ε = 12 and u = 1.4 ms⁻¹. As a further application of this idea we demonstrate the generation of a simulated atmospheric boundary layer in a wind tunnel which has tunable properties. This method offers a great advantage over the traditional one, in which vortex-generating structures need to be placed in the wind tunnel to initiate a fat boundary layer.     

Fracture of composite laminates containing cracks due to shear loading

The responses of graphite/epoxy [0/90/±45] _s , [±45]_2s , [0/90]_2s and [0/±45]_2s composite laminates with and without center cracks were studied under shear loading using the three-rail shear test. The shear stress/strain relationship, the failure mechanisms and the notched strength were analyzed. Substantial amounts of local buckling were observed in some of the laminates. The present paper shows that shear modulus can be determined accurately using the three-rail shear test with proper interpretation of data. Using the minimum strength model, only one characteristic length was needed to predict accurately the notched strength of a composite laminate under shear and tensile loadings.Paper was presented at the 1987 SEM Spring Conference on Experimental Mechanics held in Houston, TX on June 14–19.     

A network theory of constrained elastic recovery in concentrated polymer solutions

Summary The stress-strain-history relations derived byGreen andTobolsky for a relaxingGaussian molecular network with temporary junctions, generalized to allow for a distribution of junction mean lifetimes, have previously been used to calculate the elastic recovery which occurs in a polymer solution in a state of steady shear flow when all the stress components are instantaneously made zero. The equations predict that the instantaneous part of the recovery involves, in addition to the expected shear recovery, an expansion in directions normal to the previous lines of flow.The present paper contains the corresponding calculations for the case in which this expansion is not allowed to take place, the liquid being constrained so that planesx ₂=const. move rigidly in directions parallel to thex ₁-axis during recovery as well as during the steady shear flow;x ₁,x ₂,x ₃ denote coordinates relative to a rectangularCartesian coordinate system fixed in space. It is shown that the relation between shear stress and the time integral of shear strain is of the linear type much studied in the literature; the calculation of the behaviour ofp ₁₁–p ₂₂, the difference of two normal stress components, is new. It is shown that ifp ₂₁, the shear stress component, be instantaneously made zero (following steady shear flow),p ₁₁–p ₂₂ also decreases instantaneously but in general takes time to reach the value zero; this is true even for the case in which all junctions have the same mean lifetime when the instantaneous shear recovery is not followed by any delayed recovery. The magnitudes of both instantaneous and ultimate shear recovery are calculated; it is found that the latter differs by a factor of two from that calculated on the hypothesis that the stress tensor in steady shear flow is an isotropic function of the strain tensor describing the ultimate shear recovery. The transient behaviour associated with the start of steady shear flow is also considered. Inertial forces are neglected throughout.

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Zusammenfassung Die Beziehungen zwischen der Spannung und dem Zeitintegral der Verformung, die vonGreen undTobolsky für ein sich entspannendes Gaußsches Netzwerk mit zeitweiligen Vernetzungen abgeleitet wurden, werden mit Berücksichtigung der Verteilung der mittleren Lebensdauer der Vernetzungen verallgemeinert. Diese Beziehungen sind schon früher benutzt worden, um die elastische Erholung zu berechnen, die erfolgt, wenn in einer Polymerlösung, die sich in einem Zustand stationärer Scherströmung befindet, alle Kräfte momentan gleich Null gesetzt werden. Die Gleichungen sagen aus, daß der Momentanteil der Erholung nicht nur eine voraussichtliche Schererholung, sondern auch eine Ausbreitung in den Richtungen normal zu den Strömungslinien einschließt.Die vorliegende Abhandlung enthält die diesbezügliche Rechnung für den Fall, wo diese Ausbreitung nicht stattfinden kann, da die Flüssigkeit so umschlossen ist, daß die Ebenen normal zurx ₂-Achse sich unnachgiebigerweise in den Richtungen parallel mit derx ₁-Achse während der Erholung als auch während der stationären Scherströmung bewegen.x ₁,x ₂, undx ₂ bezeichnen die Koordinaten bezüglich eines räumlich festen rechtwinkeligen, cartesischen Achsenkreuzes. Man erhält eine lineare Beziehung zwischen der Schubspannung und dem Zeitintegral der Schiebung, die im Schrifttum schon viel behandelt worden ist. Die Berechnung des Verhaltens der Differenz der zwei Normalspannungskomponentenp ₁₁ undp ₂₂ ist neu. Es wird gezeigt, daß wenn, bei stationärer Scherströmung, die Schubspannungskomponentep ₂₁ momentan Null gesetzt wird, dannp ₁₁ bisp ₂₂ auch momentan abnimmt, aber im allgemeinen ist eine gewisse Zeit erforderlich, den Nullwert zu erreichen. Das trifft sogar dann zu, wenn alle Vernetzungen dieselbe mittlere Lebensdauer haben und wo der momentanen Schuberholung nicht eine verzögerte Erholung folgt. Die Beträge der momentanen und endgültigen Schererholung wurden berechnet. Die so ermittelte endgültige Schererholung beträgt die Hälfte derer, die man, von der Annahme ausgehend, daß der Spannungstensor der stationären Scherströmung eine isotrope Funktion des Verzerrungstensors, der die endgültige Schererholung beschreibt, sei, berechnet. Das vorübergehende Verhalten am Anfang der stationären Scherströmung ist auch in Betracht gezogen. Dagegen sind Trägheitskräfte durchweg vernachlässigt.