The flow of a variable viscosity fluid between parallel plates with shear heating

A model for the flow of a fluid through a channel with parallel plates is investigated. The channel is narrow, so that the lubrication approximation may be applied. The channel walls are maintained at a constant temperature. Shear heating effects are included and the fluid viscosity decreases exponentially with temperature. When the flow is driven solely by shear stress or imposed velocity at the top, analytical progress is possible. When pressure gradient also drives the flow the problem is solved numerically.     

Group classification of unsteady boundary layer equations of a class of non-Newtonian fluids

Two dimensional unsteady boundary layer equations of a general model of non-Newtonian fluids were investigated in this study. In this model, the shear stress is taken as an arbitrary function of the velocity gradient. Group classification of the equations with respect to shear stress is done using two different approaches: (1) classical theory (2) equivalence transformations. Both approaches yield identical results. It is found that the principle Lie Algebra extends only for cases of Newtonian and Power-Law flows.     

Unsteady MHD flow of a non-Newtonian fluid on a porous plate

This paper deals with the study of the MHD flow of non-Newtonian fluid on a porous plate. Two exact solutions for non-torsionally generated unsteady hydromagnetic flow of an electrically conducting second order incompressible fluid bounded by an infinite non-conducting porous plate subjected to a uniform suction or blowing have been analyzed. The governing partial differential equation for the flow has been established. The mathematical analysis is presented for the hydromagnetic boundary layer flow neglecting the induced magnetic field. The effect of presence of the material constants of the second order fluid on the velocity field is discussed.     

Coverings of Abelian groups and vector spaces

We study the question how many subgroups, cosets or subspaces are needed to cover a finite Abelian group or a vector space if we have some natural restrictions on the structure of the covering system. For example we determine, how many cosets we need, if we want to cover all but one element of an Abelian group. This result is a group theoretical extension of the theorem of Brouwer, Jamison and Schrijver about the blocking number of an affine space. We show that these covering problems are closely related to combinatorial problems, including the so-called additive basis conjecture, the three-flow conjecture, and a conjecture of Alon, Jaeger and Tarsi about nowhere zero vectors.     

A characterization of network representable polymatroids

Meggido [1974] showed that the maximum flow through sets of sources in a multiple sink flow network is a polymatroidal function. Recently, Federgruen and Groenevelt [1985], [1986] have considered polymatroids that can be represented by a multiple source flow network and have improved the runnung time of an important scheduling application.We give a characterization of network representability and relate representable polymatroids to the theory of gammoids.

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Zusammenfassung Meggido [1974] hat gezeigt, daß ein maximaler Fluß durch ein Netzwerk mit mehrfachen Senken eine polymatroidale Funktion beschreibt. Federgruen und Groenvelt [1985], [1986] haben kürzlich solche Polymatroide betrachtet, die durch Flüsse in derartigen Netzwerken repräsentiert werden können und haben so die Laufzeit einer wichtigen Schedulinganwendung verbessern können.Wir geben eine Charakterisierung von Funktionen, die durch derartige Netzflußnetzwerke realisierbar sind. Dabei stellen wir eine Verbindung her zwischen Repräsentierbarkeit und Gammoidtheorie.

     

Algorithms for flows with parametric capacities

We consider the problem of finding maximal flows with respect to capacities which are linear functions of a parametert [0,T]. Since this problem is a special case of a parametric linear program the classichorizontal approach can be applied in which optimal solutions are computed for successive subintervals of [0,T]. We discuss an alternative algorithm which approximates in each iteration the optimal solution for allt [0,T]. Thisvertical algorithm is a labeling type algorithm where the flow variables are piecewise linear functions. Flow augmentations are done alongconditional flow augmenting paths which can be found by modified path algorithms. The vertical algorithm can be used to solve the parametric flow problem optimally as well as to compute a good approximation for allt if the computation of the optimal solution turns out to be too time consuming.Partially supported by NSF Grants ECS-8412926 and INT-8521433, and NATO Grant RG 85/0240.     

Convergence of the Finite Difference Scheme for a Fast Diffusion Equation in Porous Media

We are concerned with an implicit scheme for the finite difference solution to a nonlinear parabolic equation with a multivalued coefficient that describes the fast diffusion in a porous medium. The boundary conditions contain the multivalued function as well. We prove the stability and the convergence of the scheme, emphasizing the precise nature of convergence in this specific case, and compute the error level of the approximating solution. The method is aimed to simplify the numerical computations for the solutions to equations of this type, without performing an approximation of the multivalued function. The theory is illustrated by numerical results.     

Existence for evolutionary problems with linear growth by stability methods

We establish that solutions to the Cauchy–Dirichlet problem

${?}_{t}u?\mathrm{div}(D_{\xi }f(x,Du))=0$

for functionals $f:\mathrm{\Omega }\times {\mathbb{R}}^{N\times n}\to [0,\infty )$ of linear growth can be obtained as limits of solutions to flows with p-growth in the limit $p\downarrow 1$. The result can be interpreted on the one hand as a stability result. On the other hand it provides an existence result for general flows with linear growth.     

A study of heat and mass transfer of Non-Newtonian fluid with surface chemical reaction

Considering the significance of non-Newtonian fluid usage in manufacturing such as molten plastics, polymeric materials, pulps, and so on, significant efforts have been made to investigate the phenomenon of non-Newtonian fluids. In this article the influences of heat and mass transfer on non-Newtonian Walter's B fluid flow over uppermost catalytic surface of a paraboloid is encountered. An elasticity of the fluid layer is considered in the freestream together with heat source/sink and has the tendency to cause heat flow in the fluid saturated domain. The flow problem of two-dimensional Walter's B fluid is represented using Law of conservation of mass, momentum, heat, and concentration along with thermal and solutal chemical reactive boundary conditions. The governing equations are non-linear partial differential equation and are non-dimensionalized by employing stream function and similarity transformation. The final dimensionless equations yielded are coupled non-linear ordinary differential equations. Furthermore, shooting technique along with RK-4th order method is used to get the numerical results. Graphs and tables are modeled by using MATLAB software to check the effects of Walter's B parameter, Chemical reaction parameter and Thickness parameter on temperature, velocity, and concentration profiles. Tabular analysis shows the results of some physical parameters like skin friction coefficient, Nusselt number and Sherwood number due to the variation of Walter's B parameter, thickness parameter and chemical reactive parameter.     

Hydromagnetic parametric resonance instability of two superposed conducting fluids in porous medium

Rayleigh–Taylor instability of a heavy fluid supported by a lighter one through porous medium, in the presence of a uniform, horizontal and oscillating magnetic field is studied. The fluids are taken as viscous (obeying Darcy's law), uniform, incompressible, and infinitely conducting. The amplitude of the oscillating part of the field is taken to be small compared with its steady part. The dispersion relation is obtained in the form of a third-order differential equation, with time as the independent variable and with periodic coefficients, for the vertical displacement of the surface of separation of the two fluids from its equilibrium position. The oscillatory magnetic field of frequency ω$\omega$ and steady part _H0$H_0$ has a stabilizing influence on a mode of disturbance which is unstable in a steady magnetic field of strength _H0$H_0$. It is found that the oscillatory magnetic field and porosity of the porous medium have stabilizing effects, while the medium permeability has a destabilizing influence on the considered system. For a constant value of any of these physical parameters, the system has been found to be unstable (for small wavenumbers) as well as stable afterwards after a definite wavenumber value. The marginal stability case of parametric resonance holds when _M1=_M2=0$M_1=M_2=0$ (and hence m=0$m=0$), in which the characteristic exponents, and the corresponding solutions for u$u$ break down, is also investigated in detail. It is found, to order ?$?$, that the effect of an oscillating magnetic field has no stabilizing influence on a disturbance which is marginally stable in the steady magnetic field; while to order ?²${?}^2$, and when the magnetic field oscillates, a resonance between this mode of disturbances and the oscillating field leads to instability when _ρ2>_ρ1${\rho }_2>{\rho }_1$. It is found also, in this resonant case, that all the constant or varied physical parameters, mentioned above, have destabilizing influences on the considered system. Finally, the other two resonance points appear in non-porous media (i.e., when m=±iω$m=\pm \mathrm{i}\omega$ and m=±2iω$m=\pm 2\mathrm{i}\omega$), are disappeared here due to the presence of the porous medium.