MHD boundary layer equations for power-law fluids with variable electric conductivity

The boundary layer in a power-law fluid flowing in the presence of a transverse variable magnetic field is investigated. Assuming the electric conductivity of the fluid is dependent on its velocity, Meksyn's method is used to get an analytical solution for the velocity field and the coefficient of friction. The effect of the magnetic field is then discussed.

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Sommario Lo scopo di questo lavoro è di studiare lo strato limite laminare con relazione costitutiva a legge di potenza bidimensionale di un fluido non-newttoniano incompressibile elettroconduttore che scorre lungo una parete piana in presenza di un campo magnetico trasversale e di una pressione esterna. La conducibilità elettrica del fluido viene assunta come funzione della velocità nella forma =₀ u, dove ₀ è costante eu è la velocità del flusso parallela alla parete. L'equazione base è stata risolta applicando il metodo di Meksyn per ottenere una soluzione analitica per la velocità ed il coefficiente di attrito. Viene inoltre discusso l'effetto del campo magnetico e la variazione della conducibilità elettrica.

     

Symmetries of boundary layer equations of power-law fluids of second grade

A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows, and from which the boundary layer equations are derived. Symmetries of the boundary layer equations are found by using Lie group theory, and then group classification with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical boundary layer conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.     

Similarity solutions of three-dimensional boundary layer equations of non-Newtonian fluids

An analysis of possibility of finding similaarity solutions to the three-dimensional, steady, incompressible, boundary layer equations in rectangular co-ordinates for a power law fluid has been discussed in the literature. In the present paper a similarity analysis is made of the steady, three-dimensional, incompressible, Iaminar, boundary layer flow of all time independent non-Newtonian fluids. The important conclusion drawn from this analysis in that for a non-Newtonian fiuid of any model, a similarity solution exists for the fluid for which shearing stress and rate of strain can be related by an arbitrary continuous function.     

Similarity solutions of a class of laminar three-dimensional boundary layer equations of power law fluids

An analysis of the possibility of finding similarity solutions to the three-dimensional, steady, incompressible, boundary layer equations in rectangular coordinates for a power law fluid is investigated. It is found that, in general, the two components of the mainstream flow must differ by at most a multiplicative constant and that these components are powers or exponentials of the x'-coordinate.

By assuming small cross-flows, the cross flow component may be generalized and found to be representable by a polynomial in the through flow variable, x'.     

A class of solutions of the boundary layer equations

Exact solutions of the boundary layer equations can be obtained in closed form only in rare cases. These generally involve self-similar solutions for which the corresponding ordinary differential equation can be integrated exactly. In this paper solutions of more general form, containing additive functions of the longitudinal x coordinate in the expression's for the stream function and the self-similar variable, are considered in relation to two-dimensional steady boundary layers. This makes it possible to enlarge the class of problems whose solutions are analytic expressions and in a number of cases can be obtained in the form of expressions containing arbitrary functions of x, which makes possible various interpretations of the solution. In order to introduce arbitrary functions into the solutions of the equations of the axisymmetric boundary layer the problem is reduced to an overdetermined system of ordinary differential equations. This method is also capable of being applied more widely.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 45–51, March–April, 1990.     

Variational solutions for the two-stream mixing of power-law fluids

The variational principle of Lebon-Lambermont, originally proposed for Newtonian fluids, is seen to be applicable to generalized Newtonian fluids. As an example, it is applied to obtain approximate solutions of the laminar boundary-layer equations for the two-stream mixing of power-law fluids. The flow along a flat plate is obtained as a particular case when the consistency of one of the fluids diverges.     

Similarity solutions of energy and momentum boundary layer equations for a power-law shear driven flow over a semi-infinite flat plate

The problem of a steady forced convection thermal boundary-layer driven by a power-law shear is investigated. The search for similarity solutions reduces the problem to a couple of ordinary differential equations containing three parameters: the exponent of the decaying exterior velocity profile, the exponent of the power-law prescribing the thermal condition on the wall and Prandtl number. The effects of these parameters on the existence and form of similarity solution are investigated and the functional dependence of the local Nusselt number on these parameters is reported and discussed. An analysis of the assumptions usually accepted to derive similarity solutions is also reported in order to show the range of values of the exterior velocity power-law exponent for which such solutions may exist.     

The Stokes boundary layer for a power-law fluid

We develop semi-analytical, self-similar solutions for the oscillatory boundary layer (‘Stokes layer’) in a semi-infinite power-law fluid bounded by an oscillating wall (the so-called Stokes problem). These solutions differ significantly from the classical solution for a Newtonian fluid, both in the non-sinusoidal form of the velocity oscillations and in the manner at which their amplitude decays with distance from the wall. In particular, for shear-thickening fluids the velocity reaches zero at a finite distance from the wall, and for shear-thinning fluids it decays algebraically with distance, in contrast to the exponential decay for a Newtonian fluid. We demonstrate numerically that these semi-analytical, self-similar solutions provide a good approximation to the flow driven by a sinusoidally oscillating wall.     

Group analysis and similarity solutions of the compressible boundary layer equations

Summary In this paper the application of Lie's methods to the equations of the laminar boundary layer is discussed. The momentum and energy equations in Prandtl's form are considered for a steady, viscous, compressible laminar flow with non zero pressure gradient, variable viscosity and thermal conductivity. Group analysis yields similarity solutions for given pressure distributions and particular values of the invariance group parameters (group classification). Crocco's transformation is obtained for the infinite-dimensional group of the Lie's algebra admitted by the equations.

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Sommario In questa nota si applicano i metodi di Lie alle equazioni dello strato limite laminare nella forma di Prandtl per un fluido viscoso, compressibile, con gradiente di pressione non nullo e con viscosità e conducibilità termica variabili. L'analisi gruppale fornisce soluzioni di similarità per date distribuzioni di pressione e valori particolari dei parametri del gruppo di invarianza. La trasformazione di Crocco si ottiene in corrispondenza della parte infinito-dimensionale dell'algebra di Lie ammessa dalle equazioni.

     

Exact solutions of multi-term fractional diffusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multi- term time-fractional diffusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial differential equations are converted into time-fractional ordinary differ- ential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-diffusion problems are given to validate the proposed analytical method.     

Exact solutions of multi-term fractional difusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multiterm time-fractional difusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial diferential equations are converted into time-fractional ordinary diferential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-difusion problems are given to validate the proposed analytical method.     

Exact solutions for extended boundary value problems

Summary Extended definition of a stress tensor for a non-Newtonian fluid brings in higher degree derivatives with coefficients as powers of non-Newtonian parameter in the differential equations of motion. Yet, these differential equations need to be solved subject to the same boundary conditions as in the corresponding Newtonian flow problem. A technique is developed to obtain exact solutions for such an extended boundary value problem. Some flow problems forWalters liquidB are considered.     

The new solutions for two kinds of axially symmetrical laminar boundary layer equations

The transformations, which are similar to Mangier's transformation, are given in this paper, and make the two kinds of entrance region flow of axially symmetrical laminar boundary layer in internal way into the flow of two-dimensional boundary layer, and simplify the proboems. The simplified equations can be solved by the 2-D boundary layer theory. Therefore, a new way is opened up to solve the axially symmetrical flow in the entrance region of internal way.Project Supported by the Scientific Fund of Chinese Academy of Science.     

Discontinuous solutions of boundary layer equations with a positive pressure gradient

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 53–62, September–October, 1991.     

Similarity solutions of unsteady laminar incompressible boundary layer equations for flow,heat and mass transfer in non-Newtonian fluids around axisymmetric bodies

Summary The possible existence of similarity solutions for the unsteady three-dimensional boundary layer flows with heat and mass transfer around stationary axisymmetric bodies which are fully immersed in purely viscous moving non-Newtonian fluids has been searched in general by the application of transformations, involving a single linear parameter. In particular, the cases involving rotational flows around stationary bodies and rotating bodies have been discussed as corollaries of the main analysis. The main analysis shows that the similarity solutions are possible only for the bodies for which where is a cross-sectional radius; and is the longitudinal distance from the nose point to the cross section. In case of rotating bodies, similarity solutions exist only for cones and disks. The analysis, as an example, has successfully been applied to the Powell-Eyring model. It is seen that for the same rate of shear, expenditure of energy for maintaining the rotation of the solid body is comparatively higher for a flow with a higher Eyring number where the Eyring numberEy=1/µBE. µ, B, andE are the material functions of the Powell-Eyring fluid.

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Zusammenfassung Es wird die mögliche Existenz von Ähnlichkeitslösungen für instationäre drei-dimensionale Grenzschichtströmungen rein-viskoser nicht-newtonscher Flüssigkeiten mit Wärme- und Stoffübertragung um voll eingetauchte stationäre achsensymmetrische Körper in allgemeiner Weise untersucht. Hierbei werden Transformationen verwendet, die einen einzigen linearen Parameter enthalten. Als Spezialfälle der allgemeinen Analyse werden Rotationsströmungen um ruhende und rotierende Körper diskutiert. Die Hauptanalyse ergibt, daß Ähnlichkeitslösungen nur existieren für Körper mit , wo den Abstand von der Achse und den longitudinalen Abstand auf der Oberfläche von der Nase des Körpers aus bedeuten. Im Falle rotierender Körper existieren solche Lösungen nur für Kegel und Kreisscheiben. Die Analyse läßt sich erfolgreich auf das Beispiel einer Powell-Eyring-Flüssigkeit anwenden. Man findet, daß bei gleicher Schergeschwindigkeit der Energieverbrauch zur Aufrechterhaltung der Körperrotation mit wachsender Eyring-Zahl Ey= 1/µBE ansteigt, wobeiµ, B undE Materialfunktionen der Powell-Eyring-Flüssigkeit bedeuten.

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Self-similar solutions of nonstationary equations of a plane laminar magnetohydrodynamic boundary layer

Self-similar solutions of nonstationary equations of the boundary layer in ordinary hydrodynamics are discussed in [1, 2]. In this paper self-similar solutions of nonstationary equations of a plane magnetohydrodynarnic boundary layer are sought. In this case, a transformation to curvilinear coordinates of a certain special form is employed. Its choice is determined by the requirements essential to reducing the equations of the boundary layer to a system of ordinary equations. H. Weyl's iterative method is used to solve the equations describing the flow over a plate suddenly set in motion.