A remark on similarity analysis of boundary layer equations of a class of non-Newtonian fluids

A similarity analysis of three-dimensional boundary layer equations of a class of non-Newtonian fluid in which the stress, an arbitrary function of rates of strain, is studied. It is shown that under any group of transformation, for an arbitrary stress function, not all non-Newtonian fluids possess a similarity solution for the flow past a wedge inclined at arbitrary angle except Ostwald-de-Waele power-law fluid. Further it is observed, for non-Newtonian fluids of any model only $90°$ of wedge flow leads to similarity solutions. Our results contain a correction to some flaws in Pakdemirli׳s [14] (1994) paper on similarity analysis of boundary layer equations of a class of non-Newtonian fluids.     

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'.     

MHD flow of a power-law fluid over a rotating disk

Magnetohydrodynamic flow of an electrically conducting power-law fluid in the vicinity of a constantly rotating infinite disk in the presence of a uniform magnetic field is considered. The steady, laminar and axi-symmetric flow is driven solely by the rotating disk, and the incompressible fluid obeys the inelastic Ostwald de Waele power-law model. The three-dimensional boundary layer equations transform exactly into a set of ordinary differential equations in a generalized similarity variable. These ODEs are solved numerically for values of the magnetic parameter m up to 4.0. The effect of the magnetic field is to reduce, and eventually suppress, the radially directed outflow. An accompanying reduction of the axial flow towards the disk is observed, together with a thinning of the boundary layer adjacent to the disk, thereby increasing the torque required to maintain rotation of the disk at the prescribed angular velocity. The influence of the magnetic field is more pronounced for shear-thinning than for shear-thickening fluids.     

The axisymmetrical free laminar jet of an incompressible pseudoplastic fluid

Summary The equations of momentum and continuity of a free, axisymmetrical jet for an incompressible, inelastic, non-Newtonian fluid in isothermal flow are solved. Using the boundary layer approximations and a similarity transformation the resulting non-linear third order ordinary differential equation is solved numerically. Velocity profiles are obtained and compared to previously known solutions for the two-dimensional jet. The significance of the results is discussed in detail.At present on leave at the Department of Chemical Engineering, Stanford University, Stanford (Cal.), U.S.A.     

Blasius flow and heat transfer of fourth-grade fluid with slip

This investigation deals with the effects of slip, magnetic field, and non- Newtonian flow parameters on the flow and heat transfer of an incompressible, electrically conducting fourth-grade fluid past an infinite porous plate. The heat transfer analysis is carried out for two heating processes. The system of highly non-linear differential equations is solved by the shooting method with the fourth-order Runge-Kutta method for moderate values of the parameters. The effective Broyden technique is adopted in order to improve the initial guesses and to satisfy the boundary conditions at infinity. An exceptional cross-over is obtained in the velocity profile in the presence of slip. The fourth-grade fluid parameter is found to increase the momentum boundary layer thickness, whereas the slip parameter substantially decreases it. Similarly, the non-Newtonian fluid parameters and the slip have opposite effects on the thermal boundary layer thickness.     

Analytic study on non-Newtonian natural convection boundary layer flow with variable wall temperature on a horizontal plate

The similarity transform for the steady free convection boundary layer flow of a non-Newtonian fluid (Casson model) with variable wall temperature on a horizontal plate gives a system of nonlinear ordinary differential equations which is solved analytically by applying a newly developed method namely the homotopy analysis method (HAM). The velocity and temperature profiles are obtained and the influence of Prandtl number and various physical parameters of the problem on these distributions are discussed in detail and are illustrated graphically through a set of graphs. The validity of our solutions is verified by the numerical results.     

A note on the falkner-skan flows of a non-newtonian fluid

Non-similar solutions are established for the boundary layer flow of a homogeneous incompressible fluid of second grade past a wedge placed symmetrically with respect to the flow direction. The variation of the skin-friction with respect to the non-Newtonian parameters is discussed.     

Nonequivalent similarity reductions of steady thermal boundary layer equations for an incompressible laminar flow over a continuous moving hot surface

This paper uses symmetry methods to obtain nonequivalent similarity reductions of the steady two-dimensional thermal boundary layer equations of an incompressible laminar flow. New similarity reductions are uncovered and results obtained through the classical dimensional analysis are recovered.     

Boundary layer flow of Reiner-Philippoff fluids

An analysis is made of the boundary layer flow of Reiner-Philippoff fluids. This work is an extension of a previous analysis by Hansen and Na [A.G. Hansen and T.Y. Na, Similarity solutions of laminar, incompressible boundary layer equations of non-Newtonian fluids. ASME 67-WA/FE-2, presented at the ASME Winter Annual Meeting, November (1967)], where the existence of similar solutions of the boundary layer equations of a class of general non-Newtonian fluids were investigated. It was found that similarity solutions exist only for the case of flow over a 90° wedge and, being similar, the solution of the non-linear boundary layer equations can be reduced to the solution of non-linear ordinary differential equations. In this paper, the more general case of the boundary layer flow of Reiner-Philippoff fluids over other body shapes will be considered. A general formulation is given which makes it possible to solve the boundary layer equations for any body shape by a finite-difference technique. As an example, the classical solution of the boundary layer flow over a flat plate, known as the Blasius solution, will be considered. Numerical results are generated for a series of values of the parameters in the Reiner-Philippoff model.     

Approximate solution of the problem of the three-dimensional boundary layer in an incompressible fluid

A method of successive approximations is proposed for the solution of the equations of the three-dimensional incompressible boundary layer on bodies of arbitrary shape. A coordinate system connected with the streamlines of the external nonviscous flow is used. It is assumed that the velocity across the external streamlines is small. When the intensity of secondary flow is low the equations describing the boundary layer in an incompressible fluid are reduced to a form analogous to the equations for the boundary layer on axially symmetrical bodies. An approximate analytical solution is obtained for the velocity and for the friction in the form of equations which can be used for any problems of a three-dimensional incompressible boundary layer. The method developed was applied to the problem of the three-dimensional boundary layer at a plate with a cylindrical obstacle in the presence of a slip angle.     

Laminar swirling power law non-newtonian fluid jet impinging on a normal plane

An analysis is presented for a steady, laminar, incompressible, swirling, Ostwald—De Waele type non-Newtonian fluid jet impinging normally over a horizontal plane with a free surface. A similarity solution is obtained in a region of radial distance away from the central stagnation point. Numerical solutions for the radial and swirling velocities have been obtained for the flow behavior index, n, varying from 0.1 to 2.5. Expressions for the free surface radial velocity, growth of the free surface and skin friction coefficient are given.     

Singularity in the solution of the equations of a three-dimensional boundary layer due to the collision of wall jets

The problem of the propagation of a three-dimensional jet of viscid incompressible fluid flowing from a narrow curved slot into a fluid-filled space along a rigid plane is considered within the framework of the equations of a steady laminar boundary layer. A class of initial conditions at the slot outlet which generates in the jet a velocity field without secondary flows is identified. Within this class the boundaryvalue problem for the three-dimensional boundary layer can be divided into two problems of lower dimensionality: a dynamic and a kinematic problem. As a result of the analysis of the kinematic problem the general structure of the regions of existence and uniqueness of the solution is determined. An investigation of the dynamic problem shows that as the boundaries of the region of existence are approached a singularity characterized by an infinite increase in the thickness of the jet is formed in the solution of the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 75–81, July–August, 1991.     

Rotating flow of a second-order fluid on a porous plate

A study is made of the unsteady flow engendered in a second-order incompressible, rotating fluid by an infinite porous plate exhibiting non-torsional oscillation of a given frequency. The porous character of the plate and the non-Newtonian effect of the fluid increase the order of the partial differential equation (it increases up to third order). The solution of the initial value problem is obtained by the method of Laplace transform. The effect of material parameters on the flow is given explicitly and several limiting cases are deduced. It is found that a non-Newtonian effect is present in the velocity field for both the unsteady and steady-state cases. Once again for a second-order fluid, it is also found that except for the resonant case the asymptotic steady solution exists for blowing. Furthermore, the structure of the associated boundary layers is determined.     

Natural convection of non-Newtonian power-law fluid over axisymmetric and two-dimensional bodies of arbitrary shape in fluid-saturated porous media

Numerical analysis of the free convection coupled heat and mass transfer is presented for non-Newtonian power-law fluids with the yield stress flowing over a two-dimensional or axisymmetric body of an arbitrary shape in a fluid-saturated porous medium. The governing boundary layer equations and boundary conditions are cast into a dimensionless form by the similarity transformation. The resulting system of equations is solved by a finite difference method. The parameters studied are the rheological constants, the buoyancy ratio, and the Lewis number. Representative velocity, temperature, and concentration profiles are presented and discussed. It is found that the results depend strongly on the values of the yield stress parameter and the power-law index of the non-Newtonian fluid.     

Natural convection of non-Newtonian power-law fluid over axisymmetric and two-dimensional bodies of arbitrary shape in fluid-saturated porous media

Numerical analysis of the free convection coupled heat and mass transfer is presented for non-Newtonian power-law fluids with the yield stress flowing over a two-dimensional or axisymmetric body of an arbitrary shape in a fluid-saturated porous medium. The governing boundary layer equations and boundary conditions are cast into a dimensionless form by the similarity transformation. The resulting system of equations is solved by a finite difference method. The parameters studied are the rheological constants, the buoyancy ratio, and the Lewis number. Representative velocity, temperature, and concentration profiles are presented and discussed. It is found that the results depend strongly on the values of the yield stress parameter and the power-law index of the non-Newtonian fluid.     

Stability of dual solutions in stagnation-point flow and heat transfer over a porous shrinking sheet with thermal radiation

An analysis is carried out to study the steady two-dimensional stagnation-point flow and heat transfer of an incompressible viscous fluid over a porous shrinking sheet in the presence of thermal radiation. A set of similarity transformations reduce the boundary layer equations to a set of non-linear ordinary differential equations which are solved numerically using fourth order Runge-Kutta method with shooting technique. The analysis of the result obtained shows that as the porosity parameter β increases, the range of region of existence of similarity solution increases. It is also observed that multiple solutions exist for a certain range of the ratio of the shrinking velocity to the free stream velocity (i.e., α) which again depends on β. We then discuss the stability of the unsteady solutions about each steady solution, showing that one steady state solution corresponds to a stable solution whereas the other corresponds to an unstable solution. The stable solution corresponds to the physically relevant solution. Further we obtain numerical results for each solution, which enable us to discuss the features of the respective solutions.     

Fluctuating flow of a second-order fluid near a stagnation point

Summary The boundary layer equations for axisymmetric flow of an incompressible second-order fluid have been deduced. The flow of such a fluid near a stagnation point when the main stream outside the boundary layer fluctuates in magnitude but not in direction has been discussed. The velocity distribution is found for various values of the steady mean in two limiting cases of small and large values of the frequency of the oscillation of the main stream. The frequency for which two approximate solutions overlap has been calculated in each case.     

The unsteady boundary layer flow in a convergent channel

An incompressible fluid is at rest in a converging channel between intersecting planes when, at some moment of time, the flow towards a sink at the point of intersection is set up impulsively. The motion is taken to be of the boundary layer type, and it is found that the transient part of the flow decays algebraically as the steady state motion becomes dominant. The final decay takes place at the edge of the boundary layer, with the viscous forces affecting the unsteady flow in a region that eventually lies outside the steady state boundary layer.     

MHD stagnation-point flow and heat transfer towards stretching sheet with induced magnetic field

The problem of the steady magnetohydrodynamic (MHD) stagnation-point flow of an incompressible viscous fluid over a stretching sheet is studied. The effect of an induced magnetic field is taken into account. The nonlinear partial differential equations are transformed into ordinary differential equations via the similarity transformation. The transformed boundary layer equations are solved numerically using the shooting method. Numerical results are obtained for various magnetic parameters and Prandtl numbers. The effects of the induced magnetic field on the skin friction coefficient, the local Nusselt number, the velocity, and the temperature profiles are presented graphically and discussed in detail.     

Conjugate heat transfer with radiation from a vertical circular pin in a non-newtonian ambient medium

An analysis is presented for the heat transfer characteristics of a steady, laminar, mixed convective flow of power-law type non-Newtonian fluid over a circular pin by the conjugate convection-conduction theory including radiative effects under optically thick limit approximation. Numerical results are presented for the dimensionless heat transfer coefficients, local and overall heat fluxes and temperature distribution of the pin by a simultaneous solution of the convective boundary layer equations of the fluid and the energy equation of the pin.