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A modification of Hiemenz's two‐dimensional outer potential stagnation‐point flow of strain rate a is obtained by adding periodic radial and azimuthal velocities of the form brsin2θ and brcos2θ, respectively, where b is a shear rate. This leads to the discovery of a new family of three‐dimensional viscous stagnation‐point flows depending on the shear‐to‐strain‐rate ratio γ=b/a that exist over the range −∞<γ<∞ with reflectional symmetry about γ=0. Numerical solutions for the wall shear stress parameters and the displacement thicknesses are given and compared with their large‐γ asymptotic behaviors. Sample similarity velocity profiles are also presented. It is noted that the results presented here are in many ways similar to the results reported for non‐axisymmetric Homann stagnation‐point flow. [ABSTRACT FROM AUTHOR]

The boundary layer flow above a disk in torsional motion with azimuthal velocity proportional to rm is investigated for all values of m ≥ 1; here, r is the radial coordinate measured from the center of the disk. The resulting flow is a fully three-dimensional exact solution to the steady, axisymmetric boundary layer equations in the form of similarity solutions. We compute wall shear stresses in the radial and azimuthal directions as a function of the torsional exponent m, as well as the flow induced in the far field. The large m asymptotics of the problem are computed and compared with numerical solutions. The induced radial velocity profiles have a "wall-jet" structure, and it is found that both the radial and azimuthal velocity components become confined close to the surface of the disk as m increases. [ABSTRACT FROM AUTHOR]

We present solutions for the boundary layer between two uniform shear flows flowing in the same direction. In the upper layer, the flow has shear strength a, fluid density ρ1, and kinematic viscosity ν1, while the lower layer has shear strength b, fluid density ρ2, and kinematic viscosity ν2. Similarity transformations reduce the boundary-layer equations to a pair of ordinary differential equations governed by three dimensionless parameters: the shear strength ratio γ = b/a, the density ratio ρ = ρ2/ρ1, and the viscosity ratio ν = ν2/ν1. Further analysis shows that an affine transformation reduces this multi-parameter problem to a single ordinary differential equation which may be efficiently integrated as an initial-value problem. Solutions of the original boundary-value problem are shown to agree with the initial-value integrations, but additional dual and quadruple solutions are found using this method. We argue on physical grounds and through bifurcation analysis that these additional solutions are not tenable. The present problem is applicable to the trailing edge flow over a thin airfoil with camber. [ABSTRACT FROM AUTHOR]

The normal impingement of Hiemenz stagnation-point flow on a surface executing biaxial stretching is studied. The problem is governed by two parameters, α the stretching rate in the streamwise direction of the stagnation-point flow and β the stretching rate in the cross-flow direction. Numerical solutions as a function of β are given for values of α over the range -2≤α≤2. Dual solutions are found everywhere and triple solutions exist for all β>0 over a range of α and the boundaries for the dual and triple solutions are shown in graphical form. The points of zero wall shear stress in the streamwise and cross-flow directions are also plotted in (α, β) parameter space. Sample similarity velocity profiles for the dual and triple solutions are also presented. [ABSTRACT FROM AUTHOR]

Laminar boundary layers with fully developed uniform shear cross flows are considered. The first streamwise laminar flow is a Blasius boundary layer flow, the second is uniform shear flow over a semi-infinite plate, and the third is the flow induced by a power-law stretching surface. In the first two cases, the effect of streamwise plate motion is taken into account by the parameter λ. In each case, the similarity solutions reduce the governing boundary layer equations to a primary ordinary differential equation for the streamwise flow and a secondary linear equation coupled to the primary solution for the cross flow. It is found that an infinity of solutions exist in each problem and the unique solution in each case is found by applying the Glauert criterion. In some instances, a simple exact solution for the cross flow is presented. Results for the wall shear stresses and velocity profiles are given in graphical form. [ABSTRACT FROM AUTHOR]

A study is conducted to determine all possible prescriptions of frequency with fluid depth H for sloshing in planar geometries using the inverse method of solution for potential functions of separable form. Known solutions for the rectangular well, the wedge, a hyperbolic basin and the isochronous container are recovered. In addition, three new families of containers, here denoted C1, C2 and C3, parameterized by streamfunction separation constant B are found. Containers C1 exist for correspond to a family of hyperbolic cotangent variations of frequency with depth. Containers C2 exist for correspond to a family of hyperbolic tangent variations of frequency with depth. Containers C3 exist for correspond to a family of cotangent variations of frequency with depth. There is an affinity between the C1 and C2 families in that their limiting container profiles, found as , take the shape of identical isochronous containers. Also there is an affinity between the C1 and C3 families in that their shapes asymptotically tend to a 90 wedge at low depth. Plots of the shapes of the containers and their frequency variations are given. [ABSTRACT FROM AUTHOR]

The problem of stagnation-point flow impinging radially on a linearly twisting cylinder is considered. This advances previous work on the motion outside a cylinder undergoing linear torsional motion. The problem is governed by a Reynolds number R and a dimensionless torsion rate σ. Numerical calculations are carried out using the ODEINT program, and convergence of the shooting method is obtained using the MNEWT program. The radial and azimuthal wall shear stresses are found over a range of R and σ, and radial and azimuthal velocity profiles at σ={0,1,2} are presented for various values of R. The interesting feature is that the axial wall shear stress parameter f''(1) is a very weak function of σ while the azimuthal wall shear stress parameter g'(1) is a strong function of σ although both stress parameters are a strong function of R. [ABSTRACT FROM AUTHOR]

Boundary-layer solutions for the flow induced by an exponentially stretching surface also sheared in its own plane are given. Prior to this study no similarity solutions have been reported for flows generated by exponentially sheared surfaces concomitant with surface stretching in any form. The method of solution is self-similarity. The results found here are intimately related to those of Magyari and Keller ["Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface," J. Phys. D: Appl. Phys. 32, 577-585 (1999)] who studied the motion and heat transfer induced by an exponentially stretching plate. In addition to two particular cases reported here, a third situation is found where an exponentially stretching surface admits a concomitant arbitrary streamwise shearing motion. [ABSTRACT FROM AUTHOR]

Two new families of axisymmetric isochronous containers for first-mode sloshing about nodal circles are found using the inverse method of solution. These containers support standing waves in annular geometries which separate naturally into even (and) and odd () series with increasing radius. It is convenient to present results by fixing, whereis the isochronous frequency andgis gravity. In this formulation, the maximum asymptotic widthsof the containers monotonically decrease fromto. For, both the even and odd containers take the shape of the planar isochronous container. A simple analysis reveals that mass is conserved for first-mode sloshing about the respective nodal circles. [ABSTRACT FROM PUBLISHER]

The flow induced between parallel plates separated by a distance h executing different types of in-plane motion is investigated. The upper plate radially stretches at strain rate a and the lower plate rotates at angular velocity Ω about a common axis. A similarity solution form reduces the Navier-Stokes equations to a coupled pair of ordinary differential equations in two parameters: R = |a| h2/ν and σ = Ω/|a|, where ν is the kinematic viscosity of the fluid. Solutions are obtained for both stretching and shrinking upper plates and numerical results for pressure gradient and wall shear stress parameters are found and compared with low-R series solutions and large-R asymptotic behaviors. Sample radial and azimuthal velocity profiles reveal regions of zero radial and azimuthal wall shear stress which are studied in detail. This work represents the first study of the flow induced between parallel plates for which each plate executes a different type of in-plane motion. [ABSTRACT FROM AUTHOR]

The flow induced above an impermeable sheet undergoing linear radial stretching in a system rotating at angular velocity Ω is investigated. The problem is governed by the single parameter σ = Ω/a, where a is the stretching rate of the membrane. A similarity reduction of the Navier-Stokes equations leads to a pair of nonlinearly-coupled ordinary differential equations which are numerically solved over the range 0 ≤ σ ≤ 10. Coriolis forces acting on the radially stretching flow in the vicinity of the wall induces azimuthal flow in the boundary layer. A large-σ analysis of the system leads to the Ekman equations describing the balance of Coriolis and viscous forces for which an exact solution is presented. Numerical results for the radial and azimuthal shear stresses, along with the magnitude of the velocity induced into the boundary layer, are shown to be in good agreement with the large-σ asymptotic results. [ABSTRACT FROM AUTHOR]