Showing papers on "Velocity gradient published in 1996"

Topology of fine-scale motions in turbulent channel flow

Abstract: An investigation of topological features of the velocity gradient field of turbulent channel flow has been carried out using results from a direct numerical simulation for which the Reynolds number based on the channel half-width and the centreline velocity was 7860. Plots of the joint probability density functions of the invariants of the rate of strain and velocity gradient tensors indicated that away from the wall region, the fine-scale motions in the flow have many characteristics in common with a variety of other turbulent and transitional flows: the intermediate principal strain rate tended to be positive at sites of high viscous dissipation of kinetic energy, while the invariants of the velocity gradient tensor showed that a preference existed for stable focus/stretching and unstable node/saddle/saddle topologies. Visualization of regions in the flow with stable focus/stretching topologies revealed arrays of discrete downstream-leaning flow structures which originated near the wall and penetrated into the outer region of the flow. In all regions of the flow, there was a strong preference for the vorticity to be aligned with the intermediate principal strain rate direction, with the effect increasing near the walls in response to boundary conditions.

Toward a model for airflow on the lee side of aeolian dunes

Abstract: The interaction between dunes and the primary wind results in a complex pattern of secondary airflow on the lee side of dunes. From 15 dunes studied during transverse flow conditions at Padre Island in Texas, White Sands in New Mexico, and the Algodones in California, distinct flow regions can generally be recognized, with the overall flow structure comparing favourably to that proposed for subaqueous bedforms. Downwind of dunes with flow separation is a back-flow eddy that extends about four dune-brink heights downwind from the brink of the dune. Beyond the separation cell, the velocity profiles can be divided into regions based upon segments separated by ‘kinks’ in the velocity profiles. The interior is an area above the dunes of relative high wind speed but low velocity gradient. Beneath the interior is the wake, which consists of two layers. The upper wake exhibits an uppermost portion where the flow decelerates while the remainder exhibits accelerating flow, so that the overall velocity gradient decreases downwind. The lower wake exhibits low velocity gradients and wind speeds that accelerate downwind at all heights, but primarily near the top of the layer, thereby causing the velocity gradient to increase downwind. At about eight dune heights downwind, the upper and lower wakes equilibrate to a single profile with the kink between them no longer apparent. The lowest recognizable region is the internal boundary layer. It is recognized by a relatively steep velocity gradient below the wake, and never exceeds a few tens of centimetres in height for our data set. Because of acceleration and increasing shear stress within this layer, interdune flats are at least potentially erosional. Overall, the wake and internal boundary layer show a downward transfer of momentum from upper regions so that the flow recovers. Where flow separation does not occur, simple flow expansion down the lee-face causes flow deceleration.

Wave propagation in a multiple‐scattering upper mantle—observations and modelling

Abstract: SUMMARY A striking feature of Russian long-range seismic refraction data from Peaceful Nuclear Explosions is the observation of a high-frequency teleseismic P, phase, which travels with a group velocity of 8.0 km s-l out to distances of several thousands of kilometres. Modelling using the reflectivity method shows that this phase can be understood as the response of an upper mantle that contains random RMS velocity fluctuations of about 44 per cent superimposed on a positive velocity gradient. This class of model explains the existence of the teleseismic P,, its high-frequency content and its coda length. A teleseismic P, can only be generated if velocity fluctuations are strong enough to cause multiple scattering and occur on a subwavelength scale. Cross-correlation properties of P- and S-wave velocity fluctuations exert a substantial influence on the wavefield. A completely unexpected phase can be observed if the fluctuations are imposed on a negative gradient

Coupled analysis of injection molding filling and fiber orientation, including in‐plane velocity gradient effect

Abstract: On injection molding of short fiber reinforced plastics, fiber orientation during mold filling is determined by the flow field and the interactions between the fibers. The flow field is, in turn, affected by the orientation of fibers. The Dinh and Armstrong rheological equation of state for semiconcentrated fiber suspensions was incorporated into the coupled analysis of mold filling flow and fiber orientation. The viscous shear stress and extra shear stress due to fibers dominate the momentum balance in the coupled Hele-Shaw flow approximation, but the extra in-plane stretching stress terms could be of the same order as those shear stress terms, for large in-plane stretching of suspensions of large particle number. Therefore, a new pressure equation, governing the mold filling process, was derived, including the stresses due to the in-plane velocity gradients. The mold filling simulation was then performed by solving the new pressure equation and the energy equation via a finite element/finite difference method, as well as evolution equations for the second-order orientation tensor via the fourth-order Runge-Kutta method. The effects of stresses due to the in-plane velocity gradient on pressure, velocity, and fiber orientation fields were investigated in the center-gated radial diverging flow in the cases of both an isothermal Newtonian fluid matrix and a nonisothermal polymeric matrix. In particular, the in-plane velocity gradient effect on the fiber orientation was found to be significant near the gate, and more notably for the case of a nonisothermal polymer matrix.

Scalar imaging velocimetry measurements of the velocity gradient tensor field in turbulent flows. I. Assessment of errors

Abstract: The concept of flow field velocimetry based on scalar imaging measurements [Phys. Fluids A 4, 2191 (1992)] is here formulated in terms of an integral minimization implementation, where the velocity field u(x,t) is found by minimizing weighted residuals of the conserved scalar transport equation, along with the continuity condition and a smoothness condition. We apply this technique to direct numerical simulation (DNS) data for the limiting case of turbulent mixing of a Sc=1 passive scalar field. The spatial velocity fields u(x,t) thus obtained demonstrate good correlation with the exact DNS fields, as do the statistics of the velocity and the velocity gradient fields. The results from this integral minimization implementation also show significant improvement over those from the direct inversion technique reported earlier. These results are shown to be largely insensitive to noise at levels characteristic of current fully resolved scalar field measurements.

Orientation and twins separation in a micellar cubic crystal under oscillating shear

Abstract: A series of experiments has been performed in order to analyze the shear orientation effect on a micellar cubic phase using small-angle x-ray-scattering techniques. In a Couette cell, an oscillating shear of strain amplitude less than unity was applied to the crystal. Using a x-ray synchrotron beam, the reciprocal planes perpendicular to the flow velocity and velocity gradient, respectively, were probed. Length and angle correlations were well characterized in both planes and full separation of twinned fcc single crystals was evidenced in some cases, each of them being coherent at millimeter scale.

Turbulence characteristics of a boundary layer over a swept bump

Abstract: The evolution of the turbulent boundary layer over a bump defined by three tangential circular arcs and swept at 45° was examined. The flat-plate boundary layer approaching the swept bump had a momentum thickness Reynolds number of approximately 3800. The ratios of upstream boundary-layer thickness to bump height and convex radius of curvature were 1.5 and 0.06, respectively. The boundary layer was influenced by alternating signs of streamwise pressure gradient, wall curvature, and mean crossflow, which resulted in a complex boundary-layer flow that grew rapidly on the downstream side of the bump. The mean flow profiles deviated significantly from typical logarithmic layer behaviour, but the flow remained attached. The evolution of the Reynolds stress components was explained by the growth of two internal layers triggered by discontinuities in wall curvature near the leading and trailing edges of the bump. The shear stress vector was found to lag the velocity gradient vector, despite the spanwise flow changing direction above the bump. The measurements were compared to the previous results from a two-dimensional bump with the same profile shape and Reynolds number. Contrary to previous studies, the addition of mean crossflow to this complex flow field did not reduce the vertical mixing relative to the turbulent kinetic energy.

Velocity fluctuations in forced Burgers turbulence.

Abstract: Burgers turbulence: the tails of the velocity gradient distribution in the regions where there are no shocks. We evaluate the ‘‘right’’ tail through a rather simple computation, and compare it to the more sophisticated approaches developed recently @7,3,6#. We then give a conjecture on the ‘‘left’’ tail which is based on a plausible argument, requiring the system to reach a stationary state. We shall first discuss the one dimensional case, then turn to higher dimensions, and compare our results with the previously available ones.

Three-dimensional strain-rate imaging.

Abstract: Strain-rate imaging uses large velocity encoding gradients to obtain measurements of velocity that are extremely insensitive to the effects of random noise. The spatial differential of velocity yields the velocity gradient from which the strain-rate and twist-rate tensors can be determined. These tensors represent the distortion of the material and are of interest in the analysis of the dynamic behavior of living tissue (e.g., that of the myocardium). This work presents a new technique that uses the magnitude of the signal in the velocity encoded data to measure through-plane velocity variations at the resolution of the voxel size. The magnitude of the MR signal contains information about the range of phases present within a voxel. When the phase is dependent on the velocity (as in phase velocity imaging), the magnitude contains information about the range of velocities within a voxel. The method presented in this work uses unbalanced slice-refocusing gradients to sample the magnitude variation introduced by the interaction of velocity encoding gradients with spatially dependent velocities. The previously developed in-plane velocity gradient methods can be easily integrated with this new through-plane measurement to characterize the deformation of the myocardium in three spatial dimensions with high accuracy. The applicability of these methods is demonstrated theoretically, in phantoms and in vivo.

An IR Study of the Velocity Structure of the Cometary Compact HII region G29.96-0.02

Abstract: We have mapped the velocity structure of the cometary compact HII region G29.96-0.02 using long-slit echelle spectra of the HI Br gamma line. This technique detects line emission over a much wider area at the necessary spatial resolution compared to radio recombination line observations. Significant structure in both the velocity centroids and the line widths is seen over the entire nebula. Large line widths are seen ahead of the bow and in the tail which may be due to turbulent motions in shocked and interface regions respectively. We construct analytic models of the density and velocity structure in order to attempt to distinguish between the bow shock and champagne flow models which have been put forward to explain the cometary morphology of many compact HII regions. The bow shock model is unable to explain the large velocity gradient that we see right across the tail of the cometary region which can only be explained by the streaming motions towards low density regions in the champagne model. However, our approximation to the champagne model is also not able to fit all of the features of the data. More realistic versions of this model which include the effects of stellar winds and density gradients may be able to provide a better match to these data.

Images of Dissipation Layers to Quantify Mixing Within a Turbulent Jet

Abstract: Images were obtained that visualize the structure of the mixture fraction field and the structure of the scalar dissipation layers that exist near the base of a turbulent jet. To characterize the local mixing rate, measured profiles of the mean dissipation rate were compared with a general scaling analysis. In addition, the joint probability density function (£, %) was measured. Some effects of adding coaxial air and swirl are discussed. The especially large values of the mean dissipation rate (up to 30 s"1) and the instantaneous dissipation rate (up to 175 s"1) that were measured in the base region are due to the large local values of the mean velocity gradient in this region. It was found that the dissipation layers in the base region have a unique structure; the layers tend to be aligned at approximately a 45-deg angle to the flow and thus differ from layers in the far field, which tend to be oriented in an isotropic pattern, as was shown previously. Strong dissipation layers exist at the boundary of the entrained air, and the observed 45-deg alignment in the base region is believed to be due to the ordered vortex pattern in the shear layer. Dissipation layers are typically 0.3 mm thick. Dissipation rates varied in a manner consistent with the general scaling analysis, and the proper scaling constant was measured to be 9300.

Stress tensor measurement using birefringence in oblique transmission

Abstract: A method for measuring the stress tensor of liquids obeying the stress-optical rule is presented. In particular, the procedure makes it possible to determine the shear stress, and the first and second normal stress differences for rheometric flows. This technique is an extension of the procedure recently described by Burghardt and coworkers (Brown et al., 1995) wherein light is sent obliquely through a sample sheared between transparent plates. However, in the present development, the light is transmitted in the plane containing the velocity gradient and neutral directions, thereby reducing the necessary optical measurements by one. A polystyrene-tricresyl phosphate (TCP) solution is used as the test sample. The first and second normal stress differences in steady shear flow measured by this method show good agreement with the mechanical results measured by Madga et al. (1993) using a modified cone and plate rheometer. The transient behavior of the first and second normal stress differences following the start-up of shear flow is also presented.

Western Boundary Current Separation: Inferences from a Laboratory Experiment

Abstract: Observations of a laboratory model of a western boundary current, and its separation and subsequent meandering, are described The current is established by pumping fluid through a rotating channel that contains a topographic β effect and continental slope topography The observations are compared with a theoretical model of all three aspects of the current: the structure of the attached current, the process of separation, and the dynamics and path of the meandering jet This model includes a viscous boundary layer for the attached current, with a thickness of order [ν/(dvI/dy)]1/2, where ν is kinematic viscosity and dvI/dy is the velocity gradient of the inviscid (free slip) flow along the boundary Comparison between the observations and the model show that the attached boundary current is governed by potential vorticity conservation and the Bernoulli equation, and the pressure decreases along its length The separation of this current from the sidewall is then caused by the minimum pressure le

Mixing with First-Order Decay in Variable-Velocity Porous Media Flow

Abstract: The transport and mixing of solutes undergoing first-order decay is central to many problems in groundwater hydrology. Mixing in porous media flow occurs due to advective dilution, hydrodynamic dispersion, and molecular diffusion. Mixing is stronger in regions of higher velocity, and weaker in slower-moving regions. Two-dimensional numerical experiments show that concentration profiles normal to the flow direction are displaced toward regions of slower flow in a flow field with a velocity gradient. Variable-velocity flow fields occur in subsurface flow around permeability heterogeneities, between recirculation cells, and in flow driven by natural convection. We examine two-dimensional solute concentration fields rather than breakthrough curves since for many complicated flow patterns, the breakthrough curve cannot discern important details of the concentration field. For the case of a species undergoing first-order decay, the effect of parent accumulation in regions of low velocity is enhanced for the daughter species because (i) the rate of daughter production is proportional to the local concentration of parent, and (ii) mixing is proportional to the local velocity. The resulting displacement of concentration profiles toward low-velocity regions may have important consequences for subsurface radionuclide transport and also for flows in chemically reactive systems and strongly coupled systems.

Flow‐induced noise on pressure gradient hydrophones

Abstract: Moored or drifting hydrophones are subject to a variety of potential self‐noise sources. Flow‐induced self noise arises when the sensors are subjected to oceanic currents such as those due to wave motion and changing tides. Research at Penn State, in cooperation with the Naval Air Warfare Center (NAWC), has been concerned with the basic mechanisms of flow‐induced self noise on velocity gradient hydrophones of various shapes and sizes. These sensors are configured as finite‐length cylinders in cross flow and as spheres. The sensors are sensitive to acoustic particle velocity, and one of the sensors is sensitive to acoustic intensity. With the diameter of the sensor as the characteristic dimension, and for operational flow velocities in the 0.5 to 2.0 knot range, the Reynolds number range of interest is from values less than 100 (for some miniature sensors) to about 27,000 (for standard‐size sonobuoy hydrophones). Experiments are conducted for the higher ranges of Reynolds number by towing the sensors over the given range of speeds in quiet basins of water (9 meter tow tank at Penn State and a flooded quarry at the NAWC). To achieve the lower range of Reynolds numbers over the same range of velocities, but without having in hand actual miniature sensors, some of the experiments are performed in glycerine. Glycerine has a kinematic viscosity some three orders of magnitude greater than that of water; therefore, a large sensor can be subjected to a velocity in the range of interest but yield an operational Reynolds number that is three orders of magnitude smaller. In this paper, we will show the broadband spectral characteristics of finite‐length cylindrical sensors in cross flow, as well as spherical‐shaped sensors. The Reynolds number of the flow is the independent variable. The threshold of velocity‐dependent noise increase is found to correlate with the occurrence of turbulent flow not necessarily in the wake, but on the surface of the body itself. The flow field is visualized using smoke traces in wind tunnel tests and dye injection in the underwater tests. The results show a strong correlation of the sensor output with the unsteady forces created by the unsteady flow over the sensor.

Extinction of turbulent premixed flames by small-scale turbulence at Kolmogorov microscale

Abstract: By using the counterflow, premixed double flame stabilized in the forward stagnation region of the porous cyclinder, we have accurately determine the extinction limits of laminar and turbulent propane-air premixed flames as a function of the equivalence ratio and the stretch rates. The flow-field and turbulence characteristics were measured along the stagnation streamline by a hot-wire anemometer for the case without combustion. The mean velocity profile is found to be independent of the turbulence characteristics and coincides with the theoretical one obtained assuming the potential flow. The turbulence intensity and Kolmogorov microscale are nearly constant along the stagnation strealine up to the proximity of the stagnation point. For the laminar flames, the critical velocity gradient at which the extinction occurs agrees well with the previous data obtained by differents flame configurations. In the present study, the stretch rate induced by mixture turbulence is assumed to be the inverse of the Kolmogorov time scale, and the total stretch rate is estimated by the sum of the bulk stretch rate induced by the mean velocity gradient and the turbulence stretch rate. Results show that the total stretchs rate at which the extinction of the turbulent premixed flame occurs coincides with the critical velocity gradient at which the laminar premixed flame extinction occurs. Therefore, the small-size eddies with the Kolmogorov microscale play an important role in extinguishing the turbulent premixed flame. However, these eddies do not affect the combustion reaction at molecular scale.

Turbulent length scales in a swirling flame

Abstract: LDV measurements of spatial velocity correlations and integral length scales in a swirling jet and a swirling diluted hydrogen flame are described. The experimental setup consists of two conventional laser Doppler anemometers working in forward scattering mode, one fixed and the other moved by a special traversing system. From experimental observations, the validity of the assumption of self-preserving turbulence is analyzed. This assumption postulates that the two-point correlation of a physical quantity in a turbulent field can be expressed by one length scale and one timescale only. Thus, the ratio between the longitudinal and transversal length scales in a turbulent flow field should not change within the field. This is a main assumption for all statistical turbulence models, as well as for the more elaborate second-order closure models. Also addressed is the question of whether the gradient of velocity changes strongly within the turbulent length scale, which concerns the rapid term model used for calculation of the pressure strain correlation in second-order closure. The main results are the following. o 1. The length scale ratios near the nozzle change because of history effects. 2. The recirculation zone diminishes these effects. 3. Combustion alters the position of the recirculation zone and increases the length scales. 4. There is a strong change of the velocity gradient with the turbulent length scales in front of the recirculation zone, which seems to be a main difficulty in the simulation of swirling flows.

Magnetohydrodynamic flow past a semi-infinite moving plate

Abstract: The nonlinear boundary layer equations of the title problem are solved numerically for obtaining the coefficient of skin-friction. It is found that the magnitude of the normal surface velocity gradient increases with the magnetic interaction parameter.

Stress Tensor Measurement using Birefringence in Oblique Transmission

Abstract: A method for measuring the stress tensor of liquids obeying the stress-optical rule is presented. In particular, the procedure makes it possible to determine the shear stress, and the first and second normal stress differences for rheometric flows. This technique is an extension of the procedure recently described by Burghardt and coworkers (Brown et al., 1995) wherein light is sent obliquely through a sample sheared between transparent plates. However, in the present development, the light is transmitted in the plane containing the velocity gradient and neutral directions, thereby reducing the necessary optical measurements by one. A polystyrene-tricresyl phosphate (TCP) solution is used as the test sample. The first and second normal stress differences in steady shear flow measured by this method show good agreement with the mechanical results measured by Madga et al. (1993) using a modified cone and plate rheometer. The transient behavior of the first and second normal stress differences following the start-up of shear flow is also presented.

Darryl E. Metzger Memorial Session Paper: The Streamwise Development of Görtler Vortices in a Favorable Pressure Gradient

Abstract: Measurements are presented of the streamwise velocity variation within a laminar boundary layer on a concave surface of 4 m radius of curvature for which the free-stream velocity gradient factor (ν/U0 2 )dU0 /dx was approximately 1 × 10−6 . The stream velocity variation was consistent with the presence of counterrotating vortices resulting from the Gortler instability. The vortices exhibited exponential growth over the streamwise extent of the measurements to a disturbance amplitude of approximately 13 percent of the local free-stream velocity. The vortex growth rates were found to be less than those for a zero velocity gradient factor, indicating that a favorable pressure gradient stabilizes the flow with respect to the Gortler instability. Boundary layer profiles at local upwash and downwash positions are compared with the linear theory for which the mean flow was modeled using the Pohlhausen approximation to the solution of the boundary layer equations. The agreement between the measured and predicted profiles indicates that the linear stability theory can provide a fair approximation to the small amplitude growth of the Gortler instability.

Ordering and structure at interfaces of colloidal dispersions under flow

Abstract: The structure of dispersions of sterically stabilised poly(methyl methacrylate)(PMMA) particles under flow conditions are presented. Data have been obtained with a light scattering apparatus in which the sample can be sheared. The colloidal dispersions were prepared from particles with a significant density difference to the suspending medium which gives rise to sedimentation. The scattering patterns indicate fluid-like structures coexisting with ordered arrays of particles. The ordered phase, which has a higher number density than the fluid, is found to undergo significant structural changes under steady shear. Diffraction from ordered and disordered layers of particles in finite-sized stacks is discussed. Quantitative interpretation of the scattering patterns in terms of particle arrangements are presented. The sedimentation gives rise to density and velocity gradient variations across the sample with slip regions. Such important effects would go undetected in rheological measurements, leading to misinterpretation of the rheological behaviour.

Aircraft inlet noise radiation model static, flight and bellmouth effect

Abstract: A relatively simple analytical model is provided which describes the far-field radiation directivity of sound from an aircraft inlet. The model provides for the effect of inlet flow velocity and a different external flow velocity as would be found in a static or wind tunnel test or in flight. The transition from the high duct velocity to the lower external velocity is accomplished by the assumption of an unaltered group velocity vector passing through the velocity gradient field. The influence of the bellmouth shape on the inlet radiation directivity for a static test has been incorporated into the model. Calculated results using the new model are compared to results of inlet static tests and the agreement is very good. The effect of flight on the radiation directivity for both the inlet and the exhaust duct has been derived from the radiation model and for the first time the source noise properties have been incorporated into this flight effect on far-field noise directivity. A simple radiation model is useful when a complex noise source producing many duct modes, such as for broadband noise, must be considered and numerical solutions are not practical.

The Molecular Core Associated with HH 25-26: Contraction or Expansion?

Abstract: We mapped the star formation region HH 25-26 IR in the NH3(1,1) and (2, 2) transitions using the VLA in its D configuration. The study has been made with 5'' angular resolution and 0.3 km s-1 velocity resolution.As has been seen before, there is an elongated NH3 core that lies perpendicular to the molecular outflow as traced in high-velocity CO emission. In this experiment, the NH3 core is resolved, showing a central cavity and a number of distinct velocity features. Heating is seen at the position where the velocity features overlap spatially, and on the edges of the cavity, which is also seen as a reflection nebula with evidence for shock excitation. We have also detected what appears to be a new 1.3 cm continuum source on the wall of the cavity, associated with a 2.2 μm point source and jetlike structure. It is not clear at this time whether this is a truly continuum emission or high-velocity ammonia emission. The overall kinematics is complicated. A velocity gradient can be seen, together with the signatures for expanding or contracting motions.We consider here two possible models: (a) a disk or ring structure, slowly rotating and contracting, and (b) an expanding cavity.

The conformation of semi-rigid polymers during flow through a fixed fiber bed

Abstract: Linear conservative birefringence measurements of dilute and semi-dilute solutions of the semi-rigid polymer xanthan gum in a glycerine and water solvent are conducted during flow through a disordered fiber bed. Even though the plug flow through the bed contains no average velocity gradient our results show that the xanthan gum molecules stretch, on average, in the direction of flow. The polymer stretch or birefringence increases with the number of fiber interactions within the bed until a steady-state conformation is attained. The degree of stretch increases monotonically with increasing De pore (the ratio of a characteristic polymer relaxation time and the characteristic flow time in a bed pore). The results show a number of qualitative differences when compared to those for the flow of flexible polyisobutylene (PIB) (Evans et al., J. Fluid Mech., 28 (1994) 319) through the same fiber bed. In particular, xanthan gum attains a steady-state conformation after convecting 12 pore-lengths in the bed (compared to approximately 25 for the PIB solution) and this value is independent of De pore whereas it increased with increasing De pore for the flexible polymer. Additionally the steady-state birefringence of the xanthan gum solutions displays no critical De pore at which the birefringence increases dramatically (as does the flexible polymer system) and instead exhibits a slow growth with increasing De pore . These unique results are interpreted and understood using novel Brownian dynamics simulations of a semi-rigid bead-rod polymer model flowing through a simulated stochastic porous bed flow field created via a spectral expansion. Our results show that the simulations can predict most of the experimental results, at least qualitatively, and in some instances quantitatively, and therefore provide an important tool for understanding these complicated phenomena.

Low Reynolds Number Effects on the Inner Region of a Turbulent Boundary Layer

Abstract: Low-Reynolds-number effects on the inner region of a zero pressure gradient turbulent boundary layer have been investigated using a two-component laser-Doppler velocimetry (LDV) system. The momentum thickness Reynolds number R θ is in the range 400 to 1320. The wall shear stress is determined from the mean velocity gradient close to the wall, allowing scaling on wall variables to be examined unambiguously. The results indicate that, for the present R θ range, this scaling is not appropriate.

Nonsimilar MHD Boundary Layer Flow Near in Asymmetric Plane Stagnation Point

Abstract: The nonsimilar boundary layer equations of the title problem are solved using a reliable numerical scheme for obtaining the coefficient ofskin-friction. It is found that the normal surface velocity gradient increases with the magnetic interaction parameter.