Showing papers on "Hele-Shaw flow published in 2013"

Compressive Sensing Based Machine Learning Strategy For Characterizing The Flow Around A Cylinder With Limited Pressure Measurements

Abstract: Compressive sensing is used to determine the flow characteristics around a cylinder (Reynolds number and pressure/flow field) from a sparse number of pressure measurements on the cylinder. Using a supervised machine learning strategy, library elements encoding the dimensionally reduced dynamics are computed for various Reynolds numbers. Convex L1 optimization is then used with a limited number of pressure measurements on the cylinder to reconstruct, or decode, the full pressure field and the resulting flow field around the cylinder. Aside from the highly turbulent regime (large Reynolds number) where only the Reynolds number can be identified, accurate reconstruction of the pressure field and Reynolds number is achieved. The proposed data-driven strategy thus achieves encoding of the fluid dynamics using the L2 norm, and robust decoding (flow field reconstruction) using the sparsity promoting L1 norm.

On the mechanism of elasto-inertial turbulence

Abstract: Elasto-inertial turbulence (EIT) is a new state of turbulence found in inertial flows with polymer additives. The dynamics of turbulence generated and controlled by such additives is investigated from the perspective of the coupling between polymer dynamics and flow structures. Direct numerical simulations of channel flow with Reynolds numbers ranging from 1000 to 6000 (based on the bulk and the channel height) are used to study the formation and dynamics of elastic instabilities and their effects on the flow. The flow topology of EIT is found to differ significantly from Newtonian wall-turbulence. Structures identified by positive (rotational flow topology) and negative (extensional/compressional flow topology) second invariant Qa isosurfaces of the velocity gradient are cylindrical and aligned in the spanwise direction. Polymers are significantly stretched in sheet-like regions that extend in the streamwise direction with a small upward tilt. The Qa cylindrical structures emerge from the sheets of high polymer extension, in a mechanism of energy transfer from the fluctuations of the polymer stress work to the turbulent kinetic energy. At subcritical Reynolds numbers, EIT is observed at modest Weissenberg number (Wi, ratio polymer relaxation time to viscous time scale). For supercritical Reynolds numbers, flows approach EIT at large Wi. EIT provides new insights on the nature of the asymptotic state of polymer drag reduction (maximum drag reduction), and explains the phenomenon of early turbulence, or onset of turbulence at lower Reynolds numbers than for Newtonian flows observed in some polymeric flows.

Oblique laminar-turbulent interfaces in plane shear flows.

Abstract: Localized structures such as turbulent stripes and turbulent spots are typical features of transitional wall-bounded flows in the subcritical regime. Based on an assumption for scale separation between large and small scales, we show analytically that the corresponding laminar-turbulent interfaces are always oblique with respect to the mean direction of the flow. In the case of plane Couette flow, the mismatch between the streamwise flow rates near the boundaries of the turbulence patch generates a large-scale flow with a nonzero spanwise component. Advection of the small-scale turbulent fluctuations (streaks) by the corresponding large-scale flow distorts the shape of the turbulence patch and is responsible for its oblique growth. This mechanism can be easily extended to other subcritical flows such as plane Poiseuille flow or Taylor-Couette flow.

Predictions of a Supersonic Turbulent Flow in a Square Duct

Abstract: In this paper we have confirmed that eddy viscosity turbulence models are inadequate to predict secondary vortical flows developed from corners in internal flows. To remedy this shortcoming, we have added the quadratic constitutive relation (QCR) of Spalart to the one- and two-equation SA and SST turbulence models, respectively. The results of QCR with the SA and SST turbulence models have been validated against experimental data of Davis and Gessner for supersonic flow through a square duct. The approach is shown to be simple to implement and overall agreement is seen to improve with the use of QCR. Introduction: Supersonic flow through a square duct is altered by secondary vortical flow developing from the corners. These secondary flows are generated by Reynolds stress gradients acting in the corner region and appear to have similar structure to those found in subsonic flow through square ducts (1-3). Such flow is representative of various airplane inlets and therefore it is important to understand and predict the impact of this secondary flow on inlet pressure recovery and distortion. An experimental study was performed at the University of Washington to gain a better understanding of how the secondary flow associated with corners affects local flow conditions in a square duct over the development length (3). This configuration was utilized in this study to validate and improve the RANS turbulence models utilized for corner flows at Boeing. Accurate prediction of the flow around and through the aircraft is essential for design improvement, risk mitigation, and wind-tunnel and flight test reduction. At Boeing, the RANS approach is used routinely for design and analysis of our configurations. While RANS has proven to be a powerful approach for flow-field prediction, it still has some shortcomings even for steady state flow predictions. The one- and two-equation SA (4) and SST (5) eddy viscosity turbulence models, respectively, are still the workhorses for routine applications. The Reynolds

The effect of neutrally buoyant finite-size particles on channel flows in the laminar-turbulent transition regime

Abstract: The presence of finite-size particles in a channel flow close to the laminar-turbulent transition is simulated with the Force Coupling Method which allows two-way coupling with the flow dynamics. Spherical particles with channel height-to-particle diameter ratio of 16 are initially randomly seeded in a fluctuating flow above the critical Reynolds number corresponding to single phase flow relaminarization. When steady-state is reached, the particle volume fraction is homogeneously distributed in the channel cross-section (ϕ ≅ 5%) except in the near-wall region where it is larger due to inertia-driven migration. Turbulence statistics (intensity of velocity fluctuations, small-scale vortical structures, wall shear stress) calculated in the fully coupled two-phase flow simulations are compared to single-phase flow data in the transition regime. It is observed that particles increase the transverse r.m.s. flow velocity fluctuations and they break down the flow coherent structures into smaller, more numerous and sustained eddies, preventing the flow to relaminarize at the single-phase critical Reynolds number. When the Reynolds number is further decreased and the suspension flow becomes laminar, the wall friction coefficient recovers the evolution of the laminar single-phase law provided that the suspension viscosity is used in the Reynolds number definition. The residual velocity fluctuations in the suspension correspond to a regime of particulate shear-induced agitation.

Transition between regimes of a vertical channel bubbly upflow due to bubble deformability

Abstract: The effect of the deformability of viscous bubbles on the flow rate of bubbly upflow in a vertical channel is examined using direct numerical simulations. A sharp transition between two different flow regimes has been observed. At large bubble deformability, characterized by large Eotvos number (Eo), the flow rate is close to the single phase flow rate, with adjusted pressure gradient, and the bubbles are almost uniformly distributed in the middle of the channel. On the other hand, at low Eo the bubbles are concentrated near channel walls and flow rates are much smaller than the single phase flow. The transition from high flow rate to low flow rate occurs rather abruptly. It is found that the transition occurs when the less deformable bubbles enter the viscous sublayer due to the lateral lift force on the bubbles. This leads to an increase in the viscous dissipation near the wall which leads to a decrease in the flow rate.

Extension of the Darcy–Forchheimer Law for Shear-Thinning Fluids and Validation via Pore-Scale Flow Simulations

Abstract: Flow of non-Newtonian fluids through porous media at high Reynolds numbers is often encountered in chemical, pharmaceutical and food, as well as petroleum and groundwater engineering, and in many other industrial applications. Under the majority of operating conditions typically explored, the dependence of pressure drops on flow rate is non-linear and the development of models capable of describing accurately this dependence, in conjunction with non-trivial rheological behaviors, is of paramount importance. In this work, pore-scale single-phase flow simulations conducted on synthetic two-dimensional porous media are performed via computational fluid dynamics for both Newtonian and non-Newtonian fluids and the results are used for the extension and validation of the Darcy–Forchheimer law, herein proposed for shear-thinning fluid models of Cross, Ellis and Carreau. The inertial parameter β is demonstrated to be independent of the viscous properties of the fluids. The results of flow simulations show the superposition of two contributions to pressure drops: one, strictly related to the non-Newtonian properties of the fluid, dominates at low Reynolds numbers, while a quadratic one, arising at higher Reynolds numbers, is dependent on the porous medium properties. The use of pore-scale flow simulations on limited portions of the porous medium is here proposed for the determination of the macroscale-averaged parameters (permeability K, inertial coefficient β and shift factor α), which are required for the estimation of pressure drops via the extended Darcy–Forchheimer law. The method can be applied for those fluids which would lead to critical conditions (high pressures for low permeability media and/or high flow rates) in laboratory tests.

Bending of elastic fibres in viscous flows: the influence of confinement ‡

Abstract: We present a mathematical model and corresponding series of microfluidic experiments examining the flow of a viscous fluid past an elastic fibre in a three-dimensional channel. The fibre’s axis lies perpendicular to the direction of flow and its base is clamped to one wall of the channel; the sidewalls of the channel are close to the fibre, confining the flow. Experiments show that there is a linear relationship between deflection and flow rate for highly confined fibres at low flow rates, which inspires an asymptotic treatment of the problem in this regime. The three-dimensional problem is reduced to a two-dimensional model, consisting of Hele-Shaw flow past a barrier, with boundary conditions at the barrier that allow for the effects of flexibility and three-dimensional leakage. The analysis yields insight into the competing effects of flexion and leakage, and an analytical solution is derived for the leading-order pressure field corresponding to a slit that partially blocks a two-dimensional channel. The predictions of our model show favourable agreement with experimental results, allowing measurement of the fibre’s elasticity and the flow rate in the channel.

Three-dimensionality in the wake of a rotating cylinder in a uniform flow

Abstract: The wake of a rotating circular cylinder in a free stream is investigated for Reynolds numbers Re6 400 and non-dimensional rotation rates of 6 2:5. Two aspects are considered. The first is the transition from a steady flow to unsteady flow characterized by periodic vortex shedding. The two-dimensional computations show that the onset of unsteady flow is delayed to higher Reynolds numbers as the rotation rate is increased, and vortex shedding is suppressed for > 2:1 for all Reynolds numbers in the parameter space investigated. The second aspect investigated is the transition from two-dimensional to three-dimensional flow using linear stability analysis. It is shown that at low rotation rates of 6 1, the three-dimensional transition scenario is similar to that of the non-rotating cylinder. However, at higher rotation rates, the threedimensional scenario becomes increasingly complex, with three new modes identified that bifurcate from the unsteady flow, and two modes that bifurcate from the steady flow. Curves of marginal stability for all of the modes are presented in a parameter space map, the defining characteristics for each mode presented, and the physical mechanisms of instability are discussed.

Unsteady peristaltic transport in curved channels

Abstract: The paper presents a generalized mathematical model describing the unsteady peristaltic flow of a viscous fluid in a two-dimensional curved channel. The flow is investigated in a laboratory frame of reference and the unsteady flow nature is studied by the condition that prescribing volumetric flow rate is equivalent to prescribing normal velocity of the fluid particles at the wall. The momentum and energy equations have been linearized by employing lubrication theory and the analysis is restricted to negligible flow Reynolds number. The expressions for stream function, pressure distribution, shear stress, temperature, and coefficient of heat transfer have been derived. The obtained expressions are utilized to discuss the influences of various emerging parameters on flow phenomenon.

Intrinsic features of flow around two side-by-side square cylinders

Abstract: The wake of two side-by-side square cylinders is investigated in detail based on flow visualization at a Reynolds number (Re) of 300. The cylinder center-to-center spacing ratio T* (= T/W, W is the cylinder width) is varied from 1.0 to 5.0. The intrinsic features of the wake are explored, including the gap vortices, flow switch, stability, merging of two streets into one, etc. The qualitative information on these features is further complemented by the quantitative information extracted from hotwire data at Re = 4.7 × 104 using both spectral and cross-wavelet analyses. Four flow regimes are identified: (i) the single bluff body regime (T* 2.4). The gap flow is found to switch at two distinct time scales, referred to as macro and micro switches. Macro switch occurs at 1.2 < T* < 2.1, where the gap flow is slim in width and biased for a long duration ranging...

Study of instabilities and quasi-two-dimensional turbulence in volumetrically heated magnetohydrodynamic flows in a vertical rectangular duct

Abstract: We consider magnetohydrodynamic (MHD) rectangular duct flows with volumetric heating. The flows are upward, subject to a strong transverse magnetic field perpendicular to the temperature gradient, such that the flow dynamics is quasi-two-dimensional. The internal volumetric heating imitates conditions of a blanket of a fusion power reactor, where a buoyancy-driven flow is imposed on the forced flow. Studies of this mixed-convection flow include analysis for the basic flow, linear stability analysis and Direct Numerical Simulation (DNS)-type computations. The parameter range covers the Hartmann number (Ha) up to 500, the Reynolds number (Re) from 1000 to 10 000, and the Grashof number (Gr) from 105 to 5 × 108. The linear stability analysis predicts two primary instability modes: (i) bulk instability associated with the inflection point in the velocity profile near the “hot” wall and (ii) side-wall boundary layer instability. A mixed instability mode is also possible. An equation for the critical Hartmann number has been obtained as a function of Re and Gr. Effects of Ha, Re, and Gr on turbulent flows are addressed via nonlinear computations that demonstrate two characteristic turbulence regimes. In the “weak” turbulence regime, the induced vortices are localized near the inflection point of the basic velocity profile, while the boundary layer at the wall parallel to the magnetic field is slightly disturbed. In the “strong” turbulence regime, the bulk vortices interact with the boundary layer causing its destabilization and formation of secondary vortices that may travel across the flow, even reaching the opposite wall. In this regime, the key phenomena are vortex-wall and various vortex-vortex interactions. Flow and magnetic field effects on heat transfer are also analyzed.

Multiscale considerations in direct numerical simulations of multiphase flowsa)

Abstract: Direct Numerical Simulations of multiphase flows have progressed rapidly over the last decade and it is now possible to simulate, for example, the motion of hundreds of deformable bubbles in turbulent flows. The availability of results from such simulations should help advance the development of new and improved closure relations and models of the average or large-scale flows. We review recent results for bubbly flow in vertical channels, discuss the difference between upflow and downflow and the effect of the bubble deformability and how the resulting insight allowed us to produce a simple description of the large scale flow, for certain flow conditions. We then discuss the need for the development of numerical methods for more complex situations, such as where the flow creates spontaneous thin films and threads, or where additional physical processes take place at a rate that is very different from the fluid flow. Recent work on capturing localized small-scale processes using embedded analytical models,...

Numerical studies of the effects of large neutrally buoyant particles on the flow instability and transition to turbulence in pipe flow

Abstract: The effects of large neutrally buoyant particles on the flow instability and turbulence transition in pipe flow are investigated with the fictitious domain method. The periodic boundary condition is introduced in the streamwise direction. The work comprises two parts. In the first part, the pressure gradient is kept constant, and the purpose is to study the particle-induced flow instability. In our previous study [X. Shao, Z. Yu, and B. Sun, Phys. Fluids 20, 103307 (2008)10.1063/1.3005427], it was observed that a particle of a/R = 0.1 (a and R being the radii of the particle and the tube, respectively) induced the flow structure characterized by two pairs of weak and stable streamwise vortices at the Reynolds number of 1000. In the present study, our results show that the flow structure loses stability at the Reynolds number of 1500. However, it is interesting that the system eventually reaches a stable state: the particle spirals forward along the tube wall, accompanied by a stable flow structure for the...

Relaminarisation of Re_{\tau} = 100 channel flow with globally stabilising linear feedback control

Abstract: The problems of nonlinearity and high dimension have so far prevented a complete solution of the control of turbulent flow. Addressing the problem of nonlinearity, we propose a flow control strategy which ensures that the energy of any perturbation to the target profile decays monotonically. The controller's estimate of the flow state is similarly guaranteed to converge to the true value. We present a one-time off-line synthesis procedure, which generalises to accommodate more restrictive actuation and sensing arrangements, with conditions for existence for the controller given in this case. The control is tested in turbulent channel flow ($Re_\tau=100$) using full-domain sensing and actuation on the wall-normal velocity. Concentrated at the point of maximum inflection in the mean profile, the control directly counters the supply of turbulence energy arising from the interaction of the wall-normal perturbations with the flow shear. It is found that the control is only required for the larger-scale motions, specifically those above the scale of the mean streak spacing. Minimal control effort is required once laminar flow is achieved. The response of the near-wall flow is examined in detail, with particular emphasis on the pressure and wall-normal velocity fields, in the context of Landahl's theory of sheared turbulence.

Modeling Non-Darcian Single- and Two-Phase Flow in Transparent Replicas of Rough-Walled Rock Fractures

Abstract: While it is generally assumed that in the viscous flow regime, the two-phase flow relative permeabilities in fractured and porous media depend uniquely on the phase saturations, several studies have shown that for non-Darcian flows (i.e., where the inertial forces are not negligible compared with the viscous forces), the relative permeabilities not only depend on phase saturations but also on the flow regime. Experimental results on inertial single- and two-phase flows in two transparent replicas of real rough fractures are presented and modeled combining a generalization of the single-phase flow Darcy’s law with the apparent permeability concept. The experimental setup was designed to measure injected fluid flow rates, pressure drop within the fracture, and fluid saturation by image processing. For both fractures, single-phase flow experiments were modeled by means of the full cubic inertial law which allowed the determination of the intrinsic hydrodynamic parameters. Using these parameters, the apparent permeability of each fracture was calculated as a function of the Reynolds number, leading to an elegant means to compare the two fractures in terms of hydraulic behavior versus flow regime. Also, a method for determining the experimental transition flow rate between the weak inertia and the strong inertia flow regimes is proposed. Two-phase flow experiments consisted in measuring the pressure drop and the fluid saturation within the fractures, for various constant values of the liquid flow rate and for increasing values of the gas flow rate. Regardless of the explored flow regime, two-phase flow relative permeabilities were calculated as the ratio of the single phase flow pressure drop per unit length divided by the two-phase flow pressure drop per unit length, and were plotted versus the measured fluid saturation. Results confirm the dependence of the relative permeabilities on the flow regime. Also the proposed generalization of Darcy’s law shows that the relative permeabilities versus fluid saturation follow physical meaningful trends for different liquid and gas flow rates. The presented model fits correctly the liquid and gas experimental relative permeabilities as well as the fluid saturation.

Investigation of the Flow Structures in Supersonic Free and Impinging Jet Flows

Abstract: A numerical study of compressible jet flows is carried out using Reynolds averaged Navier-Stokes (RANS) turbulence models such as k-E and k-ω-SST. An experimental investigation is performed concurrently using high-speed optical methods such as Schlieren photography and shadowgraphy. Numerical and experimental studies are carried out for the compressible impinging at various impinging angles and nozzle-to-wall distances. The results from both investigations converge remarkably well and agree with experimental data from the open literature. From the flow visualizations of the velocity fields, the RANS simulations accurately model the shock structures within the core jet region. The first shock cell is found to be constraint due to the interaction with the bow-shock structure for nozzle-to-wall distance less than 1.5 nozzle diameter. The results from the current study show that the RANS models utilized are suitable to simulate compressible free jets and impinging jet flows with varying impinging angles. © 2013 by ASME.

Fluid drainage from the edge of a porous reservoir

Abstract: We report theoretical and experimental studies to describe buoyancy-driven fluid drainage from a porous medium for configurations where the fluid drains from an edge. We first study homogeneous porous systems. To investigate the influence of heterogeneities, we consider the case where the permeability varies transverse to the flow direction, exemplified by a V-shaped Hele-Shaw cell. Finally, we analyse a model where both the permeability and the porosity vary transverse to the flow direction. In each case, a self-similar solution for the shape of these gravity currents is found and a power-law behaviour in time is derived for the mass remaining in the system. Laboratory experiments are conducted in homogeneous and V-shaped Hele-Shaw cells, and the measured profile shapes and the mass remaining in the cells agree well with our model predictions. Our study provides new insights into drainage processes such as may occur in a variety of natural and industrial activities, including the geological storage of carbon dioxide.

A Voronoï analysis of preferential concentration in a vertical channel flow

Abstract: We use three-dimensional Voronoi analysis and results from a direct numerical simulation to study the preferential concentration of inertial particles in a vertical channel flow at Reynolds number 395. By comparing results in upward and downward flows with results from a channel flow without gravity, we are able to determine how gravity affects the particle clustering. Gravity increases the drift of particles towards the walls in an upward flow, while in the downward flow more particles are transported to the centre of the channel. For particles with Stokes number 100, the mean wall-normal particle velocity is positive in the entire core region. A significant increase in variance of the Voronoi probability distribution in the core region is observed in downward flow for Stokes numbers 30 and 100, indicating stronger particle clustering than in upward flow or flow without gravity. The increased clustering in the downward flow is believed to be partly caused by the reversed wall-normal drift assisting in bringing particles close together in the centre of the channel.

Physics of Gas-Liquid Flows

Abstract: Presenting tools for understanding the behaviour of gas-liquid flows based on the ways large scale behaviour relates to small scale interactions, this text is ideal for engineers seeking to enhance the safety and efficiency of natural gas pipelines, water-cooled nuclear reactors, absorbers, distillation columns and gas lift pumps. The review of advanced concepts in fluid mechanics enables both graduate students and practising engineers to tackle the scientific literature and engage in advanced research. It focuses on gas-liquid flow in pipes as a simple system with meaningful experimental data. This unified theory develops design equations for predicting drop size, frictional pressure losses and slug frequency, which can be used to determine flow regimes, the effects of pipe diameter, liquid viscosity and gas density. It describes the effect of wavy boundaries and temporal oscillations on turbulent flows, and explains transition between phases, which is key to understanding the behaviour of gas-liquid flows.

Bubble extinction in Hele-Shaw flow with surface tension and kinetic undercooling regularization

Abstract: We perform an analytic and numerical study of an inviscid contracting bubble in a two-dimensional Hele-Shaw cell, where the effects of both surface tension and kinetic undercooling on the moving bubble boundary are not neglected. In contrast to expanding bubbles, in which both boundary effects regularize the ill-posedness arising from the viscous (Saffman–Taylor) instability, we show that in contracting bubbles the two boundary effects are in competition, with surface tension stabilizing the boundary, and kinetic undercooling destabilizing it. This competition leads to interesting bifurcation behaviour in the asymptotic shape of the bubble in the limit it approaches extinction. In this limit, the boundary may tend to become either circular, or approach a line or 'slit' of zero thickness, depending on the initial condition and the value of a nondimensional surface tension parameter. We show that over a critical range of surface tension values, both these asymptotic shapes are stable. In this regime there exists a third, unstable branch of limiting self-similar bubble shapes, with an asymptotic aspect ratio (dependent on the surface tension) between zero and one. We support our asymptotic analysis with a numerical scheme that utilizes the applicability of complex variable theory to Hele-Shaw flow.

High-precision Taylor-Couette experiment to study subcritical transitions and the role of boundary conditions and size effects

Abstract: A novel Taylor-Couette system has been constructed for investigations of transitional as well as high Reynolds number turbulent flows in very large aspect ratios The flexibility of the setup enables studies of a variety of problems regarding hydrodynamic instabilities and turbulence in rotating flows The inner and outer cylinders and the top and bottom endplates can be rotated independently with rotation rates of up to 30 Hz, thereby covering five orders of magnitude in Reynolds numbers (Re = 101–106) The radius ratio can be easily changed, the highest realized one is η = 098 corresponding to an aspect ratio of 260 gap width in the vertical and 300 in the azimuthal direction For η < 098 the aspect ratio can be dynamically changed during measurements and complete transparency in the radial direction over the full length of the cylinders is provided by the usage of a precision glass inner cylinder The temperatures of both cylinders are controlled independently Overall this apparatus combines an unmatched variety in geometry, rotation rates, and temperatures, which is provided by a sophisticated high-precision bearing system Possible applications are accurate studies of the onset of turbulence and spatio-temporal intermittent flow patterns in very large domains, transport processes of turbulence at high Re, the stability of Keplerian flows for different boundary conditions, and studies of baroclinic instabilities

A new prediction of wavelength selection in radial viscous fingering involving normal and tangential stresses

Abstract: We reconsider the radial Saffman-Taylor instability, when a fluid injected from a point source displaces another fluid of higher viscosity in a Hele-Shaw cell, where the fluids are confined between two neighboring flat plates. The advancing fluid front is unstable and forms fingers along the circumference. The so-called Brinkman equation is used to describe the flow field, which also takes into account viscous stresses in the plane of the confining plates and, unlike the Darcy equation, not only viscous stresses due to the confining plates. We show why in-plane stresses cannot always be neglected and how they appear naturally in the potential flow problem. The dispersion relation obtained with the Brinkman equation agrees better with the experimental results than the classical linear stability analysis of radial fingering in Hele-Shaw cells that uses Darcy's law as a model for the fluid motion.

Unsteady separated stagnation-point flow of an incompressible viscous fluid on the surface of a moving porous plate

Abstract: Using group-theoretic method, an analysis is presented for a similarity solution of boundary layer equations which represents an unsteady two-dimensional separated stagnation-point (USSP) flow of an incompressible fluid over a porous plate moving in its own plane with speed u0(t). It is observed that the solution to the governing nonlinear ordinary differential equation for the USSP flow admits of two solutions (in contrast with the corresponding steady flow where the solution is unique): one is the attached flow solution (AFS) and the other is the reverse flow solution (RFS). A novel result of the analysis is that in the case of stationary plate (u0(t) = 0), after a certain value of the magnitude of the blowing d ( 0), there are two stagnation-points in the flow but in the presence of blowing d (<0), there is only one stagnation-point in the flow which moves further and further up with increase in |d|. Suction is shown to increase the wall shear stress while blowing has an opposite effect. Streamlines for an USSP flow when u0(t) ≠ 0 are also plotted. It is found that in this case, the USSP flow is not in general separated.