Showing papers on "Similarity solution published in 2007"

Gravity currents in horizontal porous layers: transition from early to late self-similarity

Abstract: We investigate the evolution of a finite release of fluid into an infinite, two-dimensional, horizontal, porous slab saturated with a fluid of different density and viscosity. The vertical boundaries of the slab are impermeable and the released fluid spreads as a gravity current along a horizontal boundary. At early times the released fluid fills the entire height of the layer, and the governing equation admits a self-similar solution that is a function of the viscosity ratio between the two fluids. This early similarity solution describes a tilting interface with tips propagating as x ∝ t 1/2 . At late times the released fluid has spread along the boundary and the height of the current is much smaller than the thickness of the layer. The governing equation simplifies and admits a different similarity solution that is independent of the viscosity ratio. This late similarity solution describes a point release of fluid in a semi-infinite porous half-space, where the tip of the interface propagates as x ∝ t 1/3 . The same simplification of the governing equation occurs if the viscosity of the released fluid is much higher than the viscosity of the ambient fluid. We have obtained an expression for the time when the solution transitions from the early to the late self-similar regime. The transition time increases monotonically with increasing viscosity ratio. The transition period during which the solution is not self-similar also increases monotonically with increasing viscosity ratio, for mobility ratios larger than unity. Numerical computations describing the full evolution of the governing equation show good agreement with the theoretical results. Estimates of the spreading of injected fluids over long times are important for geological storage of CO 2 , and for the migration of pollutants in aquifers. In all cases it is important to be able to anticipate when the spreading regime transitions from x ∝ t 1/2 to Χ ∝ t 1/3 .

The transient start of supersonic jets

Abstract: This study investigates the initial transient hydrodynamic evolution of highly under-expanded slit and round jets. A closed-form analytic similarity solution is derived for the temporal evolution of temperature, pressure and density at the jet head for vanishing diffusive fluxes, generalizing a previous model of Chekmarev using Chernyi's boundary-layer method for hypersonic flows. Two-dimensional numerical simulations were also performed to investigate the flow field during the initial stages over distances of ~ 1000 orifice radii. The parameters used in the simulations correspond to the release of pressurized hydrogen gas into ambient air, with pressure ratios varying between approximately 100 and 1000. The simulations confirm the similarity laws derived theoretically and indicate that the head of the jet is laminar at early stages, while complex acoustic instabilities are established at the sides of the jet, involving shock interactions within the vortex rings, in good agreement with previous experimental findings. Very good agreement is found between the present model, the numerical simulations and previous experimental results obtained for both slit and round jets during the transient establishment of the jet. Criteria for Rayleigh–Taylor instability of the decelerating density gradients at the jet head are also derived, as well as the formulation of a model addressing the ignition of unsteady expanding diffusive layers formed during the sudden release of reactive gases.

Behaviour of a conducting drop in a highly viscous fluid subject to an electric field

Abstract: We consider the slow deformation of a relatively inviscid conducting drop surrounded by a viscous insulating fluid subject to a uniform electric field. The general behaviour is to deform and elongate in the direction of the field. Detailed numerical computations, based on a boundary integral formulation, are presented. For fields below a critical value, we obtain the evolution of the drop to an equilibrium shape; above the critical value, we calculate the drop evolution up to breakup. At breakup it appears that smaller droplets are emitted from the ends of the drop with a charge greater than the Rayleigh limit. As the electric field strength is increased the ejected droplet size decreases. A further increase in field strength results in the mode of breakup changing to a thin jet-like structure being ejected from the end. The shape of all drops is very close to spheroidal up to aspect ratios of about 5. Also, for fields just above the critical value there is a period of slow deformation which increases in duration as the critical field strength is approached from above. Slender-body theory is also used to model the drop behaviour. A similarity solution for the slender drop is obtained and a finite-time singularity is observed. In addition, the general solution for the slender-body equations is presented and the solution behaviour is examined. The slender-body results agree only qualitatively with the full numerical computations. Finally, a spheroidal model is briefly presented and compared with the other models.

Early-Time Solution for a Radial Hydraulic Fracture

Abstract: The small-time asymptotic solution for a penny-shaped fluid-driven fracture is obtained semianalytically. Scaling considerations indicate that the portion of the fracture that is filled with fluid increases with time according to a power law. The problem is shown to be self-similar at the length scale of the small fluid-filled region and to depend on only the mean fluid pressure at the length scale of the fracture. This similarity solution is unusual as the two length scales of the problem — the radius of the fracture and the radius of the fluid front — evolve according to two different power laws of time.

Two-point similarity in the round jet

Abstract: The governing equations for the two-point velocity correlations in the far field of the axisymmetric jet are examined and it is shown that these equations can have equilibrium similarity solutions for jets with finite Reynolds number that retain a dependence on the growth rate of the jet. The two-point velocity correlation can be written as the product of a scale that depends on the downstream position of the two points and a function that only depends on the similarity variables. Physically, this result implies that the turbulent processes producing and dissipating energy at the different scales of motion, as well as transferring energy between the different scales of motion, are in equilibrium as the flow evolves downstream. A particularly interesting prediction from the analysis is that the two-point similarity solutions depend only on the separation distance between the points in the streamwise similarity coordinate (i.e. v=xi'-xi), that is, the logarithm of the streamwise coordinate itself (i.e. xi=ln x(1), where x, is measured from a virtual origin). Thus, the measures of the turbulence are homogeneous in the streamwise similarity coordinate. The predictions from the similarity analysis for the streamwise two-point velocity correlation were compared with combined hot-wire and LDA measurements on the centreline of a round jet at a Reynolds number of 33000, and with two-point velocity correlations computed from PIV measurements in a round jet at a Reynolds number of 2000 performed by Fukushima et al. In both cases, the measured two-point velocity correlations in the streamwise direction collapsed when they were scaled in the manner predicted by the similarity analysis. The results provide further evidence that the equilibrium similarity hypothesis does describe the development of the flow in fully developed turbulent round jets and that the two-point correlations are statistically homogeneous in the streamwise similarity coordinate.

Non-similar solution for the axisymmetric flow of a third-grade fluid over a radially stretching sheet

Abstract: The problem of axisymmetric flow of a third grade fluid over a radially stretching sheet is studied. By means of similarity transformation, the governing non-linear partial differential equations are reduced to a non-linear ordinary differential equation. The ordinary differential equation is analytically solved using homotopy analysis method (HAM). The solution for the velocity is obtained. The series solution is developed and the convergence of the results is discussed. Finally, the results are discussed with various graphs.

Surface tension dominated impact

Abstract: We study the impact of a line mass onto a liquid-gas interface. At early times we find a similarity solution for the interfacial deformation and show how the resulting surface tension force slows the fall of the mass. We compute the motion beyond early times using a boundary integral method, and find conditions on the weight and impact speed of the mass that determine whether it sinks or is trapped by the interface. We find that for given impact speed there is a critical weight above which the mass sinks, and we investigate the asymptotic behavior of this critical weight in the limits of small and large impact speeds. Below this critical weight, the mass is trapped by the interface and subsequently floats. We also compare our theoretical results with some simple tabletop experiments. Finally, we discuss the implications of our work for the vertical jumps of water-walking arthropods.

Axisymmetric stagnation-point flow over a lubricated surface

Abstract: Axisymmetric stagnation-point flow is considered. A Newtonian fluid impinges orthogonally on a plane surface lubricated by a thin non-Newtonian liquid film of variable thickness. A slip-flow boundary condition is deduced, which allows for partial slip at the surface. The amount of slip, from full slip to no-slip, is controlled by a dimensionless slip coefficient. Similarity solutions are generally prohibited by the slip-flow boundary condition, except for one particular value of the power-law index of the lubricant. Solutions are presented for this case in order to demonstrate the influence of partial slip on the stagnation point flow. With increasing slip and reduced surface stress, a thinning of the viscous boundary layer is observed. The classical Homann flow is recovered in the no-slip limit.

Unstable sixth-order thin film equation: II. Global similarity patterns

Abstract: We continue the study begun in part I (Evans J D, Galaktionov V A and King J R 2007 Unstable sixth-order thin film equation: I. Blow-up of similarity solutions Nonlinearity 20 1799–841) of asymptotic large time behaviour of global solutions of the sixth-order thin film equation 0, \quad p>1,\end{eqnarray*} \] SRC=http://ej.iop.org/images/0951-7715/20/8/003/non237043ude001.gif/> with bounded integrable initial data. We show that for the first critical exponent, where N is the space dimension, the free-boundary problem with zero-contact-angle, zero-moment and zero-flux conditions at the interface admits continuous families (branches) of radially symmetric self-similar solutions defined for all t > 0, We also study the Cauchy problem, for which we construct global similarity solutions of the maximal regularity, these being oscillatory near the interfaces for n (0, nh), where is a 'heteroclinic bifurcation' point for a related nonlinear ordinary differential equation. We use various concepts based on the branching of sufficiently small solutions from the known eigenfunctions of the linear rescaled operator corresponding to n = 0.

An approximate analytical solution for counter-current spontaneous imbibition

Abstract: An approximate analytical solution is provided for one-dimensional, counter- current, spontaneous imbibition of a wetting phase (water) into a semi-infinite porous medium. The solution is based on the assumption that a similarity solution exists for the displacement process. This assumption, in turn, rests on the assumption that the set of relative permeability and capillary pressures curves are unique functions of saturation and do not depend on the nature of the displacement. It further rests on the assumption that the saturation at the imbibition face does not vary with time. It is demonstrated that the solution is in agreement with results obtained from experiments and also numerical analyses of these experiments. The experiments utilize cylindrical samples with the radial surface and one end-face sealed, and with counter-current imbibition occurring at the open end-face. The stage of the experiment that is modeled by the present solution is the period before the imbibition front contacts the sealed end-face. An important finding of the present analysis is that the pressure upstream of the advancing invasion front is a constant. A second, improved solution is also presented; this solution is an iterative, series solution of an integral-differential equation. It converges to a stable solution in very few terms.

Two-dimensional liquid column and liquid droplet impact on a solid wedge

Abstract: The hydrodynamic impact due to a two-dimensional (2D) liquid column or a 2D liquid droplet hitting on a solid wedge is analysed. The problem is solved using the complex velocity potential together with the boundary element method. A stretched coordinate system is used, which is defined through the ratio of the normal Cartesian coordinate system to an appropriately chosen time-varying length scale. Numerical simulations are first made for impact by a liquid wedge. The results from the time-domain method are found to be in a good agreement with the similarity solution. Simulations are also made for impact by an elliptic droplet. A condition for bisection of the droplet is introduced, which is found to provide stable and converged results.

A Coupled Variable Coefficient Modified KdV Equation Arising from a Two-Layer Fluid System

Abstract: A quite general coupled variable coefficient modified KdV (VCmKdV) equation in a two-layer fluid system is derived by means of the reductive perturbation method. Making use of the CK's direct method, some similarity reductions of the coupled VCmKdV equation are obtained and their corresponding group explanations are discussed. Some exact solutions of the coupled equations are also presented.

Behaviour of a conducting drop in a highly viscous fluid subject to an electric field

Abstract: We consider the slow deformation of a relatively inviscid conducting drop surrounded by a viscous insulating fluid subject to a uniform electric field. The general behaviour is to deform and elongate in the direction of the field. Detailed numerical computations, based on a boundary integral formulation, are presented. For fields below a critical value, we obtain the evolution of the drop to an equilibrium shape; above the critical value, we calculate the drop evolution up to breakup. At breakup it appears that smaller droplets are emitted from the ends of the drop with a charge greater than the Rayleigh limit. As the electric field strength is increased the ejected droplet size decreases. A further increase in field strength results in the mode of breakup changing to a thin jet-like structure being ejected from the end. The shape of all drops is very close to spheroidal up to aspect ratios of about 5. Also, for fields just above the critical value there is a period of slow deformation which increases in duration as the critical field strength is approached from above. Slender-body theory is also used to model the drop behaviour. A similarity solution for the slender drop is obtained and a finite-time singularity is observed. In addition, the general solution for the slender-body equations is presented and the solution behaviour is examined. The slender-body results agree only qualitatively with the full numerical computations. Finally, a spheroidal model is briefly presented and compared with the other models.

Dual solutions in mixed convection boundary-layer flow with suction or injection

Abstract: The effect of suction/injection on the laminar mixed convection boundary-layer flow about a vertical wall in an incompressible viscous fluid is considered. The similarity solutions are obtained for some values of the suction/injection parameter as well as the mixed convection parameter for three particular cases: uniform temperature, uniform heat flux and stagnation flow. The resulting system of non-linear ordinary differential equations is solved numerically for both assisting and opposing flow regimes using a finite-difference scheme known as the Keller box method. Numerical results are obtained for the skin friction coefficient and local Nusselt number as well as velocity and temperature profiles. The effects of the involved parameters on the skin friction coefficient and the local Nusselt number characteristics are discussed. It is found that dual solutions exist for assisting flow, besides that usually reported in the literature for opposing flow.

Two-point similarity in temporally evolving plane wakes

Abstract: The governing equations for the two-point correlations of the turbulent fluctuating velocity in the temporally evolving wake were analysed to determine whether they could have equilibrium similarity solutions. It was found that these equations could have such solutions for a finite-Reynolds-number wake, where the two-point velocity correlations could be written as a product of a time-dependent scale and a function dependent only on similarity variables. It is therefore possible to collapse the two-point measures of all the scales of motions in the temporally evolving wake using a single set of similarity variables. As in an earlier single-point analysis, it was found that the governing equations for the equilibrium similarity solutions could not be reduced to a form that was independent of a growth-rate dependent parameter. Thus, there is not a single 'universal' solution that describes the state of the large-scale structures, so that the large-scale structures in the far field may depend on how the flow is generated. The predictions of the similarity analysis were compared to the data from two direct numerical simulations of the temporally evolving wakes examined previously. It was found that the two-point velocity spectra of these temporally evolving wakes collapsed reasonably well over the entire range of scales when they were scaled in the manner deduced from the equilibrium similarity analysis. Thus, actual flows do seem to evolve in a manner consistent with the equilibrium similarity solutions.