Earth and Environmental Sciences (12)
Physics And Astronomy (12)
Journal of Fluid Mechanics (12)
Ryan Test (12)
The role of gravity in the prediction of the circular hydraulic jump radius for high-viscosity liquids
Yunpeng Wang, Roger E. Khayat
Journal: Journal of Fluid Mechanics / Volume 862 / 10 March 2019
The free-surface flow formed by a circular jet impinging on a stationary disk is analysed theoretically. We develop a simple and coherent model to predict the location and height of the jump for high-viscosity liquids. The study explores the effect of gravity in the supercritical flow. The formulation reduces to a problem, involving only one parameter: $\unicode[STIX]{x1D6FC}=Re^{1/3}Fr^{2}$ , where $Re$ and $Fr$ are the Reynolds and Froude numbers based on the flow rate and the jet radius. We show that the jump location coincides with the singularity in the thin-film equation when gravity is included, suggesting that the jump location can be determined without the knowledge of downstream flow conditions such as the jump height, the radius of the disk, which corroborates earlier observations in the case of type I circular hydraulic jumps. Consequently, there is no need for a boundary condition downstream to determine the jump radius. Our results corroborate well existing measurements and numerical simulation. Our predictions also confirm the constancy of the Froude number $Fr_{J}$ based on the jump radius and height as suggested by the measurements of Duchesne et al. (Europhys. Lett., vol. 107, 2014, 54002). We establish theoretically the conditions for $Fr_{J}$ to remain independent of the flow rate. The subcritical flow and the height of the hydraulic jump are sought subject to the thickness at the edge of the disk, comprising contributions based on the capillary length and minimum flow energy. The thickness at the edge of the disk appears to be negligibly small for high-viscosity liquids.
Impinging jet flow and hydraulic jump on a rotating disk
The free-surface flow formed by a circular jet impinging on a rotating disk is analysed theoretically. The study explores the effects of rotation and inertia on the thin-film flow. Both boundary-layer height and film thickness are found to diminish with rotation speed. A maximum film thickness develops in the supercritical region, which reflects the competition between the convective and centrifugal effects. Unlike the flow on a stationary disk, an increase in the wall shear stress along the radial direction is predicted, at a rate that strengthens with rotating speed. Our results corroborate well existing measurements. The location and height of the hydraulic jump are determined subject to the value of the thickness at the edge of the disk, which is established first for a stationary disk based on the capillary length, and then for a rotating disk using existing analyses and measurements in spin coating. The case of a stationary is revisited in an effort to predict the location and height of the jump uniquely. The formulated value of the height at the edge of the disk seems to give excellent results for a jet at moderately high flow rate (or low viscosity) where the jump structure is well identifiable in reality.
Initial development of a free-surface wall jet at moderate Reynolds number
Roger E. Khayat
Journal: Journal of Fluid Mechanics / Volume 826 / 10 September 2017
Published online by Cambridge University Press: 03 August 2017, pp. 235-269
Print publication: 10 September 2017
The steady laminar flow of a moderately inertial wall jet is examined theoretically near the exit of a channel. The free-surface jet emerges asymmetrically from the channel as it adheres to an infinite (upper) wall subject to a pressure gradient. The problem is solved using the method of matched asymptotic expansions. The small parameter involved in the expansions is the inverse cubic power of the Reynolds number. The flow field is obtained by matching the inviscid rotational core flow separately with the free-surface and the two wall layers. The upstream influence is examined as well as the break in the symmetry between the two wall layers. The wall jet exhibits a contraction near the channel exit that is independent of inertia, and eventually expands for any Reynolds number. Unlike the flow of a wall jet emerging into the same ambient fluid, the free-surface jet experiences a limited weakening in shear stress along the infinite wall, suggesting the possibility of separation for a jet with relatively low inertia. Significant shearing and elongation ensue at the exit, accompanied by flattening of the velocity profile near the upper wall.
Impinging planar jet flow on a horizontal surface with slip
Journal: Journal of Fluid Mechanics / Volume 808 / 10 December 2016
Published online by Cambridge University Press: 28 October 2016, pp. 258-289
Print publication: 10 December 2016
The flow of a planar jet (sheet) impinging onto a solid flat plate with slip is examined theoretically. The jet is assumed to spread out in a thin layer bounded by a hydraulic jump, and draining at the edge of the plate. In contrast to an adhering jet, a slipping jet does not admit a similarity solution. Taking advantage of the different scaling in each region, series expansions are used in the developing and fully viscous layers, which are matched at the transition point. We show that a slipping film exhibits a singularity in the normal stress at the leading edge of the boundary layer, as opposed to the singularity in velocity and shear stress for an adhering film. The boundary-layer and film heights are both found to decrease with slip relative to a smooth substrate, roughly like $\sqrt{30x/Re}-2S$ , whereas the slip velocity intensifies like $S\sqrt{Re/30x}$ with slip. Here, $x$ is the distance along the plate, $S$ is the slip length and $Re$ is the Reynolds number (in units of the jet width). The transition is delayed by slip. Guided by the measurements of Duchesne et al. (Europhys. Lett., vol. 107, 2014, p. 54002) for a circular adhering jet, the hydraulic-jump height and location are determined for a planar jet, and are found to increase with the Froude number (flow rate) like $Fr^{1/4}$ and $Fr^{5/8}$ respectively, essentially independently of slip length.
Slipping free jet flow near channel exit at moderate Reynolds number for large slip length
The flow of a slipping fluid jet is examined theoretically as it emerges from a channel at moderate Reynolds number. The ratio of the slip length to the channel width $S$ is assumed to be of order one, one order of magnitude larger than the perturbation parameter ${\it\varepsilon}=Re^{-1/2}$ , $Re$ being the Reynolds number. Poiseuille flow conditions are assumed to prevail far upstream from the exit. The problem is solved using the method of matched asymptotic expansions. A similarity solution is obtained in the inner layer of the free surface, with the outer layer extending to the jet centreline. The inner-layer thickness grows like $\sqrt{x/Re\,S}$ . A slipping jet is found to contract like $x/Re$ very near and far from the channel exit, but does not have a definite behaviour in between compared to $(x/Re)^{1/3}$ for an adhering jet, $x$ being the distance from the channel exit. Eventually, the jet reaches uniform conditions far downstream. As in the case of entry flow, there is a rapid departure in flow behaviour for a slipping jet from the $S=0$ limit. This rapid change is notably observed in the drop of boundary-layer thickness, increase in exit and relaxation lengths as well as in jet width with slip length. Finally, the connections with microchannel and hydrophobic flows are highlighted.
Free-surface jet flow of a shear-thinning power-law fluid near the channel exit
Journal: Journal of Fluid Mechanics / Volume 748 / 10 June 2014
Print publication: 10 June 2014
The jet flow of a shear-thinning power-law fluid is examined theoretically as it emerges from a channel at moderate Reynolds number. Poiseuille flow conditions are assumed to prevail far upstream from the exit. The problem is solved using the method of matched asymptotic expansions. A similarity solution is obtained in the inner layer near the free surface, with the outer layer extending to the jet centreline. An inner thin viscous sublayer is introduced to smooth out the singularity in viscosity at the free surface, allowing the inner algebraically decaying solutions to be matched smoothly with the solution near the free surface. A Newtonian jet is found to contract more than a shear-thinning jet. While both the inner-layer thickness and the free-surface height are $O(\mathit{Re}^{-1/3})$ , and grow with downstream distance, the sublayer thickness is smaller, $O(\mathit{Re}^{-(1+n)/3})$ , growing with distance for $n < 0.5$ , and decaying for $n > 0.5$ . The relaxation downstream distance for the jet is found to grow logarithmically with $\mathit{Re}$ .
Pattern selection in the thermal convection of non-Newtonian fluids
BASHAR ALBAALBAKI, ROGER E. KHAYAT
The thermogravitational instability in a fluid layer of a non-Newtonian medium heated from below is investigated. Linear and weakly nonlinear analyses are successively presented. The fluid is assumed to obey the Carreau–Bird model. Although the critical threshold is the same as for a Newtonian fluid, it is found that non-Newtonian fluids can convect in the form of rolls, squares or hexagons, depending on the shear-thinning level. Similar to Newtonian fluids, shear-thickening fluids convect only in the form of rolls. The stability of the convective steady branches is carried out to determine under which specific conditions a pattern is preferred. The influence of the rheological and physical parameters is examined and discussed in detail.
Flow of viscoelastic jet with moderate inertia near channel exit
AMIR SAFFARI, ROGER E. KHAYAT
Journal: Journal of Fluid Mechanics / Volume 639 / 25 November 2009
Published online by Cambridge University Press: 07 October 2009, pp. 65-100
Print publication: 25 November 2009
The steady-state moderately inertial jet flow of a viscoelastic liquid of the Oldroyd-B type, emerging from a two-dimensional channel, is examined theoretically in this study. Poiseuille flow conditions are assumed to prevail far upstream from the exit. The problem is solved using the method of matched asymptotic expansions. The small parameter involved in the expansions is the inverse Reynolds number. The flow and stress fields are obtained as composite expansions by matching the flow in the boundary-layer region near the free surface and the flow in the core region. The influence of elasticity on the shape of the free surface, the profiles of velocity and stress and the interplay between inertia and elasticity are explored. It is found that even for a jet with moderate inertia, elastic effects play a significant role, especially close to the channel exit near the free surface. It is also found that similar to the Newtonian case, the viscoelastic jet contracts downstream from the channel exit. However, in contrast to Newtonian jet, a viscoelastic jet is preceded by a flat region very close to the channel exit at which elastic and inertial effects are in balance. The extent of this region increases with elasticity. A momentum integral balance is applied to validate the theory and obtain the jet contraction ratio explicitly in terms of the Deborah number, viscosity ratio and Reynolds number.
Spread of a non-Newtonian liquid jet over a horizontal plate
JIANGANG ZHAO, ROGER E. KHAYAT
Journal: Journal of Fluid Mechanics / Volume 613 / 25 October 2008
Print publication: 25 October 2008
The flow of an impinging non-Newtonian jet onto a solid flat plate is examined theoretically in this study. Similarity solutions are sought for both shear-thinning and shear-thickening fluids of the power-law type. The jet is assumed to spread out in a thin layer bounded by a hydraulic jump. In addition to the stagnation-flow region, the flow domain is divided into three main regions: a developing boundary layer, fully viscous boundary layer and hydraulic jump. The anomalous behaviour of power-law fluids at small shear rate is remedied by seeking a two-layer solution in each domain. Such anomalies include the singularity of viscosity for shear-thinning fluids, and the vanishing of viscosity as well the overshoot in velocity for shear-thickening fluids. Although the rate of shear-thinning appears to affect significantly the film profile and velocity, only the overall viscosity influences the position of the hydraulic jump.
Thin-film flow of a viscoelastic fluid on an axisymmetric substrate of arbitrary shape
ROGER E. KHAYAT, KYU-TAE KIM
Published online by Cambridge University Press: 29 March 2006, pp. 37-71
The interplay between inertia and elasticity is examined in this study for the transient axisymmetric flow of a thin film. The fluid is assumed to emerge from an annulus, as it is driven by axial pressure gradient and/or gravity. The substrate is assumed to be stationary and of arbitrary shape. The boundary-layer equations are generalized for a viscoelastic film obeying the Oldroyd-B constitutive model. These equations are solved by expanding the flow field in terms of orthonormal shape functions in the radial direction and using the Galerkin projection, combined with a time-stepping implicit scheme, and integration along the flow direction. It is found that the viscosity ratio and fluid elasticity can have a significant effect on steady state as well as transient behaviour. It is also found that low-inertia and/or highly elastic fluids tend to accumulate near the annulus, exhibiting a standing wave that grows with time. This behaviour clearly illustrates the difficulty associated with coating viscoelastic high-viscosity fluids. A criterion for film rupture is also established, which is based on the steepening of flow and stress gradients. The topography of the substrate has a drastic effect on the flow as well.
Finite-amplitude Rayleigh–Bénard convection and pattern selection for viscoelastic fluids
ZHENYU LI, ROGER E. KHAYAT
Published online by Cambridge University Press: 01 April 2005, pp. 221-251
The influence of inertia and elasticity on the onset and stability of Rayleigh–Bénard thermal convection is examined for highly elastic polymeric solutions with constant viscosity. These solutions are known as Boger fluids, and their rheology is approximated by the Oldroyd-B constitutive equation. The Galerkin projection method is used to obtain the departure from the conduction state. The solution is capable of displaying complex dynamical behaviour for viscoelastic fluids in the elastic and inertio-elastic ranges, which correspond to ${\it Ra} \,{<}\, {\it Ra}_c^s$ and ${\it Ra} \,{>}\, {\it Ra}_c^s $, respectively, ${\it Ra}_c^s $ being the critical Rayleigh number at which stationary thermal convection emerges. This behaviour is reminiscent of that observed experimentally for viscoelastic Taylor–Couette flow. For a given ${\it Ra}$ in the pre-critical range, finite-amplitude periodic oscillatory convection emerges when the elasticity number, $E$, exceeds a threshold. Periodicity is lost as $E$ increases, leading to a $T^{2}$ quasi-periodic behaviour, and the breakup of the torus as $E$ increases further. Although no experimental data are available for direct comparison, this scenario is reminiscent of the flow sequence observed by Muller et al. (1993) in the Taylor–Couette flow of a Boger fluid. Stationary thermal convection emerges, via a supercritical bifurcation, when ${\it Ra}$ exceeds ${\it Ra}_c^s $. The amplitude of motion is found to be little influenced by fluid elasticity or retardation time, especially as the Rayleigh number increases. However, the range of stability of the stationary thermal convection narrows considerably for viscoelastic fluids. In this case, oscillatory thermal convection is favoured. The onset and the stability of other steady convective patterns, namely hexagons and squares, are studied in the inertio-elastic range by using an amplitude equation approach. The range of stability of each pattern is examined, simultaneously allowing the validation of the two-dimensional picture.
Finite-amplitude Taylor-vortex flow of viscoelastic fluids
Published online by Cambridge University Press: 10 December 1999, pp. 33-58
The influence of inertia and elasticity on the onset and stability of Taylor-vortex flow (TVF) is examined for an Oldroyd-B fluid. The Galerkin projection method is used to obtain the departure from Couette flow (CF). Only axisymmetric flow is examined. The solution is capable of capturing the dynamical behaviour observed experimentally for viscoelastic fluids in the inertio-elastic and purely elastic ranges. For flow with dominant inertia, the bifurcation picture is similar to that for a Newtonian fluid. However, transition from CF to TVF is oscillatory because of fluid elasticity. Steady TVF sets in, via supercritical bifurcation, as Re reaches a critical value, Rec. The critical Reynolds number decreases with fluid elasticity, and is strongly influenced by fluid retardation. As elasticity exceeds a critical level, a subcritical bifurcation emerges at Rec, similar to that predicted by the Landau–Ginzburg equation. It is found that slip along the axial direction tends to be generally destabilizing. The coherence of the formulation is established under steady and transient conditions through comparison with exact linear stability analysis, experimental measurements, and flow visualization. Good agreement is obtained between theory and the measurements of Muller et al. (1993) in the limit of purely elastic overstable TVF.