UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)
Drops and jets of complex fluids
Javadi, A.
Publication date
2013
Link to publication
Citation for published version (APA):
Javadi, A. (2013). Drops and jets of complex fluids.
General rights
It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons).
Disclaimer/Complaints regulations
If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.
5.
The Non-Newtonian
Hydraulic Jump
5.1
Introduction
A hydraulic jump is the local and abrupt rise of fluid surface, flowing from a region where the fluid is shallow and flows with a high velocity to a deeper, lower velocity zone. The most common place to see this is probably the kitchen sink. When a smooth jet of water impinges a horizontal plane it spreads out radially in a thin layer up to a “jump” after which the fluid is deeper and slower Fig. 5.1a. The boundary between these two regions corresponds to a transition of the flow velocity U (r) from a supercritical value higher than the local velocity c(h) of the surface waves in the central region, to a subcritical value further out. The gravitational wave velocity when the free surface of the fluid is at a height h from the surface beneath is c(h) =√gh. There appears a rather abrupt transition since, as the fluid becomes deeper, the fluid velocity decreases while the wave velocity increases. The Froude number F r = U (r)/c(h), then changes from a value greater than unity in the center to a value less than one in the outer region (77).
The hydraulic jump for Newtonian fluids has long been of interest for physicists, a wealth of work have been done to consider, both theoretically and experimentally, the influence of different parameters such as viscosity, surface tension, radius and height of the jump. We will first introduce the Rayleigh’s pioneering work on the topic (78). Based on an inviscid theory, he achieved an analytic relation for the radius of the jump (originally for the case of a two dimensional jump along a channel with constant width). The inviscid theory is known to be inadequate for laboratory scale circular jumps, since the depth of the central region is small enough to make the diffusion of vorticity from the lower boundary significant. Subsequently Watson (79) proposed a solution for the circular jump by taking into account the viscous boundary layer in the central thin layer. Bush (80) added corrections due to the surface tension to Watson’s formula to achieve more accurate results.
(a)
(b)
Figure 5.1: a) Hydraulic jump in a kitchen sink. b) Large scale hydraulic jump at St. Anthony falls on the Mississippi river.
In contrast to the Newtonian fluids that have been studied extensively, the non-Newtonian hydraulic jump has not been explored in the literature, particularly there are scarce experimental investigations of the problem.
5.2
Newtonian hydraulic jump
5.2.1
Inviscid jump
A cylindrical laminar jet of radius a and flux Q, falls vertically onto a solid surface, forming a thin layer of height h(r), where r is the redial distance from the axis of the jet. The hydraulic jump occurs at a distance Rj from the center. When r is
large compared to a, h would be small and the motion is almost radial with the same speed with which the jet strikes the plane U0. Hence Q = πa2U0= 2πrhU0,
so that
h = a
2
2r, r < Rj (5.1)
The depth after the jump is H and the velocity after the jump is assumed to be uniform and equal to U1. The first relation is given by continuity as
U1=
Q 2πRjH
. (5.2)
We select a control volume V , of unit length in the direction of azimuthal angle increase, bounded by surface S. The cross section of the control volume is depicted in Fig. 5.2. The conservation of momentum over this volume, neglecting viscosity,
5.2. Newtonian hydraulic jump
is in general written as Z S (ρv(v.ˆn) + pˆn) dS = Z V ρgdV, (5.3)where v is the velocity field of the fluid, ˆn is the normal to the surface and p is the scalar hydrostatic pressure. Thus, The above integrals are simplified to our second relation U12H − U02h = g(h 2 2 − H2 2 ). (5.4)
Combining (5.2) and (5.4), we arrive at: Q 2πRj 2 1 h− 1 H = g 2 H 2− h2 , (5.5) in which h = a2/2R
j. Solving the above equation one finds
H2ga2 Q2 Rj= 1 π2 − gHa4 2Q2 . (5.6)
When h d, this reduces to
H2ga2
Q2 Rj=
1
π2. (5.7)
Figure 5.2: Schematic view of circular hydraulic jump in cylindrical coordinates. The contour S is the surface of the control volume.
5.2.2
Viscous Theory
Watson applied the viscous boundary layer theorem to the problem and distin-guished four distinct regions for the flow (Fig. 5.3), (i) the stagnation region: when r = O(a), the speed outside the boundary layer rises rapidly from 0 at the stagnation point to U0, and the boundary layer thickness is δ = O(νa/U0)1/2; (ii)
the developing boundary layer : the surface speed is that of the incoming jet U0, a
Blasius sublayer develops from the lower boundary (a r < r0), the thickness of
the boundary layer is δ(r) = O(νr/U0)1/2; (iii) fully viscous thin film region: when
the viscous stresses become appreciable right up to the surface the whole flow is of boundary layer type. The velocity changes as r increases, from the Blasius type to the similarity profile (r > r0); (iv) Ultimately the way in which the flow originated
becomes unimportant and the similarity solution is valid.
Figure 5.3: Schematic view of the four regions in a typical hydraulic jump. Considering the boundary layer approximation for the flow in a thin layer, one finds ∂ ∂rru + ∂ ∂zrw = 0, (5.8) u∂u ∂r + w ∂u ∂z = ν ∂2u ∂z2 (5.9)
with the conditions
u = w = 0 at z = 0, (5.10)
5.2. Newtonian hydraulic jump
2πr Z h(r)
0
udz = Q. (5.12)
Here r, z are cylindrical coordinates (as shown in Fig. 5.3) and u, w are the corresponding velocity components. In (5.9) the gravitational pressure gradient has been ignored. Equation (5.11) assert that the shear stress is zero at the free surface, since the viscosity of air is negligible, and (5.12) is the condition of constant flux. By assuming similarity solutions as
u = U (r)f (ζ), ζ = z/h(r), (5.13) with U (r), the speed at the free surface, Watson solved the equations to arrived at expressions for u(r, z).
The position of the hydraulic jump is determined, as previously, by equating the rate of loss of momentum to the thrust of the pressure. This is legitimate provided that the width (measured radially) of the jump is small, so that skin friction can be ignored. This will hold if the depth H outside the jump is small compared with Rj. By neglecting the pressure thrust on the inward side of the wave 12ρgh2, which
is only O(h2/H2) compared with the thrust on the outward side, The condition of
momentum balance will be 1 2ρgH 2= ρZ h 0 u2dz − ρU1H, (5.14) so that H2ga2R j Q2 + a2 2π2HR j = 2a 2R j Q2 Z h 0 u2dz. (5.15)
The momentum outside the wave, which is O(h/H) compared with that inside, is included only approximately since it is assumed that the velocity U1 immediately
outside the jump is uniform (5.2).
Watson calculated the rhs. of (5.15), by solving the equations (5.8-5.12), by as-suming h H and Rj a. He evaluated the rhs. of (5.15) separately for Rj> r0
and Rj < r0, since the jump may occur at any point in the development of the
boundary layer. When Rj> r0
H2ga2Rj Q2 + a2 2π2HR j = 0.01676 " Rj a 3 Re−1+ 0.1826 #−1 , (5.16)
rovided that (Rj/a)Re−1/3 > 0.3155. Re = Qa/ν is the jet Reynolds number.
When Rj< r0 H2ga2Rj Q2 + a2 2π2HR j = 0.10132 − 0.1297 Rj a 3/2 Re−1/2, (5.17)
Figure 5.4: Comparison of experiment and theory for jump relation (laminar flow, equations (5.16) and (5.17)) (79).
Watson’s theory was tested in a number of experimental investigations including those of himself (Fig. 5.4), Craik et al. (1981) (81), Bush and Aristoff (2003) (80). The separation of the flow and recirculation eddy beyond the jump was identified by Tani (82). Fig. 5.5 is a schematic illustration of the two distinct types of laminar circular jumps that arise in water: Type I marked as unidirectional surface flow, but boundary layer separation beyond the jump; Type II, marked by reverse surface flow adjoining the jump. In the vicinity of the hydraulic jump, the agreement of theory and experiments has ranged from good to poor (Fig. 5.4), being generally good when the jump radius more than ten times the depth beyond the jump (Type II), and poor in the opposite limit of small jump radius (Type I). It is observed that Watson’s predictions are least satisfactory in the limit of relatively weak jump, specifically when the ratio of the layer depths after and before the jump is small (83).
5.2.3
Surface tension
Bush (80) further took into account the influence of the radial component of the surface tension force associated with the azimuthal curvature of the circular jump, for the Type I. He expressed the radial curvature force as
5.3. Non-Newtonian hydraulic jumps
Figure 5.5: A schematic illustration of the two principal types of steady laminar hydraulic jumps. The progression from Type I to II arises as the outer layer depth is increased.
where ∆R represents the difference of the inner and outer radial bounds on the jump, defined as the nearest points up- and downstream of the jump at which the slope ∂h/∂r vanishes. s is the arclength of the jump surface between the inner and outer bounds. In the limit of abrupt jump ∆R → 0, the arclength s approaches the jump height ∆H, so that
Fc = −2πγ∆H. (5.19)
Inserting this force to the momentum equation Bush found the corrections to (5.16) and (5.17) due to the surface tension
H2ga2R j Q2 1 + 2 Bo + a 2 2π2HR j = 0.01676 " Rj a 3 Re−1+ 0.1826 #−1 , Rj> r0 (5.20) H2ga2R j Q2 1 + 2 Bo + a 2 2π2HR j = 0.10132 − 0.1297 Rj a 3/2 Re−1/2, Rj< r0 (5.21) where Bo = ρgRj∆H/γ is the jump Bond number. (5.20) and (5.21) differ from
those of Watson only through the inclusion of the O(Bo−1). They rest on the same assumptions concerning the flow profiles, specifically that h/H 1, the radial flow speed is constant beyond the jump, and that radial gradients in the hydrostatic pressure prior to the jump are negligible relative to viscous stresses. The influence of surface tension is most significant for jumps of small radius and small height, and serves to decrease the jump radius by as much as 30% for the smallest jumps examined by Bush.
5.3
Non-Newtonian hydraulic jumps
As mentioned before few investigations has been done so far when it comes to the problem of non-Newtonian hydraulic jump. The most extensive work is probably the one by Zhao and Khayat (84). They theoretically considered the problem for power-law fluids of both shear thinning and shear thickening types. Specifically, in their numerical and analytical solutions, They engaged the values of consistency coefficient m and power-law index n, for xanthane as a shear thinning fluid and
Figure 5.6: Schematic illustration of the symmetric plane jet flow impinging on a flat solid plate (84). Note that the flow variables are dimensionless.
ethylene glycol as a shear thickening fluid. As in the Fig. 5.3 Zhao also separated the flow into 4 distinct regions. The assumption of laminar flow was also preserved. Zhao (84) assumed that the jet is planar (Fig. 5.6). In their wok, the fluid is described by the Ostwald-de Waele power law model (85), and the excess stress tensor is given by
τ = µD = m 1
2(D : D)
(n−1)/2
D, (5.22)
where D = ∇v + (∇v)T is the rate of stain tensor, with v being the velocity vector,
and T denotes matrix transposition. In this case, the viscosity µ is given explicitly by
µ = m2(ux)2+ 2(wy)2+ (uy+ wx)2 (n−1)/2
. (5.23)
where u and w are velocity components in the horizontal and vertical directions. Note that the subscripts x or y denote partial differentiation. The film thickness is assumed to be small everywhere, and the classical thin film theory is assumed to hold. aRe and a are taken as length scales in the x and y directions, respectively, where Ren= ρU02−na
n/m is the generalized Reynolds number for the jet flow. The
velocity scales in the x and y directions are U0 and U0/Ren, respectively. Hence,
the dimensionless viscosity is given by ˜ µ = ˜u2y˜+ 2ε2(˜uy˜w˜x˜+ ˜u2x˜+ ˜w 2 ˜ y) + ε 4w˜2 ˜ x (n−1)/2 , (5.24)
5.3. Non-Newtonian hydraulic jumps
where ε = 1/Ren, and ˜u and ˜w are dimensionless velocity components. The
non-dimensionalized conservation conservation equations are written ˜ ux˜+ ˜wy˜= 0, (5.25) ˜ u˜u˜x+ ˜w ˜uy˜= 2ε2(µ˜u˜x)x˜+µ(˜uy˜+ ε2w˜x˜) ˜ y. (5.26)
Since Ren is assumed to be moderately large, ε will be the small parameter in the
problem. If only leading-order terms in ε are retained, the momentum equation (5.26) reduces to ˜ u˜u˜x+ ˜w ˜uy˜= |˜uy˜|n−1u˜y˜ ˜ y. (5.27)
At the plate the no-slip and no-penetration conditions are assumed to hold, so that u(x, 0) = w(x, 0) = 0. (5.28) The flow field is sought separately in the developing boundary layer region (x < x0),
fully developed boundary layer region (x > x0), and hydraulic jump region.
Figure 5.7: Outer- (I) and inner-layer (II) domains in the developing (x < x0)
and fully developed viscous (x > x0) boundary-layer flow regions schematically
illustrated for shear-thinning fluid. The boundary layer is shown as dashed curve. From (84).
In the fully developed viscous boundary-layer region, a similarity solution is ob-tained by Zhao and Khayat for both the free-surface velocity and film thickness, and the flow is shown to develop into a two-layer structure (Fig. 5.7). The outer thin viscous layer is required 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. Similarly, Zhao found numerical and ap-proximate solutions for the velocity field in the developing boundary-layer region. A viscous adjustment layer was required in order to ensure a smooth asymptotic matching to the far outer-field flow solution. They obtained the flow in the entire
physical domain upon matching the flows between the developing and fully viscous boundary-layer regions.
Again, the position of the hydraulic jump, x = xj, is determined by equating the
rate of change of momentum to the force associated with difference in pressure that arises from the elevation change. Since, the velocity after the jump is ¯u = aU0
H , we have Z h(x) 0 u2dy −a 2U2 0 H = gH2 2 . (5.29)
Having h(x) and u(x) for both developing boundary layer (i) and the fully viscous region (ii), Zhao calculated the integral in the lhs. of (5.29) to derive the equations for the place of the jump. When xj> x0,
a H + gH2 2U2 0a = (xj+ l) aRenJn 1/(1−2n) , (5.30) where Jn = (n + 1)2n/((3n)2n(2n − 1)Fn3n) and l ≈ (n + 1) 2K n− 3n(2n − 1)cn+1 3n(n + 1)(2n − 1)KnFnn+1 . (5.31)
c, Fn and Kn are defined as below
c = Γ( 1 3)Γ(n/(n + 1)) 3Γ((4n + 1)/(3n + 3)), Fn= Γ(23)Γ(n(n + 1)) 3Γ((5n + 2)/(3n + 3)), Kn = 3ncn+1 3nFn− n − 1 . (5.35) The transition location, which is found by matching the solutions in the regions (i) and (ii), is approximately
x0=
cn+1
(n + 1)KnFnn+1
aRen. (5.36)
Finally, for xj< x0one obtains
a H + gH2 2U2 0a = Nn x j aRen 1/(n+1) + 1, (5.37)
where Nn = −(n + 1)cn[(n + 1)Kn]1/(n+1). As far as we know no experimental
study to date has examined the validity of (5.30) and (5.37).
5.4
Rheological properties
The non-Newtonian fluids we have chosen are the dilute solutions of PEO (Poly ethyleneoxide, Mw= 3 × 106a.m.u.). Depending on the structure of the flow these
5.5. Non-Newtonian experiments
some general rheological properties of Polyacrylamide solutions, in order to get a feeling of its response under different shear circumstances. Kreiba (86) studied the rheological properties of PAA solutions of weight concentrations 2-5% and moleculare weight of Mw= 5 × 106(a.m.u.). Some useful statements are as follows:
1- Strong shear thinning behavior for all concentrations
2- As the polymer concentration increases the apparent viscosity increases. 3- PAA solutions exhibit a yield stress. The yield stress increases with an increase in concentration.
4- PAA solutions behave like viscoelastic materials at all concentrations. However, the viscous part is always higher than the elastic part.
Figure 5.8: Diameter of the circular hydraulic jump as a function of volume flux compared with water. For PEO solution of 0.1 g/L concentration, height = 10 cm, diameter of the nozzle d=5mm.
5.5
Non-Newtonian experiments
We measured the dependence of the hydraulic jump diameter on the flow rate for PEO solutions of different concentrations and for different jet diameters.
The flowrate was generated by using a container filled with the fluid and placed higher than the impingement plane, so that the fluid would flow under gravity. Different evacuation rates were achieved by using an adjustable valve at the bottom of the container and also by changing the height of the surface of the fluid in the
Figure 5.9: Diameter of the circular hydraulic jump as a function of volume flux. For PEO solution of 0.5 g/L concentration, height = 6.3 cm, diameter of the nozzle d=5mm.
Figure 5.10: A schematic illustration of a typical experimental apparatus for hy-draulic jump. Fluids are pumped through the source nozzle with a prescribed flux Q. The outer depth H is controlled by an outer wall whose height is adjustable.
5.5. Non-Newtonian experiments
container. As the fluid is leaving the tank the surface of the fluid goes down and the evacuation rate changes in time. The area of the cross section of the container was large enough in order to have a reasonably constant flow rate over fixed periods of time (∼ 1 min) as the container was running empty.
In most experimental investigations done on the Newtonian hydraulic jump in the literature, a pump is used to recirculate the fluid and generate high flow rates. Due to the circulating of the fluid with the pump one can have the constant flow rate over desirable long periods of time. Fig. 5.10 shows a typical setup for hydraulic jump experiments.
Figure 5.11: Diameter of the hydraulic jump as a function of volume flux compared with water. For PEO solution of 1 g/l concentration, height = 3 cm, diameter of the nozzle d=1.4 mm.
Unfortunately, we are not able to use pumping devices, for recirculation of the fluid, since they will damage our fragile polymers due to generation of high shear rates which can affect the rheological properties significantly.
The impingement plane was a circular glass plate of radius 15 cm. It was leveled with three screws touching the bottom at the same distances from the center of it. There was no barrier at the edges of the plate to adjust the height of the hydraulic jump with, and the fluid was evacuated from the edges.
We did the experiments with PEO solutions mostly of 1 g/l concentration (Fig. 5.8-5.8). We first used a nozzle of diameter 5 mm, but we did not observe any significant deviations from results for water, see the graphs for different concentrations of the PEO solution with the Diameter of the nozzle = 5 mm. For smaller nozzles we did the same experiments. For d = 1.4 mm we observed a small deviation at high
Figure 5.12: Diameter of the hydraulic jump as a function of flux compared with water. For PEO solution of 1 g/l concentration, height = 4.5 cm, diameter of the nozzle d=2.5 mm.
Figure 5.13: Diameter of the circular hydraulic jump as a function of flux. For PEO solution of 1 g/l concentration, height = 6.3 cm, diameter of the nozzle d=5 mm.
5.6. Conclusion
flow rates, but we could not reach higher flow rates. For lower nozzle Diameters (< 1 mm) the diameter of the jump is very small and therefore very unstable to the small inclinations of the surface. The data here are very noisy.
5.6
Conclusion
There is a small effect of the polymers, but more detailed experiments are needed. In order to do this, we need to think of a setup that can recirculate the fluid from the reservoir without damaging the fragile polymers and hence change the rheological properties of the solution.
