Drop formation and break-up of rotating viscoelastic liquid jets in the Giesekus model

Drop formation and break-up of rotating viscoelastic liquid jets in the Giesekus model Abstract A curved liquid jet is widely used in various practical applications, such as the prilling process for generating small spherical pellets (fertilizer) and inkjet printing. A deep understanding of the mechanisms of the break-up of liquid jets and the associated flow dynamics are heavily dependent upon the nature of the fluid. In this paper, we model the viscoelastic liquid jet by using the Giesekus model. In addition, the governing equations have been reduced to 1-D by using an asymptotic approach. Then, we determine the trajectory of viscoelastic liquid curved jets. Furthermore, the nonlinear evolution equations for the jet radius and the axial velocity are solved numerically using finite differences scheme based on the Lax–Wendroff method. 1. Introduction The mechanisms of capillary instability of liquid jets rupture from a nozzle have received much interest since the last century. This phenomenon has many applications in different industrial and engineering processes, such as inkjets printing, fertilizers, roll coating and paint levelling. The first theoretical linear instability of inviscid liquid jets was investigated by Rayleigh (1878). He found that the primary mechanism of the jet break-up is surface tension. Weber (1931) followed the same analysis as for the inviscid liquid jet to examine the linear instability of viscous liquid jets and he found that the wavelength of the most unstable waves is increased by the viscosity. A fuller asymptotic analysis has been applied by Papageorgiou (1995) on the governing system, which describes the evolution of nonlinear waves along the jet. He also investigated the break-up behaviour for viscous liquid threads. Goren & Gottlieb (1982) examined the linear instability on the break-up of viscoelastic liquid jets by using the Oldroyd’s eight-constant model. A historical overview on the break-up and drop formation of liquid jets can be found in Eggers (1997) and Eggers & Villermaux (2008). Since then, the capillary instability of non-Newtonian liquid jets has been performed by many authors. For example, a numerical study has been used by Renardy (1995) to find the break-up of Newtonian case and viscoelastic liquid jets for the Giesekus model and upper convected Maxwell model. The stability of viscoelastic jets has been discussed by Middleman (1965). Goldin et al. (1969) compared the linear stability between inviscid, Newtonian and viscoelastic liquid jets. Magda & Larson (1988) used the Oldroyd-B model for ideal elastic liquids (called Boger fluids) for investigating the rheological behaviour of polyisobutylene and polystyrene when the shear rates are low. Schummer & Thelen (1988) studied the break-up of a viscoelastic liquid jet by using Jeffrey’s model and investigated linear stability. The nonlinear temporal instability of capillary liquid jets was investigated by Ashgriz & Mashayek (1995) and it was found that when the Reynolds number is very small, satellite droplets are not observed and can only be seen when the Reynolds number is high. Liu & Liu (2008) discussed the temporal instability of viscoelastic liquid jets for axisymmetric and asymmetric disturbances. Bazilevskii & Rozhkov (2014) conducted an experiment of capillary break-up of weakly viscoelastic jets. Feng (2003) used the Giesekus model to investigate a charged viscoelastic liquid jet and he confirmed experimentally and numerically that using the Giesekus model gives accurate predictions for the elongation rheology. The influence of viscoelasticity on drop formation has been discussed by Verhulst et al. (2009) using the Giesekus model. The trajectory and stability of inviscid curved liquid jets have been investigated by Wallwork et al. (2002). They also conducted some experiments for inviscid liquid curved jets and they found that there is a good agreement between the theoretical and experimental work. The previous work has been extended by Decent et al. (2002) to include gravity. Furthermore, Decent et al. (2009) examined the influence of viscosity on the trajectory and stability of the break-up of liquid curved jets. On the other hand, the instability of non-Newtonian liquid curved jets under gravity has been examined by Uddin & Decent (2010). Renardy (1995) determined numerically the break-up of Newtonian case and viscoelastic liquid jets for the Giesekus model and upper convected Maxwell model. The nonlinear Giesekus model has been used by Renardy (2001) to study the break-up of viscoelastic liquid jets. Renardy & Losh (2002) investigated the break-up of a viscoelastic jet in the Giesekus model using similarity solutions. The linear and nonlinear instability for Newtonian and non-Newtonian fluids have been studied by Larson (1992). The beads-on-string structure has been investigated by many authors (see e.g. Clasen et al. (2006) and Ardekani et al. (2010)). The numerical study was conducted by Li & Fontelos (2003) for the beads-on-string structure for viscoelastic liquid jets by using an explicit finite difference method. Fontelos & Li (2004) studied the evolution and break-up of viscoelastic liquid jets for two models: the Giesekus and Finit Extendable Nonlinear Elastic-Peterlin (FENE-P). Davidson et al. (2006) examined drop formation of viscoelastic liquid jets emerging from a nozzle by using the Oldroyd-B model and they compared numerical results with experimental work. Alsharif (2014) carried out the instability of non-uniform viscoelastic curved jets. The instability of a rotating viscoelastic liquid jet has been researched by Alsharif et al. (2015). They have used the Oldroyd-B model for investegating the break-up of a viscoelastic liquid curved jets, and they found from the linear instability that viscoelastic jets are more unstable than Newtonian jets and less unstable than inviscid jets. They also found from nonlinear instability that viscoelastic jets break-up before the Newtonian jets and after inviscid jets. Lorenz et al. (2014) studied a dynamic curved jet under gravitational force by using the upper convected Maxwell model. In this paper, we will extend the work of Alsharif et al. (2015) to determine the break-up lengths and drop formation of a nonlinear viscoelastic rotating liquid jet by using the Giesekus model owing to the highly nonlinear nature of this model, and it strikes a good balance between simplicity and satisfactory prediction for elongational rheology. An asymptotic approach will be applied to reduce the governing equations into a set of non-dimensional equations. Furthermore, we will find the steady state solutions. Then, we will determine numerically the break-up lengths and main and satellite droplets by using a finite difference scheme based on the two-step Lax–Wendroff method. Fig. 1. View largeDownload slide A diagram showing a plan view of rotating cylindrical drum and below a photo taken from Wallwork (2002). The Cartesian axes x and z are shown in the figure. The cylindrical drum rotates about its axis with an angular velocity Ω. Fig. 1. View largeDownload slide A diagram showing a plan view of rotating cylindrical drum and below a photo taken from Wallwork (2002). The Cartesian axes x and z are shown in the figure. The cylindrical drum rotates about its axis with an angular velocity Ω. 2. Problem formulation In the prilling process we assume that a large cylindrical container has radius s0 and rotates with angular velocity Ω (see Fig. 1). This container has a small orifice at the bottom with radius a. This radius is very small compared with the radius of the container. We examine the problem by choosing a coordinate system (X, Y, Z), which rotates with the container, and having an origin at the axis of the container. The position of the orifice is at (s0, 0, 0). Due to the rotation of the container, the liquid leaves the orifice in a curve. In this problem of the prilling process, the jet moves in the (X, Y, Z) plane, so that the centerline can be described by coordinates (X(s, t), Y (s, t), Z(s, t)), where s is the arc-length along the middle of the jet which emerges from the orifice and t is the time (see Wallwork, 2002). In any cross-section of the jet we also have plane polar coordinates (n, ϕ), which are the radial and azimuthal direction and have unit vectors which are es, en, eϕ (see Decent et al., 2002). The velocity components for this problem are (u, v, w), where u is the tangential velocity, v is the radial velocity and w is the azimuthal velocity. Fig. 2. View largeDownload slide Graph showing the trajectory of a viscoelastic curved liquid jet for different values of the Rossby number where We = 10, De = 10, αs = 0.2 and β = 0.5. Fig. 2. View largeDownload slide Graph showing the trajectory of a viscoelastic curved liquid jet for different values of the Rossby number where We = 10, De = 10, αs = 0.2 and β = 0.5. Fig. 3. View largeDownload slide Graph showing the relationship between the radius and the arc-length s of a viscoelastic curved liquid jet for different values of the Rossby number where We = 10, De = 10, αs = 0.2 and β = 0.5. Fig. 3. View largeDownload slide Graph showing the relationship between the radius and the arc-length s of a viscoelastic curved liquid jet for different values of the Rossby number where We = 10, De = 10, αs = 0.2 and β = 0.5. Fig. 4. View largeDownload slide Graph showing the relationship between $$T_{ss}^{0} -T_{nn}^{0}$$ and the arc-length s of a viscoelastic curved liquid jet for different values of the Rossby number where We = 10, De = 10, αs = 0.2 and β = 0.5. Fig. 4. View largeDownload slide Graph showing the relationship between $$T_{ss}^{0} -T_{nn}^{0}$$ and the arc-length s of a viscoelastic curved liquid jet for different values of the Rossby number where We = 10, De = 10, αs = 0.2 and β = 0.5. Fig. 5. View largeDownload slide The profile at the break-up of a viscoelastic curved liquid jet for different Rossby numbers. Here the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. Fig. 5. View largeDownload slide The profile at the break-up of a viscoelastic curved liquid jet for different Rossby numbers. Here the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. Fig. 6. View largeDownload slide (Left) The profile at the break-up of a viscoelastic curved liquid jet. Here the parameters are Re = 1000, Rb = 1, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. (Right) The profile at the break-up of a viscous curved liquid jet. Here the parameters are Re = 1000, Rb = 1, We = 10, κ = 0.7, αs = 1 and δ = 0.01. Fig. 6. View largeDownload slide (Left) The profile at the break-up of a viscoelastic curved liquid jet. Here the parameters are Re = 1000, Rb = 1, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. (Right) The profile at the break-up of a viscous curved liquid jet. Here the parameters are Re = 1000, Rb = 1, We = 10, κ = 0.7, αs = 1 and δ = 0.01. Fig. 7. View largeDownload slide The profile at the break-up of a viscoelastic curved liquid jet for two values of the Reynolds numbers which are 1000 and 100 from right to left, respectively. Here the parameters are We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. Fig. 7. View largeDownload slide The profile at the break-up of a viscoelastic curved liquid jet for two values of the Reynolds numbers which are 1000 and 100 from right to left, respectively. Here the parameters are We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. Fig. 8. View largeDownload slide The profile of the break-up of viscoelastic liquid jets for different values of the Deborah number. We observe that when these three numbers increase, we have longer break-up. Here we use Rb = 1, We = 10, k = 0.7, δ = 0.01 and αs = 0.2, Re = 80 and β = 0.05. Fig. 8. View largeDownload slide The profile of the break-up of viscoelastic liquid jets for different values of the Deborah number. We observe that when these three numbers increase, we have longer break-up. Here we use Rb = 1, We = 10, k = 0.7, δ = 0.01 and αs = 0.2, Re = 80 and β = 0.05. The flow is described by the governing equations, which are the continuity equation, the momentum equation and the constitutive equation. We used the nonlinear Giesekus model to investigate the viscoelastic curved liquid jets.   \begin{equation} \nabla \cdot u=0, \end{equation} (2.1)   \begin{equation} \rho \left(\frac{\partial u}{\partial t}+ u.\nabla u\right) = - \nabla p+ \nabla .\tau \,-2\, w\times u - w\times (w\times r), \\ \end{equation} (2.2)   \begin{equation} \tau =\mu_{s} \left(\nabla u+(\nabla u)^{T} \right)+ T, \end{equation} (2.3)   \begin{equation} \frac{\partial T}{\partial t}+( u.\nabla ) T-T \cdot \nabla u -(\nabla u)^{T} \cdot T+\frac{\beta}{\mu_{p}} T^{2}=\frac{1}{\lambda }(\mu_{p} \gamma -T), \end{equation} (2.4) where u = ues + ven + weϕ, ρ is the density, p is the pressure, μs is the viscosity of the solvent, μp is the viscosity of the polymer, β is the mobility factor and T is the extra stress tensor. The free surface of the jet is described by n − R(s, t, ϕ) = 0, where R(s, t, ϕ) is a function which gives the free surface position, and the normal vector is given by ∇(n − R(s, ϕ, t)), which gives   \begin{equation*} n=\frac{1}{E} \left(-\frac{\partial R}{\partial s} \cdot \frac{1}{h_{s}}\cdot e_{s}\,+\,e_{n}-\frac{\partial R}{\partial \phi }\cdot\frac{1}{R} \cdot e_{\phi }\right),\end{equation*} where   \begin{equation*} E\,=\,\left(1+\frac{1}{{h^{2}_{s}}} \left(\frac{\partial R}{\partial s} \right)^{2}\,+\,\frac{1}{R^{2}}\left(\frac{\partial R}{\partial \phi }\right)^{2} \right)^{\frac{1}{2}}.\end{equation*} The normal stress condition is given by n ⋅Π⋅ n = σκ, where Π is the total stress tensor given by −PI + τ, σ is the isotropic surface tension and κ is the curvature of the free surface   \begin{equation*} \kappa =\frac{1}{n h_{s}}\left(-\frac{\partial }{\partial s} \left(\frac{n}{Eh_{s}}\, \frac{\partial R}{\partial s}\right) \,+\,\frac{\partial}{\partial n}\left(\frac{n h_{s}}{E}\right) -\frac{\partial}{\partial \phi}\left(\frac{h_{s}}{En}\, \frac{\partial R}{\partial \phi} \right)\right)\nonumber. \end{equation*} The tangential stress conditions are ti ⋅ σ ⋅ n, where i = 1, 2 and $$t_{1}=\,e_{s} +\,\frac {\partial R}{\partial s}\cdot \frac {1}{h_{s}} \cdot e_{n}, \, t_{2}=\,\frac {\partial R}{\partial \phi }\cdot \frac {1}{R}\cdot e_{n}+e_{\phi }$$. The kinematic condition is given by $$ \frac {D}{Dt}(R(s,t,\phi -n)=0$$ on n = R(s, ϕ, t). We can write the previous equation as   \begin{equation*} \frac{\partial R}{\partial t}+\frac{\partial R}{\partial s} \frac{\partial s}{\partial t}+\frac{\partial R}{\partial \phi }\frac{\partial \phi }{\partial t}-\frac{\partial n}{\partial t}=0. \nonumber \end{equation*} The pressure condition on the free surface is p = σκ on n = R, and the arc-length condition is   \begin{equation*} {X_{s}^{2}}+{Z^{2}_{s}}=1.\end{equation*} These equations are similar to those found in Alsharif et al. (2015). However, the differences are in the constitutive equation of the elastic term. 3. Asymptotic analysis Here we will expand u, v, w, p in Taylor series in εn (see Eggers (1997)) and R, X, Z, Tss, Tnn, Tϕϕ, Tsn, Tsϕ, Tnϕ in ε. We suppose that the leading order of the axial component of the velocity is independent of ϕ.   \begin{align*} (u,v,w)(s,n,\phi,t)&=(u_{0},0,0)(s,t)+ \,(\varepsilon\,n )( u_{1},v_{1},w_{1})(s,\phi,t)+...\\[8pt] p(s,n,\phi,t) &=\,p_{0}(s,\phi,t)+(\varepsilon\,n)p_{1}(s,\phi,t)+...\\[8pt]R(s,n,\phi,t)\, &=\,R_{0}(s,t)+(\varepsilon)R_{1}(s,\phi,t)+...\\[8pt](X,Y,Z)(s,n,\phi,t) &=(X_{0}, Y_{0}, Z_{0})(s)+ (\varepsilon )(X_{1},Y_{1}, Z_{1} )(s,t) +...\\[8pt]\left(T_{ss},T_{nn},T_{\phi \phi } \right)(s,n,\phi,t) &= \left(T^{0}_{ss},T^{0}_{nn} \right)(s,t)+\varepsilon\, \left(T^{1}_{ss},T^{1}_{nn}, T^{1}_{\phi \phi} \right)(s,t)+...\\[8pt]\left(T_{sn},T_{s\phi },T_{n \phi } \right)(s,n,\phi,t) &= \varepsilon\, \left(T^{1}_{ss},T^{1}_{nn}, \varepsilon T^{1}_{\phi \phi} \right)(s,t)+... \end{align*} We substitute these asymptotic expansions into the governing equations. We can therefore find from the equation of continuity at the leading order   \begin{equation} v_{1}=-\frac{u_{0s}}{2}. \end{equation} (3.1) It can be also obtained that from the equation of motion in s-direction at leading order   \begin{align} u_{0t}+u_{0}u_{0s} =&-\frac{1}{We}\,\left(\frac{1}{ R_{0}}\right)_{s}+\frac{(X+1)X_{s}+ZZ_{s}}{Rb^{2}}\nonumber\\ &+\frac{3\alpha_{s}}{{R^{2}_{0}}\, Re}\left({R^{2}_{0}}\,u_{0s}\right)_{s}+\frac{1}{ {R^{2}_{0}} Re}\left( {R^{2}_{0}}\left(T^{0}_{ss}-T^{0}_{nn}\right)\right)_{s}, \end{align} (3.2)   \begin{equation} (X_{s}Z_{ss}-X_{ss}Z_{s} )\left( {u^{2}_{0}}-\frac{3\alpha_{s}}{Re}\, u_{0s}-\frac{1}{WeR_{0}}\right) - \frac{2}{Rb}\,u_{0} +\frac{(X+1)Z_{s}+ ZX_{s}}{Rb^{2}}=0, \end{equation} (3.3) from the extra stress tensor Eq. (2.4) at leading order, we get   \begin{equation} \frac{\partial T_{ss}^{0}}{\partial t}+u_{0}\,\frac{\partial T^{0}_{ss}}{\partial s}\,-2\,\frac{\partial u_{0}}{\partial s}\,T^{0}_{ss} +\frac{\beta}{1- \alpha_{s}}T_{ss}^{02}= \frac{1}{De}\,\left( 2(1-\alpha_{s})\,\frac{\partial u_{0}}{\partial s}\,-\,T^{0}_{ss}\right), \end{equation} (3.4)   \begin{equation} \frac{\partial T_{nn}^{0}}{\partial t}+u_{0}\,\frac{\partial T^{0}_{nn}}{\partial s}\,+\frac{\partial u_{0}}{\partial s}\,T^{0}_{nn}+\frac{\beta}{1- \alpha_{s}}T_{nn}^{02}= \frac{-1}{De}\,\left( (1-\alpha_{s})\frac{\partial u_{0}}{\partial s}\,+\,T^{0}_{nn}\right), \end{equation} (3.5) from the kinematic condition, we obtain at ε   \begin{equation} \frac{\partial R_{0}}{\partial t}+u_{0}R_{0s}+\frac{R_{0}}{2}u_{0s}=0, \end{equation} (3.6) where $$ Re= \frac {\rho Ua}{\mu _{0} },\, We=\frac {\rho U^{2}a}{\sigma } $$, $$ Rb=\frac {U}{s_{0}\varOmega }$$, $$ De=\frac {\lambda U}{s_{0} }$$ and $$ \alpha _{s}=\frac {\mu _{s} }{\mu _{s}+\mu _{p}} = \frac {\mu _{s} }{\mu _{0}} $$ are the Reynolds number, the Weber number, the Rossby number, the Deborah number and the viscosity ratio, respectively. We can see that when β = 0 the equations (3.2)–(3.6) recover to the linear Oldroyd-B model (see Alsharif et al. (2015)). 4. Steady state solutions From equation (3.6) we have $$ {R_{0}^{2}} u_{0}=1$$, so that the governing equations for this system are Fig. 9. View largeDownload slide The profile at the break-up of a viscoelastic curved liquid jet for different Rossby numbers. Here the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. Fig. 9. View largeDownload slide The profile at the break-up of a viscoelastic curved liquid jet for different Rossby numbers. Here the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05.   \begin{align} u_{0}u_{0s}=&-\frac{u_{0s}}{2We\,\sqrt u}+\frac{(X+1)X_{s}+ZZ_{s}}{Rb^{2}} + \frac{3\alpha_{s} }{Re}\left( u_{0ss}-\frac{u^{2}_{0s}}{u_{0}}\right) \nonumber\\ &+\frac{1}{ Re} \left( \frac{\partial }{\partial s}\left( T^{0}_{ss}-T^{0}_{nn}\right)-\frac{ u_{0s}}{u_{0}}\left(T^{0}_{ss}-T^{0}_{nn}\right)\right), \end{align} (4.1)   \begin{equation} (X_{s}Z_{ss}-X_{ss}Z_{s})\left( {u^{2}_{0}}-\frac{3}{Re}\, u_{0s}-\frac{\sqrt u}{We}\right)- \frac{2}{Rb}\,u_{0} +\frac{(X+1)Z_{s}+ZX_{s}}{Rb^{2}}=0, \end{equation} (4.2)   \begin{equation} u_{0}\frac{\partial T^{0}_{ss}}{\partial s}-2\frac{\partial u_{0}}{\partial s}T^{0}_{ss}+\frac{\beta}{1- \alpha_{s}}T_{ss}^{02}= \frac{1}{De}\left( 2(\alpha_{s}-1) \frac{\partial u_{0}}{\partial s}-T^{0}_{ss}\right), \end{equation} (4.3)   \begin{equation} u_{0}\frac{\partial T^{0}_{nn}}{\partial s}+\frac{\partial u_{0}}{\partial s}T^{0}_{nn}+ \frac{\beta}{1- \alpha_{s}} T_{nn}^{02}=\frac{-1}{De}\left( (\alpha_{s}-1) \frac{\partial u_{0}}{\partial s}+T^{0}_{nn}\right), \end{equation} (4.4) and finally the arc-length condition is   \begin{equation} {X^{2}_{s}}+{Z^{2}_{s}}=1. \end{equation} (4.5) These are a system of five equations in five unknowns which are X, Z, u0, $$T_{ss}^{0}$$ and $$T_{nn}^{0}$$. This system of nonlinear differential equations, which is stiff due to the presence of the viscous term, can be solved by using an implicit finite difference scheme as done by Parau et al. (2007) who solved the resulting set of equations using the Newtonian methods. However, the same author were able to show that the presence of the viscous term did not significantly affect the steady state obtained in the inviscid limit. Therefore, we solve these equations by using the Runge–Kutta method in the inviscid limit $$Re \rightarrow o $$ (see Parau et al. (2007)) for obtaining the trajectory of the jet with following initial conditions as u0(0) = R0(0) = Xs(0) = 1 and X(0) = Z(0) = Zs(0) = 0. In Fig. 2 we show the effect of the Rossby number on the trajectory of the liquid jet. From this figure, it can be noticed that when the Rossby number decreases (meaning the rotation rates increases) the jets coil more progressively. The relationship between the radius of the jet and the extra stress tensor, $$ T_{ss}^{0}$$, $$T_{nn}^{0} $$, and the arc-length s are plotted in Figs 3 and 4. From these two figures, we can observe that when the rotation rates Rb increase the jet becomes more thin and these two results confirm that the jet thinning occur at the nozzle and decreases along the viscoelastic liquid jet which agrees with the results of Feng (2003). 5. Nonlinear temporal solutions Linear instability analysis predicts that liquid jets break up and produce uniform drop sizes along the axis of approximately the same wavelength of initial disturbances. However, it can be observed that a number of smaller satellite droplets appeared in this case which are not equal in size. Therefore, we use nonlinear temporal analysis to examine the break-up length and the formation of satellite droplets. We replace the leading order pressure term $$ p_{0} = \frac {1}{We}\frac {1}{R_{0}}$$ in the equation (3.2) with the expression for the full curvature term which contains only R0 and is not ϕ-dependent, namely Fig. 10. View largeDownload slide Graph showing the relationship between the break-up length and Rb. Where the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. Fig. 10. View largeDownload slide Graph showing the relationship between the break-up length and Rb. Where the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05.   \begin{equation} p = \frac{1}{We}\left[ \frac{1}{R_{0}\left( 1+\varepsilon^{2} R^{2}_{0s}\right)^{1/2}}- \frac{\varepsilon^{2} R_{0ss}}{\left( 1+\varepsilon^{2} R^{2}_{0s}\right)^{3/2}} \right] . \end{equation} (5.1) For simplicity, we denote A = A(s, t), where A(s, t) = R2(s, t), and then we rewrite our equations (3.2) and (3.4)–(3.6) as   \begin{align} \frac{\partial u}{\partial t}=&-\left( \frac{u^{2}}{2}\right)_{s}- \frac{1}{We}\frac{\partial}{\partial s}\,\frac{4\left(2A+(\varepsilon A_{s})^{2}-\varepsilon^{2}AA_{ss}\right)}{\left(4A+(\varepsilon A_{s})^{2}\right)^{3/2}}\nonumber\\ &+\frac{(X+1)X_{s}+ZZ_{s}}{Rb^{2}}+ \frac{3\alpha_{s} }{Re}\,\frac{(Au_{s})_{s}}{A}+\frac{1}{ Re}\,\frac{\left( A(T_{ss}-T_{nn})\right)_{s}}{A}, \end{align} (5.2)   \begin{equation} \frac{\partial T_{ss}}{\partial t} = - \frac{\partial }{\partial s}(uT_{ss})+3\frac{\partial u}{\partial s}T_{ss} -\frac{\beta}{1- \alpha_{s}}T_{ss}^{02}+ \frac{1}{De} \left( 2(1-\alpha_{s} )\frac{\partial u}{\partial s}-T_{ss}\right), \end{equation} (5.3)   \begin{equation} \frac{\partial T_{nn}}{\partial t} = - \frac{\partial }{\partial s}(uT_{nn})-\frac{\beta}{1- \alpha_{s}}T_{nn}^{02} - \frac{1}{De} \left( (1-\alpha_{s} )\frac{\partial u}{\partial s}+T_{nn} \right), \end{equation} (5.4)   \begin{equation} \frac{\partial A}{\partial t} = -\frac{\partial }{\partial s}(Au).\end{equation} (5.5) To solve this system of equations, we use the initial conditions at t = 0 which are $$ A(s,t=0)={R^{2}_{0}}(s), u(s,t=0)=u_{0}(s), T_{ss}(s,t=0)=0, \,\,T_{nn}(s,t=0)=0 $$ as we obtained for the steady state solutions (see Section 4). We use upstream boundary conditions at the nozzle as follows   \begin{equation*} A(0, t)=1,\,\, u(0,t)=1+\delta \sin \left(\frac{\kappa t}{\varepsilon }\right), \end{equation*} where κ and δ a non-dimensional wavenumber of the perturbation of frequency and the amplitude of the initial non-dimensional velocity disturbance, respectively. In our calculations, the value of $$\varepsilon (=\frac {a}{s_{0}})$$ is 0.01 as found in experiments and the industrial problem (see Wong et al. (2004)). We have used a second-order finite difference scheme based on the two-step Lax–Wendroff method to determine the break-up lengths and drop formation of a rotating viscoelastic liquid jet. Fig. 11. View largeDownload slide Graph showing the relationship between the droplet radius and αs. Where the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, δ = 0.01 and β = 0.05. Fig. 11. View largeDownload slide Graph showing the relationship between the droplet radius and αs. Where the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, δ = 0.01 and β = 0.05. Fig. 12. View largeDownload slide Graph showing the relationship between the droplet radius and De. Where the parameters are Re = 1000, We = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. Fig. 12. View largeDownload slide Graph showing the relationship between the droplet radius and De. Where the parameters are Re = 1000, We = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. In order to display the curved jet, we are required to determine the normal vector along the centerline of the jet (X(s), Z(s)), which is in this case n = (−Zs(s), Xs(s))). Thereafter the free surface of the jet is given by (X(s), Z(s)) + R(s, t)(−Zs(s), Xs(s)) and (X(s), Z(s)) − R(s, t)(−Zs(s), Xs(s)). To determine size of main droplets, we need to integrate between two local minimums (which correspond to pinch points along the jet). If we label them as h1 and h2 then   \begin{equation*} V_{drop}=\pi \int_{h_{1}}^{h_{2}} {R^{2}_{0}} \text{ d}s , \end{equation*} where the drop radius R equates to a sphere, then   \begin{equation*} \hat{R} =\left(\frac{3V_{drop}}{4\pi }\right)^{\frac{1}{3}}. \end{equation*} We have plotted a profile (Fig. 5) to show the effects of increasing the Rossby number on the break-up of a viscoelastic liquid curved jet. It can be observed from this profile that an increase in the rotation rates (i.e. the Rossby number decreases) leads to an increase in the break-up lengths, which means that the rotation rates affect the break-up lengths. We have made a comparison between the break-up of viscoelastic and viscous curved liquid jets and we found that the viscoelastic rotating jet breaks up before the viscous rotating jet (see Fig. 6). Furthermore, we show some profiles which indicate the break-up for two different values of the Reynolds number. It can be noticed from these two profiles that when we decrease the Reynolds number, which corresponds to high viscosity, the break-up length of the liquid curved jet increases (see Fig. 7). Figure 7 shows that when we increase the Deborah number the break-up lengths and satellite drops increase. In Fig. 9 the relationship between the jet radii versus the arc-length s have been plotted. It can be noticed from this figure that when we increase the mobility factor β the liquid jet delays to break up and this result agrees with the finding of Ardekani et al. (2010). Therefore we have chosen that the break-up occurs when the non-dimensional jet radius has reached an arbitrarily small value which is for consistency with earlier works (Parau et al. (2007)) 5% of the initial jet radius. Downstream of the break-up point, where the jet is expected to breakup into droplets, our numerical solution has no real physical meaning, as it is the case in other works (see Parau et al. (2007), Decent et al. (2009) and Alsharif (2014)). In Fig. 10 we see that when we increase the mobility factor, the break-up lengths increase. We can also observe that an increase in the rotation rates (i.e. low Rb) leads to an enhance in the break-up lengths which is also agrees with spiralling Newtonian liquid jets (see Parau et al. (2007)). Furthermore, in Figs 11 and 12 we can see that the relationship between the droplet sizes and the viscosity ratio αs and the Deborah number have been plotted. From these two figures, we notice that firstly, when we increase the mobility factor β, the main and satellite droplet sizes increase. Secondly, it can be observed that satellite droplet sizes increase when the Deborah number is increased and this result is in agreement with Alsharif et al. (2015). 6. Conclusion In summary, there are many applications of break-up liquid jets which emerge from an orifice, assuming that the centreline of the jet is straight with inkjet printing. However, it can be considered that the centreline is curved, which is known as the prilling process. This case is investigated here for viscoelastic liquid curved jets using the Giesekus model. It is also important in this process to study the effect of the rotation on the break-up lengths and droplet sizes. We have made an assumption, which is that the viscosity does not affect the trajectory of the centreline (see Decent et al. (2002), Uddin & Decent (2010) and Alsharif (2014)) and this is taken as an inviscid liquid jet. We have incorporated the Giesekus model into a 1D slender-body theory and examined the role of nonlinear rheology in a rotating viscoelastic liquid curved jet. An asymptotic approach has been used to reduce the governing equations into a set of 1D equations. The effect of changing the Rossby number on the trajectory of the viscoelastic liquid curved jet has been plotted (see Fig. 2). In addition, a comparison has been made between viscoelastic and viscous liquid curved jets in terms of the break-up lengths and it can be observed that the viscoelastic liquid curved jet breaks up before the viscous one (see Fig. 7). A numerical method based on finite differences has been used to determine the break-up lengths and main and satellite droplet sizes. Our results show that in the Giesekus model for viscoelastic curved liquid jets, it can be observed that the break-up lengths increase when we increase the mobility factor (see Fig. 11). Moreover, an increase in the mobility factor leads to decrease into the main and satellite droplet sizes (see Fig. 12). For future work, we can extend this work to investigate the effects of adding surfactants on the instability of a rotating viscoelastic liquid jets (see Alsharif & Uddin (2015)). Acknowledgements Abdullah Alsharif would like to thank Taif University for their financial support. References Alsharif, A. M. ( 2014) Instability of non-uniform of viscoelastic liquid jets. Ph.D. Thesis, University of Birmingham. Alsharif, A. M. & Uddin, J. ( 2015) Instability of viscoelastic curved liquid jets with surfactants. J. Non-Newtonian Fluid Mech. , 216, 1-- 12. Google Scholar CrossRef Search ADS   Alsharif, A. M., Uddin, J. & Afzaal, M. F. ( 2015) Instability of viscoelastic curved jets. Appl. Math. Model. , 39, 3924-- 3938. Google Scholar CrossRef Search ADS   Ardekani, A. M., Sharma, V. and Mckinley, G. H. ( 2010) Dynamics of bead formation, filament thinning and breakup in weakly viscoelastic jets, J. fluid Mech. , 665, 46-- 56. Google Scholar CrossRef Search ADS   Ashgriz, N. & Mashayek, F. ( 1995) Temporal analysis of capillary jet breakup. J. Fluid Mech. , 291, 163-- 190. Google Scholar CrossRef Search ADS   Bazilevskii, A. V. & Rozhkov, A. N. ( 2014) Dynamics of capillary breakup of elastic jets. Fluid Dynamics , 49, 145-- 163. Google Scholar CrossRef Search ADS   Clasen, C., Eggers, J., Fonlelos, M. A., Li, J. & Mckinley, G. H. ( 2006) The beads-on-string structure of viscoelastic threads. J. Fluid Mech ., 556, 283-- 308. Google Scholar CrossRef Search ADS   Davidson, M. R., Harvie, J. E. & Cooper-White, J. J. ( 2006) Simulation of pendant drop formation of a viscoelastic liquid, Korea–Australia Rheol. J. , 18, 41 -- 49. Decent, S. P., King, A. C., Simmons, M. H., Părău, E. I., Wong, D. C. Y., Wallwork, I. M., Gurney, C. & Uddin, J. ( 2009) The trajectory and stability of a spiralling liquid jet: part II. Viscous theory. Appl. Math. Model. , 33, 4283-- 4302. Google Scholar CrossRef Search ADS   Decent, S. P., King, A. C. & Wallwork, I. M. ( 2002) Free jets spun from a prilling tower. J. Eng. Math. , 42, 265-- 282. Google Scholar CrossRef Search ADS   Eggers, J. ( 1997) Nonlinear dynamics and breakup of free surface flows. Rev. Mod. Physis. , 69, 865-- 929. Google Scholar CrossRef Search ADS   Eggers, J. & Villermaux, E. ( 2008) Physics of liquid jets. Rep. Prog. Phys. , 71, 036601, 79. Feng, J. J. ( 2003) Stretching of a straight electrically charged viscoelastic jet. J. Non-Newtonian Fluid Mech ., 116, 55-- 70. Google Scholar CrossRef Search ADS   Fontelos, M. A. & Li, J. ( 2004) On the evolution and rupture of filaments in Giesekus and FENE models. J. Non-Newtonian Fluid Mech ., 118, 1-- 6. Google Scholar CrossRef Search ADS   Goldin, M., Yerushalmi, J., Pfeffer, R. & Shinner, R. ( 1969) Breakup of a viscoelastic fluid. J. Fluid Mech ., 38, 689-- 711. Google Scholar CrossRef Search ADS   Goren, S. L. & Gottlieb, M. ( 1982) Surface-tension driven breakup of viscoelastic liquid threads. J. Fluid Mech ., 120, 245-- 266. Google Scholar CrossRef Search ADS   Larson, R. G. ( 1992) Instabilities in viscoelastic flows. Rheol., Acta , 31. Li, J. & Fontelos, M. A. ( 2003) Drop dynamics on the beads-on string structure for viscoelastic jets: a numerical study, Phys. Fluids , 15, 922. Liu, Z. & Liu, Z. ( 2008) Instability of viscoelastic liquid jet with axisymmetric and symmetric disturbance, International Journal of Multiphase Flow, 34, 42-- 60. Lorenz, M., Marheineke, N. & Wegner, R. ( 2014) On simulation of spinning processes with a stationary on dimensional upper convected Maxwell model. J Mathematics Industry , 4:2, 1-- 13. Magda, J. J. & Larson, R. G. ( 1988) Atransition occurring in ideal elastic liquids during shear flow. J. Non-Newtonian Fluid Mech ., 30, 1-- 19. Middleman, S. ( 1965) Stability of a viscoelastic jet. Chem. Eng. Sci. , 20, 1037-- 1040. Google Scholar CrossRef Search ADS   Papageorgiou, D. T. ( 1995) On the breakup of viscous liquid threads. Phys. Fluids , 7, 1529. Părău, E. I., Decent, S. P., Simmons, M. J. H., Wong, D. C. Y. & King, A. C. ( 2007) Nonlinear viscous liquid jets from a rotating orifice. J. Eng. Maths ., 57, 159-- 179. Rayleigh, W. S. ( 1878) On the instability of jets. Proc. Lond. Math. Soc. , 10, 4. Renardy, M. ( 1995) A numerical study of the asymptotic evolution and breakup of Newtonian and viscoelastic jets. J. Non-Newtonian Fluid Mech ., 59, 267-- 282. Google Scholar CrossRef Search ADS   Renardy, M. ( 2001) Self-similar breakup of a Giesekus jet. J. Non-Newtonian Fluid Mech ., 97, 283-- 293. Google Scholar CrossRef Search ADS   Renardy, M. & Losh, D. ( 2002) Similarity solutions for jet breakup in a Giesekus fluid with inertia. J. Non-Newtonian Fluid Mech ., 106, 17-- 27. Schummer, P. & Thelen, H. G. ( 1988) Break-up of a viscoelastic liquid jets. Rheol. Acta. , 27, 39-- 43. Google Scholar CrossRef Search ADS   Uddin, J. & Decent, S. P. ( 2010) Instability of non-Newtonian liquid jets curved by gravity. Math. Industry , 15, 597-- 602. Verhulst, K., Cardinaels, R., Moldenaers, P., Afkhami, S. & Renardy, Y. ( 2009) Influence of viscoelasticity on drop formation and orientation in shear flow. Part 2: Dynamics. J. Non-Newtonian Fluid Mech ., 156, 44-- 57. Wallwork, I. M. ( 2002) The trajectory and stability of a spiralling liquid jet. Ph.D. Thesis, Birmingham: University of Birmingham. Google Scholar CrossRef Search ADS   Weber, C. ( 1931) Zum Zerfall eines Flussigkeitsstrahles. Z. Angew. Math. Mech. , 11, 136-- 154. Wong, D. C. Y., Simmons, M. J. H., Decent, S. P., Parau, E. I. & King, A. C. ( 2004) Break-up dynamics and drop sizes distributions created from spiralling liquid jets. Int. J. Multiphase Flow , 30, 499-- 520. © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Applied Mathematics Oxford University Press

Drop formation and break-up of rotating viscoelastic liquid jets in the Giesekus model

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Oxford University Press
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© The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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Abstract

Abstract A curved liquid jet is widely used in various practical applications, such as the prilling process for generating small spherical pellets (fertilizer) and inkjet printing. A deep understanding of the mechanisms of the break-up of liquid jets and the associated flow dynamics are heavily dependent upon the nature of the fluid. In this paper, we model the viscoelastic liquid jet by using the Giesekus model. In addition, the governing equations have been reduced to 1-D by using an asymptotic approach. Then, we determine the trajectory of viscoelastic liquid curved jets. Furthermore, the nonlinear evolution equations for the jet radius and the axial velocity are solved numerically using finite differences scheme based on the Lax–Wendroff method. 1. Introduction The mechanisms of capillary instability of liquid jets rupture from a nozzle have received much interest since the last century. This phenomenon has many applications in different industrial and engineering processes, such as inkjets printing, fertilizers, roll coating and paint levelling. The first theoretical linear instability of inviscid liquid jets was investigated by Rayleigh (1878). He found that the primary mechanism of the jet break-up is surface tension. Weber (1931) followed the same analysis as for the inviscid liquid jet to examine the linear instability of viscous liquid jets and he found that the wavelength of the most unstable waves is increased by the viscosity. A fuller asymptotic analysis has been applied by Papageorgiou (1995) on the governing system, which describes the evolution of nonlinear waves along the jet. He also investigated the break-up behaviour for viscous liquid threads. Goren & Gottlieb (1982) examined the linear instability on the break-up of viscoelastic liquid jets by using the Oldroyd’s eight-constant model. A historical overview on the break-up and drop formation of liquid jets can be found in Eggers (1997) and Eggers & Villermaux (2008). Since then, the capillary instability of non-Newtonian liquid jets has been performed by many authors. For example, a numerical study has been used by Renardy (1995) to find the break-up of Newtonian case and viscoelastic liquid jets for the Giesekus model and upper convected Maxwell model. The stability of viscoelastic jets has been discussed by Middleman (1965). Goldin et al. (1969) compared the linear stability between inviscid, Newtonian and viscoelastic liquid jets. Magda & Larson (1988) used the Oldroyd-B model for ideal elastic liquids (called Boger fluids) for investigating the rheological behaviour of polyisobutylene and polystyrene when the shear rates are low. Schummer & Thelen (1988) studied the break-up of a viscoelastic liquid jet by using Jeffrey’s model and investigated linear stability. The nonlinear temporal instability of capillary liquid jets was investigated by Ashgriz & Mashayek (1995) and it was found that when the Reynolds number is very small, satellite droplets are not observed and can only be seen when the Reynolds number is high. Liu & Liu (2008) discussed the temporal instability of viscoelastic liquid jets for axisymmetric and asymmetric disturbances. Bazilevskii & Rozhkov (2014) conducted an experiment of capillary break-up of weakly viscoelastic jets. Feng (2003) used the Giesekus model to investigate a charged viscoelastic liquid jet and he confirmed experimentally and numerically that using the Giesekus model gives accurate predictions for the elongation rheology. The influence of viscoelasticity on drop formation has been discussed by Verhulst et al. (2009) using the Giesekus model. The trajectory and stability of inviscid curved liquid jets have been investigated by Wallwork et al. (2002). They also conducted some experiments for inviscid liquid curved jets and they found that there is a good agreement between the theoretical and experimental work. The previous work has been extended by Decent et al. (2002) to include gravity. Furthermore, Decent et al. (2009) examined the influence of viscosity on the trajectory and stability of the break-up of liquid curved jets. On the other hand, the instability of non-Newtonian liquid curved jets under gravity has been examined by Uddin & Decent (2010). Renardy (1995) determined numerically the break-up of Newtonian case and viscoelastic liquid jets for the Giesekus model and upper convected Maxwell model. The nonlinear Giesekus model has been used by Renardy (2001) to study the break-up of viscoelastic liquid jets. Renardy & Losh (2002) investigated the break-up of a viscoelastic jet in the Giesekus model using similarity solutions. The linear and nonlinear instability for Newtonian and non-Newtonian fluids have been studied by Larson (1992). The beads-on-string structure has been investigated by many authors (see e.g. Clasen et al. (2006) and Ardekani et al. (2010)). The numerical study was conducted by Li & Fontelos (2003) for the beads-on-string structure for viscoelastic liquid jets by using an explicit finite difference method. Fontelos & Li (2004) studied the evolution and break-up of viscoelastic liquid jets for two models: the Giesekus and Finit Extendable Nonlinear Elastic-Peterlin (FENE-P). Davidson et al. (2006) examined drop formation of viscoelastic liquid jets emerging from a nozzle by using the Oldroyd-B model and they compared numerical results with experimental work. Alsharif (2014) carried out the instability of non-uniform viscoelastic curved jets. The instability of a rotating viscoelastic liquid jet has been researched by Alsharif et al. (2015). They have used the Oldroyd-B model for investegating the break-up of a viscoelastic liquid curved jets, and they found from the linear instability that viscoelastic jets are more unstable than Newtonian jets and less unstable than inviscid jets. They also found from nonlinear instability that viscoelastic jets break-up before the Newtonian jets and after inviscid jets. Lorenz et al. (2014) studied a dynamic curved jet under gravitational force by using the upper convected Maxwell model. In this paper, we will extend the work of Alsharif et al. (2015) to determine the break-up lengths and drop formation of a nonlinear viscoelastic rotating liquid jet by using the Giesekus model owing to the highly nonlinear nature of this model, and it strikes a good balance between simplicity and satisfactory prediction for elongational rheology. An asymptotic approach will be applied to reduce the governing equations into a set of non-dimensional equations. Furthermore, we will find the steady state solutions. Then, we will determine numerically the break-up lengths and main and satellite droplets by using a finite difference scheme based on the two-step Lax–Wendroff method. Fig. 1. View largeDownload slide A diagram showing a plan view of rotating cylindrical drum and below a photo taken from Wallwork (2002). The Cartesian axes x and z are shown in the figure. The cylindrical drum rotates about its axis with an angular velocity Ω. Fig. 1. View largeDownload slide A diagram showing a plan view of rotating cylindrical drum and below a photo taken from Wallwork (2002). The Cartesian axes x and z are shown in the figure. The cylindrical drum rotates about its axis with an angular velocity Ω. 2. Problem formulation In the prilling process we assume that a large cylindrical container has radius s0 and rotates with angular velocity Ω (see Fig. 1). This container has a small orifice at the bottom with radius a. This radius is very small compared with the radius of the container. We examine the problem by choosing a coordinate system (X, Y, Z), which rotates with the container, and having an origin at the axis of the container. The position of the orifice is at (s0, 0, 0). Due to the rotation of the container, the liquid leaves the orifice in a curve. In this problem of the prilling process, the jet moves in the (X, Y, Z) plane, so that the centerline can be described by coordinates (X(s, t), Y (s, t), Z(s, t)), where s is the arc-length along the middle of the jet which emerges from the orifice and t is the time (see Wallwork, 2002). In any cross-section of the jet we also have plane polar coordinates (n, ϕ), which are the radial and azimuthal direction and have unit vectors which are es, en, eϕ (see Decent et al., 2002). The velocity components for this problem are (u, v, w), where u is the tangential velocity, v is the radial velocity and w is the azimuthal velocity. Fig. 2. View largeDownload slide Graph showing the trajectory of a viscoelastic curved liquid jet for different values of the Rossby number where We = 10, De = 10, αs = 0.2 and β = 0.5. Fig. 2. View largeDownload slide Graph showing the trajectory of a viscoelastic curved liquid jet for different values of the Rossby number where We = 10, De = 10, αs = 0.2 and β = 0.5. Fig. 3. View largeDownload slide Graph showing the relationship between the radius and the arc-length s of a viscoelastic curved liquid jet for different values of the Rossby number where We = 10, De = 10, αs = 0.2 and β = 0.5. Fig. 3. View largeDownload slide Graph showing the relationship between the radius and the arc-length s of a viscoelastic curved liquid jet for different values of the Rossby number where We = 10, De = 10, αs = 0.2 and β = 0.5. Fig. 4. View largeDownload slide Graph showing the relationship between $$T_{ss}^{0} -T_{nn}^{0}$$ and the arc-length s of a viscoelastic curved liquid jet for different values of the Rossby number where We = 10, De = 10, αs = 0.2 and β = 0.5. Fig. 4. View largeDownload slide Graph showing the relationship between $$T_{ss}^{0} -T_{nn}^{0}$$ and the arc-length s of a viscoelastic curved liquid jet for different values of the Rossby number where We = 10, De = 10, αs = 0.2 and β = 0.5. Fig. 5. View largeDownload slide The profile at the break-up of a viscoelastic curved liquid jet for different Rossby numbers. Here the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. Fig. 5. View largeDownload slide The profile at the break-up of a viscoelastic curved liquid jet for different Rossby numbers. Here the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. Fig. 6. View largeDownload slide (Left) The profile at the break-up of a viscoelastic curved liquid jet. Here the parameters are Re = 1000, Rb = 1, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. (Right) The profile at the break-up of a viscous curved liquid jet. Here the parameters are Re = 1000, Rb = 1, We = 10, κ = 0.7, αs = 1 and δ = 0.01. Fig. 6. View largeDownload slide (Left) The profile at the break-up of a viscoelastic curved liquid jet. Here the parameters are Re = 1000, Rb = 1, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. (Right) The profile at the break-up of a viscous curved liquid jet. Here the parameters are Re = 1000, Rb = 1, We = 10, κ = 0.7, αs = 1 and δ = 0.01. Fig. 7. View largeDownload slide The profile at the break-up of a viscoelastic curved liquid jet for two values of the Reynolds numbers which are 1000 and 100 from right to left, respectively. Here the parameters are We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. Fig. 7. View largeDownload slide The profile at the break-up of a viscoelastic curved liquid jet for two values of the Reynolds numbers which are 1000 and 100 from right to left, respectively. Here the parameters are We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. Fig. 8. View largeDownload slide The profile of the break-up of viscoelastic liquid jets for different values of the Deborah number. We observe that when these three numbers increase, we have longer break-up. Here we use Rb = 1, We = 10, k = 0.7, δ = 0.01 and αs = 0.2, Re = 80 and β = 0.05. Fig. 8. View largeDownload slide The profile of the break-up of viscoelastic liquid jets for different values of the Deborah number. We observe that when these three numbers increase, we have longer break-up. Here we use Rb = 1, We = 10, k = 0.7, δ = 0.01 and αs = 0.2, Re = 80 and β = 0.05. The flow is described by the governing equations, which are the continuity equation, the momentum equation and the constitutive equation. We used the nonlinear Giesekus model to investigate the viscoelastic curved liquid jets.   \begin{equation} \nabla \cdot u=0, \end{equation} (2.1)   \begin{equation} \rho \left(\frac{\partial u}{\partial t}+ u.\nabla u\right) = - \nabla p+ \nabla .\tau \,-2\, w\times u - w\times (w\times r), \\ \end{equation} (2.2)   \begin{equation} \tau =\mu_{s} \left(\nabla u+(\nabla u)^{T} \right)+ T, \end{equation} (2.3)   \begin{equation} \frac{\partial T}{\partial t}+( u.\nabla ) T-T \cdot \nabla u -(\nabla u)^{T} \cdot T+\frac{\beta}{\mu_{p}} T^{2}=\frac{1}{\lambda }(\mu_{p} \gamma -T), \end{equation} (2.4) where u = ues + ven + weϕ, ρ is the density, p is the pressure, μs is the viscosity of the solvent, μp is the viscosity of the polymer, β is the mobility factor and T is the extra stress tensor. The free surface of the jet is described by n − R(s, t, ϕ) = 0, where R(s, t, ϕ) is a function which gives the free surface position, and the normal vector is given by ∇(n − R(s, ϕ, t)), which gives   \begin{equation*} n=\frac{1}{E} \left(-\frac{\partial R}{\partial s} \cdot \frac{1}{h_{s}}\cdot e_{s}\,+\,e_{n}-\frac{\partial R}{\partial \phi }\cdot\frac{1}{R} \cdot e_{\phi }\right),\end{equation*} where   \begin{equation*} E\,=\,\left(1+\frac{1}{{h^{2}_{s}}} \left(\frac{\partial R}{\partial s} \right)^{2}\,+\,\frac{1}{R^{2}}\left(\frac{\partial R}{\partial \phi }\right)^{2} \right)^{\frac{1}{2}}.\end{equation*} The normal stress condition is given by n ⋅Π⋅ n = σκ, where Π is the total stress tensor given by −PI + τ, σ is the isotropic surface tension and κ is the curvature of the free surface   \begin{equation*} \kappa =\frac{1}{n h_{s}}\left(-\frac{\partial }{\partial s} \left(\frac{n}{Eh_{s}}\, \frac{\partial R}{\partial s}\right) \,+\,\frac{\partial}{\partial n}\left(\frac{n h_{s}}{E}\right) -\frac{\partial}{\partial \phi}\left(\frac{h_{s}}{En}\, \frac{\partial R}{\partial \phi} \right)\right)\nonumber. \end{equation*} The tangential stress conditions are ti ⋅ σ ⋅ n, where i = 1, 2 and $$t_{1}=\,e_{s} +\,\frac {\partial R}{\partial s}\cdot \frac {1}{h_{s}} \cdot e_{n}, \, t_{2}=\,\frac {\partial R}{\partial \phi }\cdot \frac {1}{R}\cdot e_{n}+e_{\phi }$$. The kinematic condition is given by $$ \frac {D}{Dt}(R(s,t,\phi -n)=0$$ on n = R(s, ϕ, t). We can write the previous equation as   \begin{equation*} \frac{\partial R}{\partial t}+\frac{\partial R}{\partial s} \frac{\partial s}{\partial t}+\frac{\partial R}{\partial \phi }\frac{\partial \phi }{\partial t}-\frac{\partial n}{\partial t}=0. \nonumber \end{equation*} The pressure condition on the free surface is p = σκ on n = R, and the arc-length condition is   \begin{equation*} {X_{s}^{2}}+{Z^{2}_{s}}=1.\end{equation*} These equations are similar to those found in Alsharif et al. (2015). However, the differences are in the constitutive equation of the elastic term. 3. Asymptotic analysis Here we will expand u, v, w, p in Taylor series in εn (see Eggers (1997)) and R, X, Z, Tss, Tnn, Tϕϕ, Tsn, Tsϕ, Tnϕ in ε. We suppose that the leading order of the axial component of the velocity is independent of ϕ.   \begin{align*} (u,v,w)(s,n,\phi,t)&=(u_{0},0,0)(s,t)+ \,(\varepsilon\,n )( u_{1},v_{1},w_{1})(s,\phi,t)+...\\[8pt] p(s,n,\phi,t) &=\,p_{0}(s,\phi,t)+(\varepsilon\,n)p_{1}(s,\phi,t)+...\\[8pt]R(s,n,\phi,t)\, &=\,R_{0}(s,t)+(\varepsilon)R_{1}(s,\phi,t)+...\\[8pt](X,Y,Z)(s,n,\phi,t) &=(X_{0}, Y_{0}, Z_{0})(s)+ (\varepsilon )(X_{1},Y_{1}, Z_{1} )(s,t) +...\\[8pt]\left(T_{ss},T_{nn},T_{\phi \phi } \right)(s,n,\phi,t) &= \left(T^{0}_{ss},T^{0}_{nn} \right)(s,t)+\varepsilon\, \left(T^{1}_{ss},T^{1}_{nn}, T^{1}_{\phi \phi} \right)(s,t)+...\\[8pt]\left(T_{sn},T_{s\phi },T_{n \phi } \right)(s,n,\phi,t) &= \varepsilon\, \left(T^{1}_{ss},T^{1}_{nn}, \varepsilon T^{1}_{\phi \phi} \right)(s,t)+... \end{align*} We substitute these asymptotic expansions into the governing equations. We can therefore find from the equation of continuity at the leading order   \begin{equation} v_{1}=-\frac{u_{0s}}{2}. \end{equation} (3.1) It can be also obtained that from the equation of motion in s-direction at leading order   \begin{align} u_{0t}+u_{0}u_{0s} =&-\frac{1}{We}\,\left(\frac{1}{ R_{0}}\right)_{s}+\frac{(X+1)X_{s}+ZZ_{s}}{Rb^{2}}\nonumber\\ &+\frac{3\alpha_{s}}{{R^{2}_{0}}\, Re}\left({R^{2}_{0}}\,u_{0s}\right)_{s}+\frac{1}{ {R^{2}_{0}} Re}\left( {R^{2}_{0}}\left(T^{0}_{ss}-T^{0}_{nn}\right)\right)_{s}, \end{align} (3.2)   \begin{equation} (X_{s}Z_{ss}-X_{ss}Z_{s} )\left( {u^{2}_{0}}-\frac{3\alpha_{s}}{Re}\, u_{0s}-\frac{1}{WeR_{0}}\right) - \frac{2}{Rb}\,u_{0} +\frac{(X+1)Z_{s}+ ZX_{s}}{Rb^{2}}=0, \end{equation} (3.3) from the extra stress tensor Eq. (2.4) at leading order, we get   \begin{equation} \frac{\partial T_{ss}^{0}}{\partial t}+u_{0}\,\frac{\partial T^{0}_{ss}}{\partial s}\,-2\,\frac{\partial u_{0}}{\partial s}\,T^{0}_{ss} +\frac{\beta}{1- \alpha_{s}}T_{ss}^{02}= \frac{1}{De}\,\left( 2(1-\alpha_{s})\,\frac{\partial u_{0}}{\partial s}\,-\,T^{0}_{ss}\right), \end{equation} (3.4)   \begin{equation} \frac{\partial T_{nn}^{0}}{\partial t}+u_{0}\,\frac{\partial T^{0}_{nn}}{\partial s}\,+\frac{\partial u_{0}}{\partial s}\,T^{0}_{nn}+\frac{\beta}{1- \alpha_{s}}T_{nn}^{02}= \frac{-1}{De}\,\left( (1-\alpha_{s})\frac{\partial u_{0}}{\partial s}\,+\,T^{0}_{nn}\right), \end{equation} (3.5) from the kinematic condition, we obtain at ε   \begin{equation} \frac{\partial R_{0}}{\partial t}+u_{0}R_{0s}+\frac{R_{0}}{2}u_{0s}=0, \end{equation} (3.6) where $$ Re= \frac {\rho Ua}{\mu _{0} },\, We=\frac {\rho U^{2}a}{\sigma } $$, $$ Rb=\frac {U}{s_{0}\varOmega }$$, $$ De=\frac {\lambda U}{s_{0} }$$ and $$ \alpha _{s}=\frac {\mu _{s} }{\mu _{s}+\mu _{p}} = \frac {\mu _{s} }{\mu _{0}} $$ are the Reynolds number, the Weber number, the Rossby number, the Deborah number and the viscosity ratio, respectively. We can see that when β = 0 the equations (3.2)–(3.6) recover to the linear Oldroyd-B model (see Alsharif et al. (2015)). 4. Steady state solutions From equation (3.6) we have $$ {R_{0}^{2}} u_{0}=1$$, so that the governing equations for this system are Fig. 9. View largeDownload slide The profile at the break-up of a viscoelastic curved liquid jet for different Rossby numbers. Here the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. Fig. 9. View largeDownload slide The profile at the break-up of a viscoelastic curved liquid jet for different Rossby numbers. Here the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05.   \begin{align} u_{0}u_{0s}=&-\frac{u_{0s}}{2We\,\sqrt u}+\frac{(X+1)X_{s}+ZZ_{s}}{Rb^{2}} + \frac{3\alpha_{s} }{Re}\left( u_{0ss}-\frac{u^{2}_{0s}}{u_{0}}\right) \nonumber\\ &+\frac{1}{ Re} \left( \frac{\partial }{\partial s}\left( T^{0}_{ss}-T^{0}_{nn}\right)-\frac{ u_{0s}}{u_{0}}\left(T^{0}_{ss}-T^{0}_{nn}\right)\right), \end{align} (4.1)   \begin{equation} (X_{s}Z_{ss}-X_{ss}Z_{s})\left( {u^{2}_{0}}-\frac{3}{Re}\, u_{0s}-\frac{\sqrt u}{We}\right)- \frac{2}{Rb}\,u_{0} +\frac{(X+1)Z_{s}+ZX_{s}}{Rb^{2}}=0, \end{equation} (4.2)   \begin{equation} u_{0}\frac{\partial T^{0}_{ss}}{\partial s}-2\frac{\partial u_{0}}{\partial s}T^{0}_{ss}+\frac{\beta}{1- \alpha_{s}}T_{ss}^{02}= \frac{1}{De}\left( 2(\alpha_{s}-1) \frac{\partial u_{0}}{\partial s}-T^{0}_{ss}\right), \end{equation} (4.3)   \begin{equation} u_{0}\frac{\partial T^{0}_{nn}}{\partial s}+\frac{\partial u_{0}}{\partial s}T^{0}_{nn}+ \frac{\beta}{1- \alpha_{s}} T_{nn}^{02}=\frac{-1}{De}\left( (\alpha_{s}-1) \frac{\partial u_{0}}{\partial s}+T^{0}_{nn}\right), \end{equation} (4.4) and finally the arc-length condition is   \begin{equation} {X^{2}_{s}}+{Z^{2}_{s}}=1. \end{equation} (4.5) These are a system of five equations in five unknowns which are X, Z, u0, $$T_{ss}^{0}$$ and $$T_{nn}^{0}$$. This system of nonlinear differential equations, which is stiff due to the presence of the viscous term, can be solved by using an implicit finite difference scheme as done by Parau et al. (2007) who solved the resulting set of equations using the Newtonian methods. However, the same author were able to show that the presence of the viscous term did not significantly affect the steady state obtained in the inviscid limit. Therefore, we solve these equations by using the Runge–Kutta method in the inviscid limit $$Re \rightarrow o $$ (see Parau et al. (2007)) for obtaining the trajectory of the jet with following initial conditions as u0(0) = R0(0) = Xs(0) = 1 and X(0) = Z(0) = Zs(0) = 0. In Fig. 2 we show the effect of the Rossby number on the trajectory of the liquid jet. From this figure, it can be noticed that when the Rossby number decreases (meaning the rotation rates increases) the jets coil more progressively. The relationship between the radius of the jet and the extra stress tensor, $$ T_{ss}^{0}$$, $$T_{nn}^{0} $$, and the arc-length s are plotted in Figs 3 and 4. From these two figures, we can observe that when the rotation rates Rb increase the jet becomes more thin and these two results confirm that the jet thinning occur at the nozzle and decreases along the viscoelastic liquid jet which agrees with the results of Feng (2003). 5. Nonlinear temporal solutions Linear instability analysis predicts that liquid jets break up and produce uniform drop sizes along the axis of approximately the same wavelength of initial disturbances. However, it can be observed that a number of smaller satellite droplets appeared in this case which are not equal in size. Therefore, we use nonlinear temporal analysis to examine the break-up length and the formation of satellite droplets. We replace the leading order pressure term $$ p_{0} = \frac {1}{We}\frac {1}{R_{0}}$$ in the equation (3.2) with the expression for the full curvature term which contains only R0 and is not ϕ-dependent, namely Fig. 10. View largeDownload slide Graph showing the relationship between the break-up length and Rb. Where the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. Fig. 10. View largeDownload slide Graph showing the relationship between the break-up length and Rb. Where the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05.   \begin{equation} p = \frac{1}{We}\left[ \frac{1}{R_{0}\left( 1+\varepsilon^{2} R^{2}_{0s}\right)^{1/2}}- \frac{\varepsilon^{2} R_{0ss}}{\left( 1+\varepsilon^{2} R^{2}_{0s}\right)^{3/2}} \right] . \end{equation} (5.1) For simplicity, we denote A = A(s, t), where A(s, t) = R2(s, t), and then we rewrite our equations (3.2) and (3.4)–(3.6) as   \begin{align} \frac{\partial u}{\partial t}=&-\left( \frac{u^{2}}{2}\right)_{s}- \frac{1}{We}\frac{\partial}{\partial s}\,\frac{4\left(2A+(\varepsilon A_{s})^{2}-\varepsilon^{2}AA_{ss}\right)}{\left(4A+(\varepsilon A_{s})^{2}\right)^{3/2}}\nonumber\\ &+\frac{(X+1)X_{s}+ZZ_{s}}{Rb^{2}}+ \frac{3\alpha_{s} }{Re}\,\frac{(Au_{s})_{s}}{A}+\frac{1}{ Re}\,\frac{\left( A(T_{ss}-T_{nn})\right)_{s}}{A}, \end{align} (5.2)   \begin{equation} \frac{\partial T_{ss}}{\partial t} = - \frac{\partial }{\partial s}(uT_{ss})+3\frac{\partial u}{\partial s}T_{ss} -\frac{\beta}{1- \alpha_{s}}T_{ss}^{02}+ \frac{1}{De} \left( 2(1-\alpha_{s} )\frac{\partial u}{\partial s}-T_{ss}\right), \end{equation} (5.3)   \begin{equation} \frac{\partial T_{nn}}{\partial t} = - \frac{\partial }{\partial s}(uT_{nn})-\frac{\beta}{1- \alpha_{s}}T_{nn}^{02} - \frac{1}{De} \left( (1-\alpha_{s} )\frac{\partial u}{\partial s}+T_{nn} \right), \end{equation} (5.4)   \begin{equation} \frac{\partial A}{\partial t} = -\frac{\partial }{\partial s}(Au).\end{equation} (5.5) To solve this system of equations, we use the initial conditions at t = 0 which are $$ A(s,t=0)={R^{2}_{0}}(s), u(s,t=0)=u_{0}(s), T_{ss}(s,t=0)=0, \,\,T_{nn}(s,t=0)=0 $$ as we obtained for the steady state solutions (see Section 4). We use upstream boundary conditions at the nozzle as follows   \begin{equation*} A(0, t)=1,\,\, u(0,t)=1+\delta \sin \left(\frac{\kappa t}{\varepsilon }\right), \end{equation*} where κ and δ a non-dimensional wavenumber of the perturbation of frequency and the amplitude of the initial non-dimensional velocity disturbance, respectively. In our calculations, the value of $$\varepsilon (=\frac {a}{s_{0}})$$ is 0.01 as found in experiments and the industrial problem (see Wong et al. (2004)). We have used a second-order finite difference scheme based on the two-step Lax–Wendroff method to determine the break-up lengths and drop formation of a rotating viscoelastic liquid jet. Fig. 11. View largeDownload slide Graph showing the relationship between the droplet radius and αs. Where the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, δ = 0.01 and β = 0.05. Fig. 11. View largeDownload slide Graph showing the relationship between the droplet radius and αs. Where the parameters are Re = 1000, We = 10, De = 10, κ = 0.7, δ = 0.01 and β = 0.05. Fig. 12. View largeDownload slide Graph showing the relationship between the droplet radius and De. Where the parameters are Re = 1000, We = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. Fig. 12. View largeDownload slide Graph showing the relationship between the droplet radius and De. Where the parameters are Re = 1000, We = 10, κ = 0.7, αs = 0.2, δ = 0.01 and β = 0.05. In order to display the curved jet, we are required to determine the normal vector along the centerline of the jet (X(s), Z(s)), which is in this case n = (−Zs(s), Xs(s))). Thereafter the free surface of the jet is given by (X(s), Z(s)) + R(s, t)(−Zs(s), Xs(s)) and (X(s), Z(s)) − R(s, t)(−Zs(s), Xs(s)). To determine size of main droplets, we need to integrate between two local minimums (which correspond to pinch points along the jet). If we label them as h1 and h2 then   \begin{equation*} V_{drop}=\pi \int_{h_{1}}^{h_{2}} {R^{2}_{0}} \text{ d}s , \end{equation*} where the drop radius R equates to a sphere, then   \begin{equation*} \hat{R} =\left(\frac{3V_{drop}}{4\pi }\right)^{\frac{1}{3}}. \end{equation*} We have plotted a profile (Fig. 5) to show the effects of increasing the Rossby number on the break-up of a viscoelastic liquid curved jet. It can be observed from this profile that an increase in the rotation rates (i.e. the Rossby number decreases) leads to an increase in the break-up lengths, which means that the rotation rates affect the break-up lengths. We have made a comparison between the break-up of viscoelastic and viscous curved liquid jets and we found that the viscoelastic rotating jet breaks up before the viscous rotating jet (see Fig. 6). Furthermore, we show some profiles which indicate the break-up for two different values of the Reynolds number. It can be noticed from these two profiles that when we decrease the Reynolds number, which corresponds to high viscosity, the break-up length of the liquid curved jet increases (see Fig. 7). Figure 7 shows that when we increase the Deborah number the break-up lengths and satellite drops increase. In Fig. 9 the relationship between the jet radii versus the arc-length s have been plotted. It can be noticed from this figure that when we increase the mobility factor β the liquid jet delays to break up and this result agrees with the finding of Ardekani et al. (2010). Therefore we have chosen that the break-up occurs when the non-dimensional jet radius has reached an arbitrarily small value which is for consistency with earlier works (Parau et al. (2007)) 5% of the initial jet radius. Downstream of the break-up point, where the jet is expected to breakup into droplets, our numerical solution has no real physical meaning, as it is the case in other works (see Parau et al. (2007), Decent et al. (2009) and Alsharif (2014)). In Fig. 10 we see that when we increase the mobility factor, the break-up lengths increase. We can also observe that an increase in the rotation rates (i.e. low Rb) leads to an enhance in the break-up lengths which is also agrees with spiralling Newtonian liquid jets (see Parau et al. (2007)). Furthermore, in Figs 11 and 12 we can see that the relationship between the droplet sizes and the viscosity ratio αs and the Deborah number have been plotted. From these two figures, we notice that firstly, when we increase the mobility factor β, the main and satellite droplet sizes increase. Secondly, it can be observed that satellite droplet sizes increase when the Deborah number is increased and this result is in agreement with Alsharif et al. (2015). 6. Conclusion In summary, there are many applications of break-up liquid jets which emerge from an orifice, assuming that the centreline of the jet is straight with inkjet printing. However, it can be considered that the centreline is curved, which is known as the prilling process. This case is investigated here for viscoelastic liquid curved jets using the Giesekus model. It is also important in this process to study the effect of the rotation on the break-up lengths and droplet sizes. We have made an assumption, which is that the viscosity does not affect the trajectory of the centreline (see Decent et al. (2002), Uddin & Decent (2010) and Alsharif (2014)) and this is taken as an inviscid liquid jet. We have incorporated the Giesekus model into a 1D slender-body theory and examined the role of nonlinear rheology in a rotating viscoelastic liquid curved jet. An asymptotic approach has been used to reduce the governing equations into a set of 1D equations. The effect of changing the Rossby number on the trajectory of the viscoelastic liquid curved jet has been plotted (see Fig. 2). In addition, a comparison has been made between viscoelastic and viscous liquid curved jets in terms of the break-up lengths and it can be observed that the viscoelastic liquid curved jet breaks up before the viscous one (see Fig. 7). A numerical method based on finite differences has been used to determine the break-up lengths and main and satellite droplet sizes. Our results show that in the Giesekus model for viscoelastic curved liquid jets, it can be observed that the break-up lengths increase when we increase the mobility factor (see Fig. 11). Moreover, an increase in the mobility factor leads to decrease into the main and satellite droplet sizes (see Fig. 12). For future work, we can extend this work to investigate the effects of adding surfactants on the instability of a rotating viscoelastic liquid jets (see Alsharif & Uddin (2015)). Acknowledgements Abdullah Alsharif would like to thank Taif University for their financial support. References Alsharif, A. M. 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Published: Feb 1, 2018

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