Local Weissenberg Number Research Articles

Boundary-layer flow and heat transfer of cross fluid over a stretching sheet

The current study is a pioneering work in presenting the boundary-layer equations for the 2-D flow and heat transfer of the Cross fluid over a linearly stretching sheet. The system of PDE is turned down into highly non-linear ODE by applying suitable similarity transformations. The stretching sheet solutions are presented via. a numerical technique namely the shooting method and graphs are constructed. The impact of the emerging parameters namely the power-law index, n, the local Weissenberg number, We, and the Prandtl number on the velocity and temperature fields are investigated through graphs. The numerical values of the local skin friction coefficient and the local Nusselt number are also presented in tabular form. Additionally, the graphs are sketched for the local skin friction coefficient and the local Nusselt number. It is observed that with growing values of the local Weissenberg number the velocity profiles exhibited a decreasing trend while opposite behavior is seen for the temperature field. Further, comparisons are made with previously available literature for some limiting cases and an excellent agreement is achieved.

Interaction between chemical species and generalized Fourier’s law on 3D flow of Carreau fluid with variable thermal conductivity and heat sink/source: A numerical approach

Abstract This paper deals with three-dimensional (3D) flow of a Carreau fluid by utilizing the impact of heterogeneous-homogeneous reactions towards the bidirectional stretched surface. The heat transfer mechanism is carried out in apparition of improved heat conduction relation. This occurrence is documented upon the notion of generalized Fourier’s law that contributes by the thermal relaxation. Additionally, temperature dependent thermal conductivity and heat sink/source are accounted. On utilization of a suitable conversions a system of nonlinear ordinary differential equation (ODEs) is attained and then inferred numerically via bvp4c approach. The delineations of velocities, temperature and concentration fields corresponding to the numerous somatic parameters are scrutinized explicitly. The impact of local Weissenberg number We 1 on f ′ ( η ) and We 2 on g ′ ( η ) are same for (n = 0.5 and 1.5). Furthermore, our inspection spectacles that the concentration of the Carreau liquid decays as the heterogeneous-homogeneous reaction k 2 , k 1 parameters boost up. It is also remarkable that for shear thinning ( n 1 ) fluid the influence of local Weissenberg numbers We 1 , We 2 are absolutely contradictory as associated with the case of shear thickening ( n > 1 ) fluid. For authentication of numerical outcomes a comparison table is prepared via benchmarking with formerly itemized limiting cases and we pledge a marvelous communication with these results. Additionally, graphically assessment is presented between numerically (bvp4c) and analytically (HAM) techniques with tremendous settlement.

Numerical simulation for heat transfer performance in unsteady flow of Williamson fluid driven by a wedge-geometry

Abstract The main concern of this communication is to investigate the two-layer flow of a non-Newtonian rheological fluid past a wedge-shaped geometry. One remarkable aspect of this article is the mathematical formulation for two-dimensional flow of Williamson fluid by incorporating the effect of infinite shear rate viscosity. The impacts of heat transfer mechanism on time-dependent flow field are further studied. At first, we employ the suitable non-dimensional variables to transmute the time-dependent governing flow equations into a system of non-linear ordinary differential equations. The converted conservation equations are numerically integrated subject to physically suitable boundary conditions with the aid of Runge-Kutta Fehlberg integration procedure. The effects of involved pertinent parameters, such as, moving wedge parameter, wedge angle parameter, local Weissenberg number, unsteadiness parameter and Prandtl number on the non-dimensional velocity and temperature distributions have been evaluated. In addition, the numerical values of the local skin friction coefficient and the local Nusselt number are compared and presented through tables. The outcomes of this study indicate that the rate of heat transfer increases with the growth of both wedge angle parameter and unsteadiness parameter. Moreover, a substantial rise in the fluid velocity is observed with enhancement in the viscosity ratio parameter while an opposite trend is true for the non-dimensional temperature field. A comparison is presented between the current study and already published works and results found to be in outstanding agreement. Finally, the main findings of this article are highlighted in the last section.

Mixed convection heat transfer to cross fluid with thermal radiation: Effects of buoyancy assisting and opposing flows

The present investigations proclaim the mixed convection flow and heat transfer of Cross fluid past a vertical stretching sheet taking into account the effects of assisting and opposing buoyancy. In addition, the heat transfer analysis is thoroughly examined corresponding the impact of thermal radiation and convective boundary condition. The modeled boundary layer equations are reduced to couple non-linear ordinary differential equations by endeavoring suitable transformations. The solution of the resulting system of equations is then numerically determined by the help of shooting technique. The influence of the emerging parameters on the flow and heat transfer is investigated with the respective ranges taken as: the Weissenberg number (1 ≤ We ≤ 5), the power-law index (0 ≤ m ≤ 1), the mixed convection parameter ( − 0.51 ≤ λ ≤ 0.5), the Prandtl number (0.7≤Pr≤70), the radiation parameter (0.1 ≤ R ≤ 1) and the Biot number (0.1 ≤ γ ≤ 1.5). Comparative plots are provided for the case of assisting as well as opposing flows. Few of the several significant findings of the paper are: (i) the growing values of local Weissenberg number reduces the local skin friction coefficient, (ii) the radiation parameter and Biot number increases the thermal boundary layer thickness and the effects are much prominent for opposing flow. The validity of the numerical results is ensured by making comparison with existing work as special cases.

Numerical simulation for magneto Carreau nanofluid model with thermal radiation: A revised model

The present work explores the magnetohydrodynamic (MHD) flow of Carreau nanoliquid by exponentially convected stretchable surface. Formulation and computations are presented for Brownian motion and thermophoresis. Concentration by zero mass condition is reported. Consideration of thermal radiation characterizes the heat transfer process. Transformation procedure is utilized for reduction of PDEs into ODEs. Highly nonlinear complex problems are computed numerically through Runge–Kutta Fehlberg technique. Salient characteristics of local Weissenberg number, Hartman number, Biot number, thermophoresis parameter, Prandtl number, thermal radiation parameter and Schmidt number on the velocity, temperature, nanoparticles concentration, surface drag force and Nusselt number are reported through graphs and Tables. The results demonstrated here reveal that the velocity distribution for local Weissenberg number in case of shear thinning liquid reduces whereas it increments for shear thickening liquid. Temperature and thermal layer thickness are increasing functions of thermal radiation. Besides this the results of presented analysis are compared with the available works in particular situations and reasonable agreement is noted.

On axisymmetric flow and heat transfer of Cross fluid over a radially stretching sheet

In this article, an analysis is made on the axisymmetric flow and heat transfer of the Cross fluid over a radially stretching sheet. The present study provides with the boundary layer equations of the Cross fluid in cylindrical polar co-ordinates. The modelled momentum and energy equations are further simplified into non-linear ordinary differential equations by applying suitable similarity transformations. The system of equation is then numerically solved by the help of well-known shooting technique. The velocity and temperature profiles are plotted for some values of the governing parameters such as power-law index, local Weissenberg number and the Prandtl number. It is found that growing values of the power-law index elevated the momentum boundary layer structures while the thermal boundary layer thickness lessened correspondingly. Further, the numerical values of the local skin friction coefficient and the local Nusselt number are tabulated for several set of physical parameters. An outstanding agreement is observed by comparing the present results with the previously reported results in the literature asa special case.

Stability analysis of constitutive equations for polymer melts in viscometric flows

The stability of two representative constitutive equations for polymer melts, the exponential Phan-Thien–Tanner (PTT) model and the Giesekus model, are investigated in planar shear flows. For the PTT equation, instabilities are predicted for both plane Couette and Poiseuille flows using transient finite-element calculations. A Chebyshev–Tau spectral method is used to confirm that these instabilities are not spurious or an artifact of the finite element formulation. Mechanisms are proposed based on an energy analysis of the most unstable mode for each flow. The stability of plane Couette flow of a Giesekus model is also probed using our spectral method and found to be stable for the range of parameters investigated. However, in pressure driven flow, the Giesekus model is unstable over a critical local Weissenberg number ( Wi) based on the shear rate at the channel wall. We present the complete eigenspectrum for this model in both Couette and Poiseuille flows and note that ‘ballooning’ of the continuous spectrum, which can cause spurious instabilities, is significantly stabilized for this constitutive equation relative to the upper convected Maxwell model or PTT model. Both Poiseuille flow instabilities occur at moderate Weissenberg numbers and, with more careful investigation, may be able to explain some of the unusual phenomena observed in slit flows of polymer melts [AIChE J. 22 (2) (1976) 209; J. Non-Newt. Fluid Mech. 77 (1998) 123; Adv. Polym. Sci. 138 (1999) 227].

Dynamical similarity and dimensionless relations for turbulent drag reduction by polymer additives

Drag reduction in turbulent flow of high-molecular-weight polymer solutions has been investigated. Experiments were run in a rotational apparatus with geometrically similar operating units of different sizes. Aqueous solutions of poly(ethylene oxide) with different molecular weights and solutions of polyisobutylene in kerosene were used. The concentration and temperature of the solutions were varied. Drag reduction was correlated with the rheological characteristics of the solutions; i.e. with the natural time and the shear-thinning viscosity. The dimensionless viscoelastic parameters have been found, which, with the Reynolds number, determine the turbulent friction. These are the Weissenberg number and the relative amplitude of the shear-thinning viscosity. Drag reduction sets in at a certain value of the local Weissenberg number calculated from turbulent flow characteristics near the wall. The critical value of this criterion is equal to several units. Friction coefficient versus Reynolds number functions are of a power nature with the exponent depending on the relative amplitude of the shear-thinning viscosity. Empirical relations were obtained that can describe fluid friction of the polymer solutions in dimensionless form.

Viscoelastic free surface flows: thin film hydrodynamics of Hele-Shaw and dip coating flows

The effect of elasticity on the meniscus shape and film thickness for the free boundary creeping flow created by injecting air to a Hele-Shaw cell initially filled with a viscoelastic fluid is studied theoretically. The theory is developed with the assumptions that the displaced viscoelastic fluid wets the walls and that capillary number Ca and the local Weissenberg number We are both small. The transition region between the advancing meniscus and the entrained film is where the fluid rheology has its greatest effect. The Oldroyd-B constitutive equation is used to model the viscoelastic fluid. The theory is formulated as a double expansion in Ca 1 3 and We Ca 1 3 . According to our asymptotic analysis, as the fluid becomes more viscoelastic, the film thickness decreases and the pressure drop at the meniscus tip increases. A detailed analysis shows that the dominant mechanisms are the resistance to stream-wise strain, tending to lower the film thickness, and the buildup of shear stress, tending to raise the film thickness, with the former being the numerically larger of the two. Our theory leads to the prediction that as viscoelasticity comes to dominate the shear resistance, the film thickness will scale with U 4 3 . The effects of shear thinning and normal stress thinning are analyzed by adapting an approximate model obtained by retaining only the dominant terms in the force balance. Our Hele-Shaw cell theory is extended to dip coating and soap film forming flows. Our theoretical results pertaining to the effects of viscoelasticity on the thickness of the film are in qualitative agreement with recent experimental data.