Showing papers on "Similarity solution published in 2021"

Investigation of mixture fluid suspended by hybrid nanoparticles over vertical cylinder by considering shape factor effect

Abstract: In this study, flow of a mixture of water and ethylene glycol (50–50%) with hybrid nanoparticles (MWCNT–Ag) over a vertical stretching cylinder has been investigated. In this research, the fluid passes through a porous media, while a magnetic field has been applied to the system. Furthermore, the effects of thermal radiation, viscous dissipation, and natural convection have been studied. As a novelty, the effects of different shape factors have been investigated. In the first step, the governing equations are extracted from partial differential equations and then converted to ordinary differential equations (ODE) using the similarity solution. In the next step, the fifth-order Runge–Kutta method has been used to solve the related ODEs. The effects of parameters such as magnetic field, radiation parameter, porosity parameter, nanofluid volume fraction, and nanofluid shape factor on dimensionless velocity and temperature profile have been presented for single and hybrid nanofluid. The results showed that at $$\eta$$ = 2.5 for hybrid nanoparticles the shape factors lamina and spherical have the largest difference; lamina is smaller by 6%, also the results demonstrated that at $$\eta$$ = 2 with increasing Ha, the radial velocity reduced 9.68% for hybrid nanoparticles.

Effect of Magnetohydrodynamics on Heat Transfer Behaviour of a Non-Newtonian Fluid Flow over a Stretching Sheet under Local Thermal Non-Equilibrium Condition

Abstract: A mathematical model is proposed to describe the flow, heat, and mass transfer behaviour of a non-Newtonian (Jeffrey and Oldroyd-B) fluid over a stretching sheet. Moreover, a similarity solution is given for steady two-dimensional flow subjected to Buongiorno’s theory to investigate the nature of magnetohydrodynamics (MHD) in a porous medium, utilizing the local thermal non-equilibrium conditions (LTNE). The LTNE model is based on the energy equations and defines distinctive temperature profiles for both solid and fluid phases. Hence, distinctive temperature profiles for both the fluid and solid phases are employed in this study. Numerical solution for the nonlinear ordinary differential equations is obtained by employing fourth fifth order Runge–Kutta–Fehlberg numerical methodology with shooting technique. Results reveal that, the velocity of the Oldroyd-B fluid declines faster and high heat transfer is seen for lower values of magnetic parameter when compared to Jeffry fluid. However, for higher values of magnetic parameter velocity of the Jeffery fluid declines faster and shows high heat transfer when compared to Oldroyd-B fluid. The Jeffery liquid shows a higher fluid phase heat transfer than Oldroyd-B liquid for increasing values of Brownian motion and thermophoresis parameters. The increasing values of thermophoresis parameter decline the liquid and solid phase heat transfer rate of both liquids.

MHD Flow and Heat Transfer of Hybrid Nanofluid over an Exponentially Shrinking Surface with Heat Source/Sink

Abstract: In nanotechnology research, nanofluid technology contributes many applications to engineering applications and industry, such as power generation, solar collection, heat exchangers for cooling, and many more. However, there are still a few constraints in terms of heat transfer enhancement, although nanofluid properties show the best heat transfer rate compared with conventional fluids. Thus, this study was conducted for the purpose of investigating the behaviors of flow and heat transfer of hybrid nanofluid with carbon nanotubes (CNTs) on a permeable exponentially shrinking surface, as well as investigating the effects of a magnetic field and heat source/sink. This study was conducted by developing a mathematical model, which was the Tiwari–Das model for momentum and energy equations, and then transforming the model’s partial differential equations (PDEs) to ordinary differential equations (ODEs) using a similarity solution. Next, these equations were solved numerically using the MATLAB bvp4c boundary value problem solver. The authors particularly explored these behaviors with a few variations. Based on the results obtained, it was found that dual solutions exist in a specific range of the shrinking case, λc 0. Therefore, this study deduced that the heat transfer rate of hybrid nanofluid (CNTs/Cu–water) is better than regular nanofluid (CNT–water) and conventional fluid (water). The present study took into consideration the problem of MHD flow and heat transfer analysis of a hybrid nanofluid towards an exponentially shrinking surface with the presence of heat source/sink and thermal radiation effects. The authors show that dual solutions exist within a specific range of values due to the shrinking case. The current work is predicted to have numerous benefits in equivalent real-world systems.

Stefan problems for the diffusion–convection equation with temperature-dependent thermal coefficients

Abstract: Different one-phase Stefan problems for a semi-infinite slab are considered, involving a moving phase change material as well as temperature dependent thermal coefficients. Existence of at least one similarity solution is proved imposing a Dirichlet, Neumann, Robin or radiative–convective boundary condition at the fixed face. The velocity that arises in the convective term of the diffusion–convection equation is assumed to depend on temperature and time. In each case, an equivalent ordinary differential problem is obtained giving rise to a system of an integral equation coupled with a condition for the parameter that characterizes the free boundary, which is solved through a double-fixed point analysis. Some solutions for particular thermal coefficients are provided.

On small-time similarity-solution behaviour in the solidification shrinkage of binary alloys

Abstract: In the one-dimensional solidification of a binary alloy undergoing shrinkage, there is a relative motion between solid and liquid phases in the mushy zone, leading to the possibility of macrosegregation; thus, the problem constitutes an invaluable benchmark for the testing of numerical codes that model these phenomena. Here, we revisit an earlier obtained solution for this problem, that was posed on a semi-infinite spatial domain and valid for the case of low superheat, with a view to extending it to the more general situation of a finite spatial domain, arbitrarily large superheat and both eutectic and non-eutectic solidification. We find that a similarity solution is available for short times which contains a boundary layer on the liquid side of the mush–liquid interface; this solution is believed to constitute the correct initial condition for the subsequent numerical solution of the full non-similar problem, which is deferred to future work.

Numerical Simulation of Laminar Boundary Layer Flow Over a Horizontal Flat Plate in External Incompressible Viscous Fluid

Abstract: In this paper, steady laminar boundary layer flow of a Newtonian fluid over a flat plate in a uniform free stream was investigated numerically when the surface plate is heated by forced convection from the hot fluid. This flow is a good model of many situations involving flow over fins that are relatively widely spaced. All the solutions given here were with constant fluid properties and negligible viscous dissipation for two-dimensional, steady, incompressible laminar flow with zero pressure gradient. The similarity solution has shown its efficiency here to transform the governing equations of the thermal boundary layer into a nonlinear, third-order ordinary differential equation and solved numerically by using 4th-order Runge-Kutta method which in turn was programmed in FORTRAN language. The dimensionless temperature, velocity, and all boundary layer functions profiles were obtained and plotted in figures for different parameters entering into the problem. Several results of best approximations and expressions of important correlations relating to heat transfer rates were drawn in this study of which Prandtl’s number to the plate for physical interest was also discussed across the tables. The same case of solution procedure was made for a plane plate subjected to other thermal boundary conditions in a laminar flow. Finally, for the validation of the treated numerical model, the results obtained are in good agreement with those of the specialized literature, and comparison with available results in certain cases is excellent.

Closed form solutions of cross flows of Casson fluid over a stretching surface

Abstract: The boundary layer flow for the Casson fluid generated by a stretching sheet with cross flows is studied. In this paper we analyzed three different cases of cross flow with streamwise flow. The cross flow in the first case, is generated due to uniform transverse free stream while the stretching surface is at rest. In second case the cross flow is caused by the motion of stretching surface with uniform transverse velocity and the cross flow in third case deals with the transverse shearing motion of the stretching surface. In all the case, possible one parameter solutions appear namely material parameter. The similarity solution for the present model is obtained from the Weidman solution by using some proper transformation. The velocity profiles and the wall shearing parameters for all type of flows are displayed in graphical form. It is depicted that Casson fluid material parameter β causes reduction in stream wise flow, uniform transverse plate motion and transverse shearing motion. Moreover, it is found that the magnitude of longitudinal wall shear stress parameter is greater than or equal to transverse wall shear stress for β.

A self-similar solution for unsteady adiabatic and isothermal flows behind the shock wave in a non-ideal gas using Lie group analysis method with azimuthal or axial magnetic field in rotating medium

Abstract: The similarity solutions using Lie group analysis for shock wave propagation in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field in the case of isothermal and adiabatic flows are obtained. All possible cases of similarity solutions are discussed using the Lie group analysis for the isothermal and adiabatic flows. The arbitrary constants involved in the generators of local Lie groups bring various possible cases of solutions with exponential law and power law shock paths. Similarity solution for isothermal and adiabatic flows with power law shock path is discussed in detail. The density of ambient medium is taken to be constant. The axial and azimuthal fluid velocities and magnetic field in the ambient medium are assumed to be varying according to power law. The effect on shock wave strength and that on the flow variables due to variation of the Alfven Mach number, adiabatic index of the gas, non-idealness parameter, rotational parameter and initial magnetic field variation exponent are investigated. It is found that these parameters have decaying effects on shock wave. The obtained results in the case of isothermal and adiabatic flows are also compared with each other.

Homann stagnation-point flow impinging on a biaxially stretching surface

Abstract: The normal impingement of axisymmetric Homann stagnation-point flow on a surface executing perpendicular, planar, biaxial stretching is studied. The flow field generated is an exact solution of the steady, three-dimensional Navier–Stokes equations in the form of a similarity solution. It is shown that two sets of dual solutions exist, forming four different branches of steady solutions. For sufficiently small stretches (including compressions of the surface) the four branches exhibit a multi-branch spiralling behaviour in the surface shear stress parameter plane. The linear stability of the solutions are also examined, identifying only one stable solution for each set of parameters.

Revisiting the flat plate laminar boundary layer flow of viscoelastic FENE-P fluids

Abstract: The laminar flat plate boundary-layer flow solution for finitely extensible nonlinear elastic model with Peterlin's closure fluids, originally derived by Olagunju [D. O. Olagunju, Appl. Math. Comput. 173, 593–602 (2006)], is revisited by relaxing some of the assumptions related to the conformation tensor. The ensuing simplification through an order of magnitude analysis and the use of similarity-like variables allows for a semi-analytical approximate similarity solution to be obtained. The proposed solution is more accurate than the original solution, and it tends to self-similar behavior only in the limit of low elasticity. Additionally, we provide a more extensive set of results, including profiles of polymer conformation tensor components, laws of decay for peak stresses and their location, as well as the streamwise variations of boundary layer thickness, displacement and momentum thicknesses. We also provide asymptotic laws for these quantities under low elasticity flow conditions. Comparisons with results from the numerical solution of the full set of governing equations show the approximate similarity solution to be valid up to a high local Weissenberg number ( W i x) between 0.2 and 0.3, corresponding to a local Weissenberg number based on the boundary layer thickness of about 10, for a wide range of values of dumbbell extensibility and solvent viscosity ratio. Above this critical condition, the semi-analytical solution is unable to describe the complex variations of the conformation tensor within the boundary layer, but it still remains accurate in its description of the velocity profiles, friction coefficient, and the variations of displacement, momentum, and boundary layer thicknesses for Weissenberg numbers at least one order of magnitude higher (within 5% up to W i x ≈ 5).

A CFD Tutorial in Julia: Introduction to Compressible Laminar Boundary-Layer Flows

Abstract: A boundary-layer is a thin fluid layer near a solid surface, and viscous effects dominate it. The laminar boundary-layer calculations appear in many aerodynamics problems, including skin friction drag, flow separation, and aerodynamic heating. A student must understand the flow physics and the numerical implementation to conduct successful simulations in advanced undergraduate- and graduate-level fluid dynamics/aerodynamics courses. Numerical simulations require writing computer codes. Therefore, choosing a fast and user-friendly programming language is essential to reduce code development and simulation times. Julia is a new programming language that combines performance and productivity. The present study derived the compressible Blasius equations from Navier–Stokes equations and numerically solved the resulting equations using the Julia programming language. The fourth-order Runge–Kutta method is used for the numerical discretization, and Newton’s iteration method is employed to calculate the missing boundary condition. In addition, Burgers’, heat, and compressible Blasius equations are solved both in Julia and MATLAB. The runtime comparison showed that Julia with for loops is 2.5 to 120 times faster than MATLAB. We also released the Julia codes on our GitHub page to shorten the learning curve for interested readers.

On the starting vortex generated by a translating and rotating flat plate

Abstract: We consider the trailing-edge vortex produced in an inviscid fluid by the start-up motion of a two-dimensional flat plate. A general starting motion is studied that includes the initial angle-of-attack of the plate (which may be zero), individual time power laws for plate translational and rotational speeds and the pivot position for plate rotation. A vortex-sheet representation for a start-up separated flow at the trailing edge is developed whose time-wise evolution is described by a Birkhoff–Rott equation coupled to an appropriate Kutta condition. This description includes convection by the outer flow, rotation and vortex-image self-induction. It admits a power-law similarity solution for the (small-time) primitive vortex, leading to an equation set where each term carries its own time-wise power-law factor. A set of four general plate motions is defined. Dominant-balance analysis of this set leads to discovery of three distinct start-up vortex-structure types that form the basis for all vortex motion. The properties of each type are developed in detail for some special cases. Numerical and analytical solutions are described and transition between solution types is discussed. Singular and degenerate vortex behaviour is discovered which may be due to the absence of fluid viscosity. An interesting case is start-up motion with zero initial angle of attack coupled to power-law plate rotation for which time-series examples are given that can be compared to high Reynolds number viscous flows.

Numerical investigation of boundary layer flow past a thin heated needle immersed in hybrid nanofluid

Abstract: The heat transfer in flow past thin needle has applications in instruments like hot wire anemometers. The objective of this article is to study the influence of hybrid nanoparticles on heat transfer distribution of the boundary layer flow over a parabolically shaped thin hot needle. The Sakiadis and Blasius 2-D flow scenarios have been analyzed by implementing a mathematical model with the Navier–Stokes and the energy equations. The resulting equations are solved numerically by using a similarity solution technique. This technique results in a differential equation in terms of a single variable, representing the curves parallel to the needle surface. The results show that using distinct nanoparticles allows us to control the heat transfer rate apart from the physical parameters, such as, needle size or velocity ratio parameter. A comparative analysis of Nusselt number, frictional drag, temperature, and velocity profiles for Ag–water nanofluid, Ag–CuO/water hybrid nanofluid, and CuO–water nanofluid has been carried out for different flow conditions, including Sakiadis and Blasius flow. The addition of nanoparticles hikes the heat transfer rate by 27–28% in Blasius flow and 6–8% in Sakiadis flow.

Non-axisymmetric three-dimensional stagnation-point flow and heat transfer under a jet impingement boiling

Abstract: The non-axisymmetric three-dimensional flow and heat transfer in the stagnation-point region of a planar jet impingement boiling on a flat surface has been investigated by using similarity solution approach, considering additional diffusivity terms in momentum and energy equations as a result of bubble-induced mixing in flow. The free jet stream along z direction impinges on the surface and produces a flow with different velocity components. This situation may happen if the flow pattern on the plate is bounded from both sides in one of the directions, because of any physical limitation or due to conditions of the surface such as moving plates or stretching sheets with different values of stretching velocities in the x and y directions. The governing equations have been transformed into ordinary differential equations by introducing appropriate similarity variables, and an exact solution has been obtained for three-dimensional boiling problem for the first time. The similarity variables have been presented based on non-axisymmetric three-dimensional and additional diffusivity effects. The bubble-induced diffusion due to bubble formation, growth, departure and collapse causes an enhancement in heat transfer rate from the surface to the bulk flow. The total heat flux transferred from the surface to the flow has been estimated as summation of the single-phase heat transfer due to forced convection and the nucleate boiling heat flux due to bubble-induced diffusion. The effects of the velocity components ratio and the ratio between the maximum total diffusivity to the molecular diffusivity on the flow field and heat transfer characteristics have been obtained and discussed and illustrated graphically. A comparison of the predicted heat flux has been made with previously published experimental data. As expected, the average deviation values show relatively more accurate results for the three-dimensional simulation than the two-dimensional one because of being closer to the experimental conditions.

On Solving the Nonlinear Falkner–Skan Boundary-Value Problem: A Review

Abstract: This article is a review of ongoing research on analytical, numerical, and mixed methods for the solution of the third-order nonlinear Falkner–Skan boundary-value problem, which models the non-dimensional velocity distribution in the laminar boundary layer.

The extended Prandtl closure model applied to the two-dimensional turbulent classical far wake

Abstract: Prandtl’s mixing length closure model has been used extensively in turbulent wake flows. Although the simplicity of this model is advantageous, it contains mathematical and physical limitations. In particular, this model results in a poor estimation of the flow on the center-line and near the wake boundary. Prandtl constructed an improved model, which will be referred to as the extended mixing length model, in an attempt to address many of the limitations of the original model. In this work, the extended Prandtl model is considered. A similarity solution that leaves both the governing equation for the stream-wise mean velocity deficit and the conserved quantity invariant is obtained. The governing partial differential equation is reduced to an ordinary differential equation. The ordinary differential equation,which must be solved subject to appropriate boundary conditions and the conserved quantity, cannot be solved analytically and thus a double-shooting method is developed to obtain the stream-wise mean velocity deficit. A plot of the mean velocity deficit is then produced.

Stability of similarity solutions of viscous thread pinch-off

Abstract: In this paper we compute the linear stability of similarity solutions of the breakup of viscous liquid threads, in which the viscosity and inertia of the liquid are in balance with the surface tension. The stability of the similarity solution is determined using numerical continuation to find the dominant eigenvalues. The stability of the first two solutions (those with the largest minimum radius) is considered. We find that the first similarity solution, which is the one seen in experiments and simulations, is linearly stable with a complex nontrivial eigenvalue, which could explain the phenomenon of breakup producing sequences of small satellite droplets of decreasing radius near a main pinch-off point. The second solution is seen to be linearly unstable. These linear stability results compare favorably to numerical simulations for the stable similarity solution, while a profile starting near the unstable similarity solution is shown to very rapidly leave the linear regime.

Exact and Numerical Solution Using Lie Group Analysis for the Cylindrical Shock Waves in a Self-Gravitating Ideal Gas with Axial Magnetic Field

Abstract: In this paper, we seek the exact and numerical solutions using Lie group analysis for one dimensional unsteady adiabatic flow in a self-gravitating ideal gas behind a cylindrical shock wave with axial magnetic field. With the help of Lie group analysis, the one-dimensional optimal system of sub-algebra is obtained for the system of equations of motion. With the help of optimal classes of infinitesimal generators, we constructed the similarity variable and transformation of the flow variables, which convert the system of partial differential equations into system of ordinary differential equations. In three particular cases, we have derived a general framework to solve the fundamental equations and exact feasible solutions are obtained. In two cases, the similarity solutions with exponential law and power law shock path are discussed. The similarity solution is obtained using numerical method in the case of power law shock path. It is obtained that the increase in the values of magnetic field strength and adiabatic index have the decaying effect on the shock wave. Also, increase in the strength of shock wave is witnessed with the rise in the gravitation parameter value. The effects of variation of magnetic field strength, adiabatic exponent and gravitational parameter on the flow variables are analyzed graphically.

Flow Dynamics of the Homogeneous and Heterogeneous Reactions with an Internal Heat Generation and Thermal Radiation between Two Squeezing Plates

Abstract: This paper explores the time dependent squeezing flow of a viscous fluid between parallel plates with internal heat generation and homogeneous/heterogeneous reactions. The motive of the present effort is to upgrade the heat transformation rate for engineering and industrial purpose with the rate of chemical reaction. For this purpose the equations for the conservation of mass, momentum, energy and homogeneous/heterogeneous reactions are transformed to a system of coupled equations using the similarity transformation. According to HAM, with the proper starting assumptions and other factors, a similarity solution may be found. On the way to verifying the validity and correctness of HAM findings, we compare the HAM solution with numerical solver programme BVP4c to see whether it matches up. The results of a parametric inquiry are summarized and presented with the use of graphs.

A Length Scale Approach to the Highest Standing Water Wave

Abstract: The highest standing surface wave at infinite depth is a classical hydrodynamic problem, illuminated by Taylor's excellent experiments [G. I. Taylor, “An experimental study of standing waves,” Proc. R. Soc. London, Ser. A 218, 44–59 (1953)]. Based on length scale arguments, we present a compact analytical approach to the highest standing wave. Our physical postulate is that the highest deep-water wave has a single length scale, i.e., its wavelength. The single-scale postulate for standing periodic deep-water waves is confronted with two distinctly different cases where zero and two length scales are postulated as follows: (i) No physical length scale for an isolated rogue-wave peak at deep water suggests a similarity solution. (ii) Two length scales for the periodic peaked surface at constant depth suggest a one-parameter family of standing waves. Moreover, the two length scales are the wavelength and average fluid depth. The deep-water limit with its single-length scale postulate confirms Grant's theory [M. A. Grant, “Standing Stokes waves of maximum height,” J. Fluid Mech. 60, 593–604 (1973)], taking the highest standing wave as a state of zero kinetic energy. The reversible motion is irrotational according to Lord Kelvin's theorem. The acceleration field for the highest deep-water wave has a single Fourier component according to our single length scale postulate. The resulting free-surface shape follows from the exact nonlinear dynamic condition. Our analytical theory confirms the ratio 0.203 for maximal wave height to wavelength, found by Grant. We test its robustness by extending the theory to a moderate spatial quasi-periodicity. Appendix A provides a simple proof for the right-angle peak, representing a regular extremal point of a locally quadratic complex function. Appendix B presents a quadrupole solution for an isolated peak of stagnant deep-water rogue waves.

Performance of similarity explicit group iteration for solving 2D unsteady convection-diffusion equation

Abstract: In this paper, a similarity finite difference (SFD) solution is addressed for thetwo-dimensional (2D) parabolic partial differential equation (PDE), specifically on the unsteady convection-diffusion problem. Structuring the similarity transformation using wave variables, we reduce the parabolic PDE into elliptic PDE. The numerical solution of the corresponding similarity equation is obtained using a second-order central SFD discretization schemeto get the second-order SFD approximation equation. We propose a four-point similarity explicit group (4-point SEG) iterative methodasa numericalsolution of the large-scale and sparse linear systems derived from SFD discretization of 2D unsteady convection-diffusion equation (CDE). To showthe 4-point SEG iteration efficiency, two iterative methods, such as Jacobiand Gauss-Seidel (GS) iterations, are also considered. The numerical experiments are carried out using three different problems to illustrate our proposed iterative method's performance. Finally, the numerical results showed that our proposed iterative method is more efficient than the Jacobiand GS iterations in terms of iteration number and execution time.

Similarity Solutions to Nonlinear Diffusion/Harry Dym Fractional Equations

Abstract: By using scalar similarity transformation, nonlinear model of time-fractional diffusion/Harry Dym equation is transformed to corresponding ordinary fractional differential equations, from which a travelling-wave similarity solution of time-fractional Harry Dym equation is presented. Furthermore, numerical solutions of time-fractional diffusion equation are discussed. Again, through another similarity transformation, nonlinear model of space-fractional diffusion/Harry Dym equation is turned into corresponding ordinary differential equations, whose two similarity solutions are also worked out.