Showing papers on "Similarity solution published in 2022"

Heat and mass transfer analysis of radiative fluid flow under the influence of uniform horizontal magnetic field and thermophoretic particle deposition

Abstract: A magnetic field is frequently employed to stabilize the flow field in real-world applications of fluid mechanics difficulties. The current study examines the effect of thermophoretic particle deposition on liquid flow across a rotating disk. In this study, a horizontal uniform magnetic field is used to regularize the flow field formed by a rotating disk. The horizontal magnetic field that is applied is not the same as the external upright magnetic field. In addition, the energy equation is investigated when exposed to thermal radiation. Using the conventional von Kármán similarity transformations, it is shown that a horizontal magnetic field leads to a similarity system of equations. Using proper transformations, the modeling equations are translated into ordinary differential equations (ODEs). Later, using the shooting technique and the Runge–Kutta–Fehlberg's–fourth–fifth order approach (RKF-45), the obtained system is numerically solved. The obtained numerical findings are then graphically shown and discussed in depth. The results reveal that, the rise in values of magnetic parameter declines the both axial and radial velocity profiles. The rise in values of thermophoretic parameter and thermophoretic coefficient decays the mass transport.

Cattaneo–Christov heat flux model for nanofluid flow over a curved stretching sheet: An application of Stefan blowing

Abstract: The present work investigates the thermophoresis and Brownian motion effects in nanofluid flow over a curved stretching sheet (CSS). Also, the Cattaneo–Christov heat flux and Stefan blowing (SB) conditions are considered for studying heat and mass transport characteristics. The present work's novelty is associated with considerations of convective boundary and SB conditions in nanomaterial flow over a CSS. The coupled partial differential equations are changed to ordinary differential equations by employing suitable similarity variables, and the resultant model is numerically handled using Runge–Kutta–Fehlberg's fourth fifth‐order method with the shooting scheme. The stimulation of the involved parameters/numbers on the flow, mass, and heat fields is broadly deliberated using suitable graphs. The present analysis's significant relevant outcomes are that the inclination in thermophoresis and Brownian motion parameters increases the heat transfer. The inclined values of the Brownian motion parameter decay the mass transfer. Furthermore, the increased values of both Schmidt number and SB parameter drop the mass transport. The increased values of the Brownian motion parameter and Schmidt number decays the rate of mass transference.

Similarity solution for induced magnetic field boundary layer flow of metallic nanofluids via convectively inclined stationary/moving flat plate: Spectral relaxation computation

Abstract: The present article describes a mathematical model for incompressible free convection flow with convective heat transport from an inclined stationary/moving flat plate under impact of induced magnetic field (IMF). The current flow model is formulated to consider different water-based nanofluids with metallic nanoparticles (Copper and Silver). Effectively a nanoscale formulation with the Tiwari-Das model deployed to study material properties for specific nanoparticles and base fluid. A similarity solution is obtained for non-dimensional form via similarity transformation rendered from basic flow equations for which numerical simulations utilizing the Spectral Relaxation Method (SRM). SRM is a simple iteration scheme that does not require any evaluation of derivatives, perturbation and linearization for solving non-linear system of equations. Graphical results for linear velocity, IMF, temperature, skin friction and Nusselt number distributions are presented for the different metallic-aqueous nanofluid cases as well as stationary/moving flat plate. The results are verified for limiting cases by comparing with various investigators for the case of stationary as well as moving flat plate and found to be in excellent agreement. Furthermore, computed numerical results for skin friction and Nusselt number for different emerging parameters in case Cu and Ag nanofluids for moving plate which are tabulated and discussed in detail.

MHD radiative ohmic heating nanofluid flow of a stretching penetrable wedge: A numerical analysis

Abstract: The present numerical investigation has focused on the magnetohydrodynamics flow of a viscous nanofluid over a stretching wedge with the boundary convective conditions, thermal radiation, and ohmic heating. Buongiorno's two‐component nonhomogeneous nanoscale model was used and a dilute nanofluid with spherical type particles is considered. Similarity transformations are used to render the system of governing partial differential equations into a system of coupled similarity equations. The transformed equations are solved numerically with the BVP4C method. Validation of solutions with previous studies based on special cases of the general model is included. The salient features of fluid velocity profile, temperature as well as concentration profiles are discussed in a graphical manner for various values of selected governing factors. The skin friction coefficient, mass, and heat transfer rates are calculated and summarized. It is worthwhile noticing that the validation results exhibit an excellent agreement with already existing reports. The modeling of the present problem is useful in the thermal processing of sheet‐like substances that is a necessary operation in paper procurement, wire drawing, drawing of plastic films, polymeric sheets, and metal spinning.

Flow and heat transfer analysis of a special third grade fluid over a stretchable surface in a parallel free stream

Abstract: This paper aims to investigate a boundary layer flow with heat and mass transfer of a special third grade fluid over a stretchable surface in a parallel free stream. Using the similarity variables generated by the Lie group analysis, the governing nonlinear partial differential equations (PDEs) are transformed into a system of nonlinear ordinary differential equations (ODEs). The transformed equations are then solved numerically using the shooting technique. In addition, an attempt is made to carry the asymptotic solution behavior for large stretching and suction parameters, and the asymptotic results of skin friction coefficient are then compared with the direct numerical solutions, which shows a good agreement. It is observed that the similarity equations exhibit dual solutions in a certain range of shrinking strength. These two solution branches show different behavior on the skin friction coefficient, local Nusselt number, Sherwood number, velocity, and temperature profiles. Thus, emphasis has been given to carrying out a stability analysis to determine the physically reliable solution. The stability analysis shows that the upper branch solution is stable. It is observed that the suction parameter increases the range of dual solutions and the magnitude of the critical point from where the dual solutions bifurcate; however, the non-Newtonian parameter shows the opposite behavior. The momentum, thermal and concentration boundary layer thicknesses in the upper branch solution are lower than the lower branch solution.

Three-Dimensional Self-Similarity of Coalescing Viscous Drops in the Thin-Film Regime.

Abstract: Coalescence and breakup of drops are classic problems in fluid physics that often involve self-similarity and singularity formation. While the coalescence of suspended drops is axisymmetric, the coalescence of drops on a substrate is inherently three-dimensional. Yet, studies so far have only considered this problem in two dimensions. In this Letter, we use interferometry to reveal the three-dimensional shape of the interface as two drops coalescence on a substrate. We unify the known scaling laws in this problem within the thin-film approximation and find a three-dimensional self-similarity that enables us to describe the anisotropic shape of the dynamic interface with a universal curve.

Wall laminar nanofluid jet flow and heat transfer

Abstract: Purpose This work aims to investigates exact solutions of the classical Glauert’s laminar wall jet mass and heat transfer under wall suction, wall contraction or dilation, and two thermal transport boundary conditions; prescribed constant surface temperature and prescribed constant surface flux in nanofluidic environment. Design/methodology/approach The flow system arranged in terms of partial dif- ferential equations is non-dimensionalized with suitable dimensionless transformation variables, and this new set of equations is reduced into ordinary differential equations via a set of similarity transformations, where they are treated analytically for closed form solutions. Findings Exact solutions of nanofluid flow for velocity distributions, momentum flux, wall shear stress and heat transfer boundary layers for commonly studied nanoparticles; namely copper, alumina, silver, and titanium oxide are presented. The flow behavior of alumina and titanium oxide is identical, and a similar behavior is seen for copper and silver, making two pairs of identical traits. The mathematical expressions as well as visual analysis of wall shear drag and temperature gradient which are of practical interest are analyzed. It is shown that wall stretching or shrinking, wall transpiration and velocity slip together influences the jet flow mechanism and extends the original Glauert’s jet solutions. The exact solutions for the two temperature boundary layer conditions and temperature gradients are analyzed analytically. It is found that the effect of nanopar- ticles concentration on thermal boundary layer is intense, causing temperature uplift, whereas the wall transpiration causes a decrease in thermal layers. Originality/value The analysis carried out in nanofluid environment is genuinely new and unique, as our work generalizes the Glauert’s classical regular wall jet fluid problem.

Water surge impingement onto a vertical wall: A new self-similarity solution for impact pressure

Abstract: The impingement process of water surge onto a vertical wall and the impact pressure are studied analytically in this work. We propose a new initial-boundary value problem particularly for the fluid motion near the corner of the horizontal bed and the vertical wall. The explicit solutions of the velocity and the pressure fields are analytically obtained using the self-similarity method under some verifiable physical assumptions. The impact pressure is found to be proportional to the product of the squared incident surge front velocity and the density of water, with a constant coefficient of around 0.867. We compare the analytical solution of the impact pressure with some existing laboratory data. The analytical solution agrees with the median value of the stochastic data of impact pressure from laboratory experiments. Subsequently, the velocity and the pressure fields from the analytical model are compared to the numerical simulation results based on OpenFOAM. The comparisons validate the physical assumptions made in the analytical derivation, demonstrating fair consistency. The analytical model successfully describes the early stage of the contact process between the surge front and the wall and provides a theoretical basis for the physics of water surge impingement.

A numerical study of rotationally symmetric nanofluid flow over a permeable surface using Buongiorno model

Abstract: This note describes boundary layer development beneath a generalized vortex flow of nanofluid using two-phase Buongiorno model. Vortex motion is characterized by a prescribed tangential flow with velocity proportional to r m , where r is radial coordinate and m denotes the power-law index. Different from previously adopted practice, the diffusion coefficients are not assumed constant here. A similarity solution is proposed, which transforms the constitutive equations into a coupled differential system whose solution is evaluated numerically. Simulations are made by assuming a power-law surface temperature distribution. Two separate situations namely (i) rigid body rotation ( m = 1 ) and (ii) potential vortex ( m = − 1 ) are carefully assessed. Furthermore, subtle fluid dynamics entities such as resisting wall shear and heat transfer rate are deliberated. Computational results reveal that there is no noticeable change in nanofluid temperature when diffusion coefficients are varied. Though, marked variations in nanoparticle concentration profile are observed whenever diffusion coefficients are varied.

Similarity solution using group theoretic method for unsteady flow behind shock wave in a self-gravitating dusty gas

Abstract: In this article, the similarity solution using the Lie group theoretic method for unsteady isothermal and adiabatic flows behind cylindrical or spherical shock wave in a mixture of self-gravitating real gas and small solid particles is discussed. The Lie group theoretic method gives out different cases, i.e., exponential law and power-law shock paths. The similarity solution exists only when the shock path varies according to an exponential law. The dispersal of the flow variables with the variations of the solid particles’ mass concentration in the mixture kp, gravitational parameter g0, the ratio of the density of solid particles to the initial density of the gas μ1, non-idealness parameter b¯ and the geometry index ν are discussed graphically. It is found that an increase in the value of μ1 or g0 or ν leads to an increase in the shock strength, but the shock wave decay with an increase in b¯. A comparison between the solutions for cylindrical and spherical symmetry, and for adiabatic and isothermal flows is also made.

Flow Characteristics of Heat and Mass for Nanofluid under Different Operating Temperatures over Wedge and Plate

Abstract: Background and Purpose: Nanofluids are a new class of heat transfer fluids that are used for different heat transfer applications. The transport characteristics of these fluids not only depend upon flow conditions but also strongly depend on operating temperature. In respect of these facts, the properties of these fluids are modified to measure the temperature effects and used in the governing equations to see the heat and mass flow behavior. Design of Model: Consider the nanofluids which are synthesized by dispersing metallic oxides (SiO2, Al2O3), carbon nanostructures (PEG-TGr, PEG-GnP), and nanoparticles in deionized water (DIW), with (0.025–0.1%) particle concentration over (30–50 °C) temperature range. The thermophysical properties of these fluids are modeled theoretically with the help of experimental data as a function of a temperature and volume fraction. These models are further used in transport equations for fluid flow over both wedge and plate. To get the solution, the equations are simplified in the shape of ordinary differential equations by applying the boundary layer and similarity transformations and then solved by the RK method. Results: The solution of the governing equation is found in the form of velocity and temperature expressions for both geometries and displayed graphically for discussion. Moreover, momentum and thermal boundary layer thicknesses, displacement, momentum thicknesses, the coefficient of skin friction, and Nusselt number are calculated numerically in tabular form. Finding: The maximum reduction and enhancement in velocity and temperature profile is found in the case of flow over the plate as compared to the wedge. The boundary layer parameters are increased in the case of flow over the plate than the wedge.

Analysis of forced convection phenomenon in boundary layer flow of nanofluid over power-law rotating disk

Abstract: In this paper, the thermal analysis is performed on the three-dimensional flow of viscous nanofluid over a rotating disk. The disk rotation is assumed as of the form [Formula: see text] where [Formula: see text], thus, there are different azimuthal velocities of the disk at different radii, and it increases with the increase in [Formula: see text]. Also, the effect of nanoparticles on the thermal conductivity of fluid is investigated by thermophoresis and Brownian motion features. For the governing problem, suitable similarity transformations are employed that convert the produced partial differential equations to ordinary differential equations. The flow field and energy transport are presented graphically and discussed in detail for the torsional exponent [Formula: see text], [Formula: see text] and [Formula: see text], respectively. It is noted that the temperature and concentration profile rise for the thermophoresis parameter while the opposite trend showed for Brownian motion. Moreover, the power-law index parameter [Formula: see text] declines the flow field temperature and concentration profiles.

Magnetohydrodynamics stagnation point flow and heat transfer over a non‐isothermal permeable shrinking sheet with variable surface temperature

Abstract: The present study is intended to analyse the effect of heat and mass transfer on boundary layer stagnation point flow of a viscous fluid over a non‐isothermal shrinking sheet subject to transverse magnetic field and variable surface temperature. The medium of flow is considered to be porous. Further, the effect of variable surface temperature and concentration are also taken care of in the present analysis. The governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) by suitable similarity transformations. The numerical simulation is carried out using Runge–Kutta method of fourth order with shooting technique. The physical significance of pertinent parameters of the flow phenomenon is studied with the help of graphs and tables. One striking outcome of the present analysis is that the unstable velocity profiles with inflection points are marked due to power law variation of temperature and concentration of the linearly stretchable bounding surface leaving aside the smooth fall in temperature and concentration (span wise) across the flow field. Further, it is noted that increase in magnetic field intensity, suction and thermal as well as mass buoyancy parameters enhance the skin friction concomitantly favouring the effective momentum transport.

Lubricated axisymmetric gravity currents of power-law fluids

Abstract: The motion of glaciers over their bedrock or drops of fluid along a solid surface can become unstable when these substrates are lubricated. Previous studies modelled such systems as coupled gravity currents (GCs), consisting of one fluid that lubricates the flow of another fluid, and having two propagating fronts. When both fluids are Newtonian and discharged at constant flux, global similarity solutions were found. However, when the top fluid is strain-rate softening, experiments have shown that each fluid front evolved with a different exponent. Here, we explore theoretically and numerically such lubricated GCs consisting of axisymmetric spreading of a power-law fluid on top of a Newtonian fluid, where each fluid volume grows in time like $t^{\alpha }$ . We find that the structure imposed by the non-Newtonian flow precludes general self-similarity, unlike purely Newtonian GCs. Consequently, we identify outstripping solutions in which the inner fluid front outstrips the outer fluid front. Despite the absence of a general global similarity solution, we find similarity solutions in several asymptotic limits. These include the purely Newtonian limit for any $\alpha$ , the case of $\alpha =5$ for a general power-law fluid, asymptotic limits in the viscosity ratio, and in the vicinity of the fluid fronts. Many of our theoretical predictions are found to be consistent with recent laboratory experiments. Discrepancies suggest the presence of hydrofracturing or wall slip near the fronts, and potentially, a progressive significance of extensional stresses as front outstripping is approached.

An extension result on similarity solutions arising during mixed convection

Abstract: In this paper, we are interested in the boundary value problem involving a third order autonomous ordinary nonlinear differential equation. Its solutions are the similarity solutions of a problem of boundary-layer theory dealing with mixed convection phenomena in a porous medium. We confirm our results by numerical illustrations using a shooting algorithm of Mathematica.

Development of heat transfer calculation methods based on the similarity criterion of deposit formation with the electrochemical number

Abstract: A similarity criterion of deposit formation with the electrochemical number was developed. Experimental studies of the saline deposits effect on heat transfer were generalized, and a new criterion equation was obtained for conditions of the saline solution natural convection. The modernized criterion of sedimentation similarity served as the basis for heat transfer calculation during the deposits formation.

Numerical Solution of Mixed Convection MHD Viscous Fluid Flow on Lower Stagnation Point of a Sliced Magnetic Sphere

Abstract: This paper considers mathematical modeling on mixed convection MHD viscous fluid flow on the lower stagnation point of a magnetic sliced sphere. The study began with transforming the governing equations which are in dimensional partial differential equations to non-dimensional ordinary differential equations by using the similarity variable. The resulting similarity equations are then solved by the Keller-Box scheme. The characteristics and effects of the Prandtl number, the sliced angle, the magnetic parameter, and the mixed convection parameter are analyzed and discussed.

Nonlinear Stefan problem for one-phase generalized heat equation with heat flux and convective boundary condition

Abstract: Abstract In this article we consider a mathematical model of an initial stage of closure electrical contact that involves a metallic vaporization after instantaneous exploding of contact due to arc ignition with power P 0 on fixed face z = 0 and heat transfer in material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component with heat flux and convective boundary conditions prescribed at the known free boundary z = α(t) . The temperature field in the liquid region of such kind of material can be modelled by Stefan problem for the generalized heat equation. A similarity variable transformation is used to solve the problem, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation and we have to determine temperature solution for the liquid phase and location of melting isotherm. Existence and uniqueness of the solution is proved by using the fixed point Banach theorem. The solution for two cases of thermal coefficients, in particular, constant and linear thermal conductivity are represented, existence and uniqueness for each type of solution is proved. Mathematics Subject Classification (2010). 80A22, 80A05.

Explicit solution for non-classical one-phase Stefan problem with variable thermal coefficients and two different heat source terms

Abstract: A one-phase Stefan problem for a semi-infinite material is studied for special functional forms of the thermal conductivity and specific heat depending on the temperature of the phase-change material. Using the similarity transformation technique, an explicit solution for these situations are shown. The mathematical analysis is made for two different kinds of heat source terms, and the existence and uniqueness of the solutions are proved.

Mathematical modelling of boundary layer flow over a permeable and time-dependent shrinking sheet – A stability analysis

Abstract: Micropolar fluid is one type of non-Newtonian fluid which consists of non-deformable spherical particles that suspended in viscous medium. In this paper, the problem of two-dimensional boundary layer flow over a permeable shrinking sheet with time dependent velocity in strong concentration micropolar fluid is studied theoretically. The mathematical model is governed by continuity, momentum and microrotation equations. Similarity variables are introduced so that, after performing the similarity transformation on the governing equations, the resulting system of nonlinear ordinary differential equations is then numerically solved using the program bvp4c in Matlab software. The effects of the micropolar material parameter, the unsteadiness parameter, the shrinking parameter and the mass suction parameter to the skin friction coefficient, velocity profiles and microrotation profiles are investigated. It is found that triple solutions exist for some values of the parameters that were considered. Based on the stability analysis that was performed, it showed that only two branches of solutions are categorized as stable, whereas one solution branch is unstable.

Scaling properties of a class of interfacial singular equations

Abstract: This paper can be considered as an introductory review of scale invariance theories illustrated by the study of the equation ∂th=−∂x∂xh1−2ν+∂xxxh, where ν>1/2. The d−dimensionals version of this equation is proposed for ν≥1 to discuss the coarsening of growing interfaces that induce a mound-type structure without slope selection (Golubović, 1997). Firstly, the above equation is investigated in detail by using a dynamic scaling approach, thus allowing for obtaining a wide range of dynamic scaling functions (or pseudosimilarity solutions) which lend themselves to similarity properties. In addition, it is shown that these similarity solutions are spatial periodic solutions for any ν>1/2, confirming that the interfacial equation undergoes a perpetual coarsening process. The exponents β and α describing, respectively, the growth laws of the interfacial width and the mound lateral size are found to be exactly β=(1+ν)/4ν and α=1/4, for any ν>12. Our analytical contribution examines the scaling analysis in detail and exhibits the geometrical properties of the profile or scaling functions. Our finding coincides with the result previously presented by Golubović for 0<ν≤3/2.

Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficients

Abstract: In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component is considered. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modelled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature solution for two phases and free boundaries which describe the position of boiling and melting interfaces. Existence and uniqueness of the solution is provided by using the fixed point Banach theorem.

An Analytic Solution of the Thermal Boundary Layer at the Leading Edge of a Heated Semi-Infinite Flat Plate under Forced Uniform Flow

Abstract: The heated flat plate under uniform flow has been vastly studied, with the Blasius and Pohlhausen solutions developed over 100 years ago. These solutions are numerical in nature. Here, an analytic solution is found for the temperature and velocity profiles at the leading edge of a heated flat plate under forced uniform flow. By defining a similarity variable the governing equations are reduced to a dimensionless equation with an analytic solution at the leading edge. This report gives justification for the similarity variable via scaling analysis, details the process of converting to similarity form, and presents a similarity solution. The analytic fluid and thermal solutions are then checked against a numerical solution obtained via computational dynamics.