A Survey of Methods for Determining Similarity Transformations

1985 ◽  
pp. 35-61 ◽  
Author(s):  
R. Seshadri ◽  
T. Y. Na
Keyword(s):  
Similarity Transformations

scholarly journals Numerical simulation for the steady nanofluid boundary layer flow over a moving plate with suction and heat generation

2021 ◽  
Vol 3 (2) ◽  
Author(s):  
M. Ferdows ◽  
MD. Shamshuddin ◽  
S. O. Salawu ◽  
K. Zaimi
Keyword(s):  
Heat Generation ◽  
Volume Fraction ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Moving Plate ◽  
Flow Equations ◽  
Similarity Transformations ◽  
Layer Flow ◽  
Matlab Package ◽  
Steady Laminar

AbstractIn the study, the steady, laminar, incompressible, convective flow of a viscous fluid over a moving plate is investigated theoretically by adopting different types of nanoparticles. Radiation, internal heat generation and viscous dissipation effects are considered in the energy modeled equation. The governing flow equations for the momentum and temperature are reduced to dimensionless form via similarity transformations. The solutions to the resultant equations alongside with the transformed boundary conditions are numerically obtained using MATLAB package bvp4c. Validation with earlier studies are done for the non-internal heat generation case for two distinct nanoparticles of type Cu-water and Al-water. Extensive visualization of flow rate and heat distributions for various emerging parameters are examined. Temperature is consistently enhanced with a rising Eckert number of both types of nanofluids, whereas it is strongly reduced with rising values of radiation term. Heat transfer coefficient is consistently increased with a nanoparticle volume fraction of high convective heat in the medium.

Dissipative effects on a chemically and thermally radiative heat fluid flow past a shrinking porous sheet

2020 ◽  
pp. 1-14
Author(s):  
A. Shahid ◽  
M. Ali Abbas ◽  
H.L. Huang ◽  
S.R. Mishra ◽  
M.M. Bhatti
Keyword(s):  
Fluid Flow ◽  
Differential Equations ◽  
Thermal Radiation ◽  
Chemical Reaction ◽  
Linearization Method ◽  
Surface Drag ◽  
Suction Parameter ◽  
Similarity Transformations ◽  
Dissipative Effects ◽  
Successive Linearization Method

The present study analyses the dissipative influence into an unsteady electrically conducting fluid flow embedded in a pervious medium over a shrinkable sheet. The behavior of thermal radiation and chemical reactions are also contemplated. The governing partial differential equations are reformed to ordinary differential equations by operating similarity transformations. The numerical outcomes for the arising non-linear boundary value problem are determined by implementing the Successive linearization method (SLM) via Matlab software. The velocity, temperature, and concentration magnitudes for distant values of the governing parametric quantities are conferred, and their conduct is debated via graphical curves. The surface drag coefficient increases, whereas the local Nusselt number and Sherwood number decreases for enhancing unsteadiness parameter across suction parameter. Moreover, the magnetic and suction parameters accelerate velocity magnitudes while by raising porosity parameter, velocity decelerates. Larger numeric of thermal radiation parameter and Eckert number accelerates the temperature profile while by enhancing Prandtl number it decelerates. Schmidt number and chemical reaction parameters slowdowns the concentration distribution, and the chemical reaction parameter influences on the point of chemical reaction that benefits the interface mass transfer. It is expected that the current achieved results will furnish fruitful knowledge in industrious utilities.

scholarly journals Numerical study of bio-convection flow of magneto-cross nanofluid containing gyrotactic microorganisms with activation energy

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Qiu-Hong Shi ◽  
Aamir Hamid ◽  
M. Ijaz Khan ◽  
R. Naveen Kumar ◽  
R. J. Punith Gowda ◽  
...  
Keyword(s):  
Radiation Effects ◽  
Numerical Study ◽  
Source Parameters ◽  
Lewis Number ◽  
Convective Boundary Condition ◽  
Convection Flow ◽  
Convective Boundary ◽  
Similarity Transformations ◽  
Cross Nanoliquid ◽  
Source Sink

AbstractIn this study, a mathematical model is developed to scrutinize the transient magnetic flow of Cross nanoliquid past a stretching sheet with thermal radiation effects. Binary chemical reactions and heat source/sink effects along with convective boundary condition are also taken into the consideration. Appropriate similarity transformations are utilized to transform partial differential equations (PDE’s) into ordinary ones and then numerically tackled by shooting method. The impacts of different emerging parameters on the thermal, concentration, velocity, and micro-rotation profiles are incorporated and discussed in detail by means of graphs. Results reveal that, the escalation in magnetic parameter and Rayleigh number slowdowns the velocity and momentum of the fluid. The increase in Biot number, radiation and heat sink/source parameters upsurges the thermal boundary but, converse trend is seen for escalating Prandtl number. The density number of motile microorganisms acts as a growing function of bioconvection Lewis number and declining function of bioconvection Peclet number.

The effects of Coriolis force and radial stretch on the convective instability characteristics of the Bödewadt flow

2021 ◽  
pp. 095440622110071
Author(s):  
Dip Mukherjee ◽  
Bikash Sahoo
Keyword(s):  
Boundary Layer ◽  
Coriolis Force ◽  
Boundary Layer Flow ◽  
Convective Instability ◽  
Type I ◽  
Similarity Transformations ◽  
Instability Modes ◽  
Neutral Curves ◽  
Lower Disk ◽  
Layer Flow

The Bödewadt boundary-layer flow is induced by the rotation of a viscous fluid rotating with a constant angular velocity over a stationary disk. In this paper, the Bödewadt boundary-layer flow has been studied in the presence of the Coriolis force to observe the effect of radial stretch of the lower disk on the flow. For the first time in the literature, a numerical investigation of the effects of both stretching mechanism and the Coriolis force on the flow behaviour and on the convective instability characteristics of the above flow has been carried out. In this paper, the Kármán similarity transformations have been considered in order to convert the system of PDEs representing the momentum equations of the flow into a system of highly non-linear coupled ODEs and solved numerically to obtain the velocity profiles of the Bödewadt flow. Then, a convective instability analysis has been performed by using the Chebyshev collocation method in order to obtain the neutral curves. From the neutral curves it is observed that radial stretch has a globally stabilising effect on both the inviscid Type-I and the viscous Type-II instability modes. This underlying physical phenomena has been verified by performing an energy analysis of the flow. The results obtained excellently supports the previous works and will be prominently treated as a benchmark for our future studies.

Magneto convective flow of casson nanofluid due to Stefan blowing in the presence of bio-active mixers

2021 ◽  
pp. 239779142110166
Author(s):  
Venkatesh Puneeth ◽  
Sarpabhushana Manjunatha ◽  
Bijjanal Jayanna Gireesha ◽  
Rama Subba Reddy Gorla
Keyword(s):  
Magnetic Field ◽  
Brownian Motion ◽  
Convective Flow ◽  
Three Dimensional ◽  
Induced Magnetic Field ◽  
Density Profiles ◽  
Casson Nanofluid ◽  
And Brownian Motion ◽  
Similarity Transformations ◽  
The Mathematical Model

The induced magnetic field for three-dimensional bio-convective flow of Casson nanofluid containing gyrotactic microorganisms along a vertical stretching sheet is investigated. The movement of these microorganisms cause bioconvection and they act as bio-active mixers that help in stabilising the nanoparticles in the suspension. The two forces, Thermophoresis and Brownian motion are incorporated in the Mathematical model along with Stefan blowing. The resulting model is transformed to ordinary differential equations using similarity transformations and are solved using [Formula: see text] method. The Velocity, Induced Magnetic field, Temperature, Concentration of Nanoparticles, and Motile density profiles are interpreted graphically. It is observed that the Casson parameter decreases the flow velocity and enhances the temperature, concentration, and motile density profiles and also it is noticed that the blowing enhances the nanofluid profiles whereas, suction diminishes the nanofluid profiles. On the other hand, it is perceived that the rate of heat conduction is enhanced with Thermophoresis and Brownian motion.

Analysis of flow and heat transfer over stretching/shrinking and porous surfaces

2021 ◽  
pp. 875608792110258
Author(s):  
Azhar Ali ◽  
Dil Nawaz Khan Marwat ◽  
Aamir Ali
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Current Model ◽  
Uniform Heat Flux ◽  
Similarity Transformations ◽  
Flow And Heat Transfer ◽  
Special Cases ◽  
Porous Surfaces

Flows and heat transfer over stretching/shrinking and porous surfaces are studied in this paper. Unusual and generalized similarity transformations are used for simplifying governing equations. Current model includes all previous cases of stretched/shrunk flows with thermal effects discussed so far. Moreover, we present three different cases of thermal behavior (i) prescribed surface temperature (ii) Variable/uniform convective heat transfer at plat surface and (iii) prescribed variable/uniform heat flux. Stretching/shrinking velocity Uw(x), porosity [Formula: see text], heat transfer [Formula: see text], heat flux [Formula: see text] and convective heat transfer at surface are axial coordinate dependent. Boundary layer equations and boundary conditions are transformed into nonlinear ODEs by introducing unusual and generalized similarity transformations for the variables. These simplified equations are solved numerically. Final ODEs represent suction/injection, stretching/shrinking, temperature, heat flux, convection effects and specific heat. This current problem encompasses all previous models as special cases which come under the scope of above statement (title). The results of classical models are scoped out as a special case by assigning proper values to the parameters. Numerical result shows that the dual solutions can be found for different possible values of the shrinking parameter. A stability analysis is accomplished and apprehended in order to establish a criterion for determining linearly stable and physically compatible solutions. The significant features and diversity of the modeled equations are scrutinized by recovering the previous problems of fluid flow and heat transfer from a uniformly heated sheet of variable (uniform) thickness with variable (uniform) stretching/shrinking and injection/suction velocities.

scholarly journals Numerical simulation of variable thermal conductivity on 3D flow of nanofluid over a stretching sheet

2020 ◽  
Vol 9 (1) ◽  
pp. 233-243 ◽  
Author(s):  
Nainaru Tarakaramu ◽  
P.V. Satya Narayana ◽  
Bhumarapu Venkateswarlu
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Stretching Sheet ◽  
Three Dimensional ◽  
Variable Thermal Conductivity ◽  
Normal Velocity ◽  
Transfer Coefficients ◽  
Similarity Transformations ◽  
Frictional Coefficients ◽  
The Impact

AbstractThe present investigation deals with the steady three-dimensional flow and heat transfer of nanofluids due to stretching sheet in the presence of magnetic field and heat source. Three types of water based nanoparticles namely, copper (Cu), aluminium oxide (Al2O3), and titanium dioxide (TiO2) are considered in this study. The temperature dependent variable thermal conductivity and thermal radiation has been introduced in the energy equation. Using suitable similarity transformations the dimensional non-linear expressions are converted into dimensionless system and are then solved numerically by Runge-Kutta-Fehlberg scheme along with well-known shooting technique. The impact of various flow parameters on axial and transverse velocities, temperature, surface frictional coefficients and rate of heat transfer coefficients are visualized both in qualitative and quantitative manners in the vicinity of stretching sheet. The results reviled that the temperature and velocity of the fluid rise with increasing values of variable thermal conductivity parameter. Also, the temperature and normal velocity of the fluid in case of Cu-water nanoparticles is more than that of Al2O3- water nanofluid. On the other hand, the axial velocity of the fluid in case of Al2O3- water nanofluid is more than that of TiO2nanoparticles. In addition, the current outcomes are matched with the previously published consequences and initiate to be a good contract as a limiting sense.

scholarly journals Non-Hermitian Hamiltonians and Similarity Transformations

2015 ◽  
Vol 55 (2) ◽  
pp. 843-850 ◽  
Author(s):  
Francisco M. Fernández
Keyword(s):  
Similarity Transformations

Falconer's formula for the Hausdorff dimension of a self-affine set in R2

1995 ◽  
Vol 15 (1) ◽  
pp. 77-97 ◽  
Author(s):  
Irene Hueter ◽  
Steven P. Lalley
Keyword(s):  
Hausdorff Dimension ◽  
Dimensional Hausdorff Measure ◽  
Similarity Transformations ◽  
Affine Mappings ◽  
Finite Set ◽  
Self Similar ◽  
Image Position ◽  
Affine Set ◽  
Small Overlap ◽  
Box Dimensions

Let A1, A2,…,Ak be a finite set of contractive, affine, invertible self-mappings of R2. A compact subset Λ of R2 is said to be self-affine with affinitiesA1, A2,…,Ak ifIt is known [8] that for any such set of contractive affine mappings there is a unique (compact) SA set with these affinities. When the affine mappings A1, A2,…,Ak are similarity transformations, the set Λ is said to be self-similar. Self-similar sets are well understood, at least when the images Ai(Λ) have ‘small’ overlap: there is a simple and explicit formula for the Hausdorff and box dimensions [12, 10]; these are always equal; and the δ-dimensional Hausdorff measure of such a set (where δ is the Hausdorff dimension) is always positive and finite.

scholarly journals Transformation of Uncertain Linear Systems with Real Eigenvalues into Cooperative Form: The Case of Constant and Time-Varying Bounded Parameters

Algorithms ◽  
2021 ◽  
Vol 14 (3) ◽  
pp. 85
Author(s):  
Andreas Rauh ◽  
Julia Kersten
Keyword(s):  
Linear Systems ◽  
Initial Conditions ◽  
A Priori ◽  
Real Life ◽  
Reachability Analysis ◽  
Interval Methods ◽  
Space Representation ◽  
Uncertain Parameters ◽  
Time Varying ◽  
Similarity Transformations

Continuous-time linear systems with uncertain parameters are widely used for modeling real-life processes. The uncertain parameters, contained in the system and input matrices, can be constant or time-varying. In the latter case, they may represent state dependencies of these matrices. Assuming bounded uncertainties, interval methods become applicable for a verified reachability analysis, for feasibility analysis of feedback controllers, or for the design of robust set-valued state estimators. The evaluation of these system models becomes computationally efficient after a transformation into a cooperative state-space representation, where the dynamics satisfy certain monotonicity properties with respect to the initial conditions. To obtain such representations, similarity transformations are required which are not trivial to find for sufficiently wide a-priori bounds of the uncertain parameters. This paper deals with the derivation and algorithmic comparison of two different transformation techniques for which their applicability to processes with constant and time-varying parameters has to be distinguished. An interval-based reachability analysis of the states of a simple electric step-down converter concludes this paper.

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