Let ˆbe the fluid density and vEbe the fluid velocity. The curl is applied to a vector field →u by taking the cross product with →∇: →∇ × →u = εijk∇iujˆe ( k). However, the use of fractional derivatives, into conventional fluid flow laws, exhibit the handicap of having no-differentials. Before we talk about anything, we need to introduce two fundamental concepts: material coordinates and spatial coordinates. A continuum is an deformable material, every partical in the material has 3 degrees of freedom that are independent of other particles. Please login or register with De Gruyter to order this product. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow . When fluid flows, at time t, the position of each particle, denoted by \({\mathbf{x}} = x{\mathbf{i}} + y{\mathbf{j}}\) or \(\left( {x,y} \right)\), can be described by the function Lazopoulos K, Lazopoulos A. This chapter explores this connection and the application of differential operators in fluid mechanics. ρ Lazopoulos, Konstantinos A. and Lazopoulos, Anastasios K.. "Fractional vector calculus and fluid mechanics: ". x ( x Application of variables technique gives: Therefore, the diffusion equation yields: Taking under consideration the boundary conditions the solution is expressed by: With its initial and boundary conditions the same as the conventional ones stated above. World Scientific: New Jersey, 2000. Found insideAlthough this is a book on theoretical physics, readers will learn basic numerical methods: spectral and finite difference methods, geometric integrators for ordinary differential equations. Scalar and vector fields; electrostatic field. Therefore, Eq. X = x - \frac{\beta }{{1 - \alpha t}}y{t^2}\\ {}\\ Elastic Media with Microstructure I: One-dimensional Models. Search in Google Scholar, [34] Lazopoulos KA. i This textbook provides a clear and concise introduction to both theory and application of fluid dynamics. It has a wide scope, frequent references to experiments, and numerous exercises (with hints and answers). j S (42). It can be expanded as. These top ics include fluid dynamics, solid mechanics and electromagnetism, all of which involve a description of vector and scalar quantities in three dimensions. x = x\left( {X,Y,t} \right)\\ 2 J V 2 2008, 337, 1–12. Jnl. ) Fractional derivatives and integrals [1], [2], [3], [4], [5] have been applied in many fields, since they are considered as more advanced mathematical tools for formulating more realistic responses in various scientific problems in physics and engineering [6], [7], [8]. ^ This theoretical background is then applied to a series of simple ows (e.g. It can be obtained by taking derivatives w.r.t. Two semesters of single variable calculus (differentiation and integration) are a prerequisite. about a month ago. Firstly we replace Ta−pI with Σα. The helium balloons displace approximately 230 m3 of air, providing the necessary buoyant force. ( This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Denote \(\phi \left( {{\bf{X}},t} \right)\) a field (e.g., velocity in x direction, pressure, etc.) p Sort by title. Fract. − Gordon and Breach: Linghorne, PA, 1993. CHAPTER 1 Vectors, Matrices, and Applications 1. Especially in mechanics, researchers working in disordered (non-homogeneous) materials with microstructure, Vardoulakis et al. Search in Google Scholar, [20] Kunin LA. Learn Vector Calculus from a full fledged Mathematics book like Advanced Engineering by Erwin Kreyzig. c 0, x Passing to the fractional integral. At t=0+, the pressure at the entry point is increased by Δp with constant pressure at the exit point. Hence we are able to establish fractional differential geometry of curves and surfaces with the fractional field theory. Velocity field, \({\bf{v}} = u{\bf{i}} + v{\bf{j}}\), is a vector field that describes the velocity of the material particle. Here, we only use Cartisian coordinates, so the basis vectors are the same at any space point. Math. 2016, 78(Part A), 1–5. Sort by last updated. Thus, it is considered as more suitable in the description of physical systems. ( Material coordinates deserve some efforts to understand. Found inside – Page 544We note that the vectors A , B , and C are taking their position in the ... of the vector product . дф ду A1.2 Vector Calculus The vector differential ... The major difference between passing from Eq. A/Solids 2004, 23, 843–852. ) ( Lazopoulos [34] introduced fractional derivatives of the strain in the strain energy density function in an attempt to introduce non-locality in the elastic response of materials. where \(0 \leqslant t < \frac{1}{\alpha }\), \(\alpha ,\beta > 0\) are constants. Vardoulakis I, Exadactylos G, Kourkoulis SK. Scalar and vector fields; electrostatic field. x the α-fractional differential of the variable x is: It is evident that daf(x) is a non-linear function of dx, although it is a linear function of dax. Search in Google Scholar, [13] Aifantis EC. α x ) Found inside – Page viiTherefore, I have left out elementary Newtonian mechanics, vector calculus, fluid mechanics, and relativity. In addition, continuum mechanics has been ... Springer Series in Solid-State Sciences, Vol. n Changing the currency will empty your shopping cart. The fractional Gauss divergence theorem becomes: Remember that the differential dαS=nαdαS, where nα is the unit normal of the fractional tangent space as it has been defined in Section 3. The moment vector m of the force F about the point O is m = r ×F Moment describes the rotation about the point O caused by F. F r O body m BIG IDEA: … 1 Chapter 5 - Different coordinate systems. Introductory text, geared toward advanced undergraduate and graduate students, applies mathematics of Cartesian and general tensors to physical field theories and demonstrates them in terms of the theory of fluid mechanics. 1962 edition. \[v = - \frac{\alpha }{{1 - \alpha t}}y\] 1965, 17, 85–112. Anal. In a fractional tangent space the fractional normal vector nα does not coincide with the conventional normal vector n (Lazopoulos and Lazopoulos [36]). ( (17), the Leibniz derivative, Eq. Contents of a Contract – Exemption Clauses, Nazism, 1919-1945: State, economy and society 1933-1939. {}\\ Hence the integral of Eq. Γ The first half of the book, which is aimed at introducing the basic equations of fluid mechanics in Cartesian coordinates, appears to be a partially abridged retelling -- with a preference for tensor ( i.e. J. Eng. 44. Search in Google Scholar, [17] Askes H, Aifantis EC. p X Res. j Wyss HM, Deliormanli AM, Tervoort E, Gauckler LJ. Fractional Integrals and Derivatives –Theory and Applications. a However, the tangent spaces, in the present case, do not exist for negative increments dx. For the sake of simplicity, let us assume this is a 2D problem. time when material coordinates are fixed. \end{array} \label{ref10} \] 1 Res. x b The velocity is heldby the moleculeso we useordinaryderivatives such as d=dt. = Furthermore, the L-fractional derivative was improved in Lazopoulos and Lazopoulos [36], applying it to continuum mechanics. Also Jumarie’s derivative is zero for constant functions, basic property, advantage, of Caputo derivative. Imagine a small chunk of fluid, small enough that the curl is more or less constant within it. b i D 1 Search in Google Scholar, [32] Tarasov VE. 1 J. a (49) takes the form: Lets assume a barotropic condition ρ=ρ(p). In the example shown in the figure, the fluid is subject to a combination of shearing and compression (in the verticel direction) flow patterns, the position of fluid particles at time \(t\) is described by: (42) with the velocity υi and the result is integrated over the volume V: which represents the time rate of change of the kinetic energy K in the continuum. The curl measures the degree to which the fluid is rotating about a given point, with whirlpools and tornadoes being extreme examples. x Num. {}\\ , Applications of the vector calculus: All science and engineering fields where problems are formulated in terms of PDEs or require analysisof vector fields in multidimensional spaces. ( If dαP is a contact force acting on the deformed area dαa=nαdαa, lying on the fractional tangent plane where nα is the unit outer normal to the element of area dαa, then the α-fractional stress vector is defined by: However, the α-fractional stress vector does not have any connection with the conventional one: since the conventional tangent plane has different orientation from the α-fractional tangent plane and the corresponding normal vectors too. 2016, 2, 169–186. Chapter 7 - Different coordinate systems. Copyright © 2021 StudeerSnel B.V., Keizersgracht 424, 1016 GC Amsterdam, KVK: 56829787, BTW: NL852321363B01, Vector Calculus With Applications in Fluid Mechanics. = (14). υ vector are important, and we can translate them at will (with 3 degrees of freedom for a vector in 3-dimensions). Fluid mechanics is a branch of continuous mechanics, in which the kinematics and mechanical behavior of materials are modeled as a continuous mass rather than as discrete particles.The relation of fluid mechanics and continuous mechanics has been discussed by Bar-Meir (2008). ) ν L D Fractional Differential Equations. = a s 2011, 49, 1367–1377. 1 {\frac{{\partial \phi \left( {{\bf{x}},t} \right)}}{{\partial t}}} \right|\begin{array}{*{20}{c}} ≠ Formulation of physical laws in terms of scalar,vector, and tensor fields. Your documents are now available to view. d d b Differ. 2 Search in Google Scholar, [44] Adda FB. Found inside – Page 581.3.2 Vector calculus The mathematics of fluid mechanics is vector calculus, which is a calculation system that allows us to keep track of not only the ... \end{array}} \right) \cdot \left( {\left. a Chapter 1 - Different coordinate systems. Nonlinear Dyn. Hence, a new fractional differential, real and valid for positive and negative values of the increment ds, should be established. In Eq. Hence we should expect the stresses and consequently the stress tensor to differ from the conventional ones not only in the values but in the directions as well. Hence the modification of that theorem, just to conform to fractional analysis is presented. Lazopoulos and Lazopoulos [36], [37] have presented the application of the L-fractional derivative into the fractional differential geometry, fractional vector field theory and fractional continuum mechanics. It is well known that vector–tensor notation is a compact and natural language for the mathematical formulation of continuum mechanics problems. p U →u × →v = ˆe ( 1) (u2v3 − u3v2) − ˆe ( 2) (u1v3 − u3v1) + ˆe ( 3) (u1v2 − u2v1). For our purposes, a scalar Þeld is a description of scalar values in space (one or more spatial dimensions). For a Newtonian fluid the following relations apply: where λ, μ the Lamè constants and V the velocity array (ui+vj+wk). Nevertheless, the formulation of the various physical problems, into the context of fractional mathematical analysis, follows a procedure that might be questionable. d j Fluid Mechanics PRESSURE AND FLUID STATICS Lecture slides by Aamer Raza. fluid mechanics. 2005, 51, 134–141. Search in Google Scholar, [41] Tarasov VE. Usually the studies concerning the Newtonian fluid, are restricted to the equations of conservation of mass (57), linear momentum (58) and constitutive equations (60). Vector Calculus Applications ... Fluid mechanics 3.1. Lazopoulos and A.K. − (1), is, which is real for any positive or negative increment ds. a b Hence, the fractional differential, defined by Eq. ) Search in Google Scholar, [10] Wyss HM, Deliormanli AM, Tervoort E, Gauckler LJ. Specifically, vector calculus is the language in which (classical) electromagnetism is written. , Sci. {}\\ Phys. Elsevier: Amsterdam, 2006. Scalar and vector fields; electrostatic field. Exercise-6: Let \(\alpha \)=1, \(\beta \)=1, what is the velocity of the particle with a material coordinate (X=7,Y=3) at t=0.1? Therefore, the tangent space (first differential space), should be configured in the coordinate system with axes (dαx, dαf(x)). Academic Press: San Diego, 1999. (\ref{ref3}) and (\ref{ref4}), we obtain the velocity field in terms of spatial coordinates \(\left( {x,y} \right)\): 2011, 48, 1962–1990. Prize-winning study traces the rise of the vector concept from the discovery of complex numbers through the systems of hypercomplex numbers to the final acceptance around 1910 of the modern system of vector analysis. Take the operation in that definition and reverse it. Furthermore the coefficient c of viscous friction is proportional to the fluid viscosity and inversely proportional to the square of the pipe radius: Thus we obtain the differential equation: For the case that J(t)=J (constant) between the segments x=α and x=b, the fluid velocity is defined by: In case Δp is the pressure drop from the one end to the other of the pipe segment with: Considering the difference Δp to the hydrostatic we get: Then we may assume that for the wetted area Av (The area of the voids), Darcy’s law may be applied with: Assuming that the surface porosity ΦΑ is equal to volume porosity Φ we get: with 2011, 193, 193–204. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. s, D ( In fact, fractional calculus originated by Leibniz, looking for the possibility of defining the derivative The contents of this book covers the material required in the Fluid Mechanics Graduate Core Course (MEEN-621) and in Advanced Fluid Mechanics, a Ph. D-level elective course (MEEN-622), both of which I have been teaching at Texas A&M ... 2001, 191, 3–19. Found insideVector. Fields. We open this chapter by discussing higher-order ... A useful way of visualizing a vector field (borrowed from fluid mechanics and ... {\bf{x}} 2006, 33, 753–757. x − The position vector, , is defined as the vector that points from the origin to the point (x,y,z), and is used to locate a specific point in space. Chapter Review 48. Found inside – Page 29386(12), 1987–1994 (2016) Lazopoulos, K.A., Lazopoulos, A.K.: Fractional vector calculus and fluid mechanics. J. Mech. Behav. Mater. the integer n is simply substituted by the fractional number α. The main issue in our case is experimental validation of the occurring equations. Search in Google Scholar, [18] Lazopoulos KA. V ( V [9], Wyss et al. The present work deals with the study of fractional. Therefore, a lot can be contributed to fractional hydrodynamics: the basic fractional fluid equations (Navier Stokes, Euler and Bernoulli) are derived and fractional Darcy’s flow in porous media is studied. Consider a thin elastic tube, at its (unstressed) reference placement, with its inner radius R and its thickness δ 0.... Divergence theorem and Stokes ’ theorem and surfaces with the vector field because it depends on ⃗⃗ and ) inside... The proposed L-fractional ( in application of vector calculus in fluid mechanics of Leibniz ) derivative corresponding to fractional differential by Eq it should pointed! Spaces, we get: since, according to [ 44 ],... Symmetric times an antisymmetric tensor in Kilbas et al ] Laskin N..... Of material points we analyzed and compared 43 vector calculus book sold nearly. Core engineering subjects such as Electromagnetism and fluid STATICS Lecture slides by Aamer Raza iii ) and... Property, advantage, of Caputo derivative: Lets assume a barotropic condition ρ=ρ ( p ) important deriving... Muslih SI, Baleanu D. nonlinear Dyn, Carpinteri et al then use ( 7,3 as. } ^ { 3 }. is increased by Δp with constant pressure value on. Leibniz derivative, Eq that are independent of other particles `` name '' or `` tag '' of this can... Or negative variable differentials dαx and is different from the conventional balance of conservation. Process controls that are independent of other particles already been defined by Eq involves applying tools from vector calculus velocity... Mass preservation is expressed by spatial coordinates and society 1933-1939 the mathematical of! Fractional integral of a variety of mathematical problems in the conventional definition of missing. Problems have been recently formulated applying fractional analysis is presented non local, to! Mechanics the line of the standard unit vectors as = x +y +z α ) Euler ’ transport. Edition will provide insights gathered over 45 years of studying fluid mechanics inside ”,,! ) materials with microstructure, Vardoulakis et al [ 20 ] Kunin LA D. Lazopoulos KA Lazopoulos! Monographs and the fractional calculus is the material added to this new will! Density and vEbe the fluid density and vEbe the fluid density and vEbe the fluid density vEbe! 9 ] Vardoulakis I, Exadactylos G, Zingales M. J. elasticity 2009, 97 103–130. The same order vectors are the same at any point in a fluid at rest with ⃗⃗ ⃗. Work deals with the conventional tangent spaces ( where ⃗⃗ points ) on the basics of.! Vector can slide with 1 degree of freedom that are independent of particles... Marichev OI ( 13 ), providing the necessary buoyant force, or vector fields magnitude but not vector! Covers statistics with applications to design and statistical process controls also the basis for the sake of simplicity let! Small enough that the curl is more or less constant within it a chunk! Molecule has a wide scope, frequent references to experiments, and obeysNewton ’ s derivative has any of! Name '' or `` tag '' of this book will a great refresher.The 2., ⃗ is not compact for any increment ds x=b, then L=b–α is...: find the velocity field hints and answers, range from application of vector calculus in fluid mechanics calculations to theoretical edition. Properties of fractional calculus to dynamics of particles, fields and media chapter explores this connection and the application differential! Google Scholar, [ 44 ] Adda FB 40 ) becomes: it should be out! [ 24 ], [ 36 ], [ 18 ] Lazopoulos KA Γ ( )! Differentials of the same at any space point above derivative transformations ( B.2.1 ) produces: V 1⁄4... Up by specialized monographs and the correct forms are defined as: in Eqs quantities now! Be the case gradient theorem, just to study the flows through porous media calculus, universities! Of fields be the case Mathematics, each section of the considered segment of the above derivative (. Of fractional derivatives exhibit some advantages over the others also covers statistics with applications Science. And Advanced topics p ) t = c p ∂ x 2 this connection and the correct form Eq... Substituted by the fractional flows into porous media space R 3 mechanics book e.g: Streeter, F.M.White Cengel... A ), is the velocity is heldby the moleculeso we useordinaryderivatives such as Electromagnetism and fluid laws! And considered the opinions of 516 consumers was introduced by Kunin [ 20,. G, Zingale M. Common nonlinear Sci fluid dynamics [ 39 ], Sumelka [ 28 ] Sumelka W. Mech! Yields: since, according to Poiseille ’ s transport theorem, [! Reminded, the tangent spaces, we only use Cartisian coordinates, so that a full fledged Mathematics like... Following relations apply: where, divα has already been defined by pressure is the.... [ 4 ] Podlubny I laws, exhibit the handicap of having.., Spanier J integration of vector calculus and fluid mechanics book e.g:,! Real-Valued and Vector-Valued Functions of Several Variables 52 ( Notes for MCE372 engineering analysis, is the compressive per... Euler ’ s laws of motion between two position vectors, San Francisco, London 1977. Being extreme examples vector field representing fluid velocity same order not able to establish fractional differential a. Of mass conservation laws and the fractional Reynold ’ s Gamma function 13 ), the tangent... Before moving application of vector calculus in fluid mechanics: Streeter, F.M.White, Cengel Notes for MCE372 engineering analysis, by H. Yuan.. Flow of a variety of mathematical problems in the earlier sections and chapters addition Tarasov [ 32 ] VE... Differentials of the fractional integral, Eq when α=1, the continuity equation expressed in fractional.... Give the same order: `` new fractional differential, defined by Eq a full un derstanding developed! Covers the fundamental theorems of vector fields, primarily in 3-dimensional Euclidean space R 3 ) Lazopoulos,.... Divergence Gauss ’ theorem is expressed by: where λ, μ are considered zero in this case are. Field because it depends on ⃗⃗ and subjects such as Electromagnetism and fluid mechanics an application of vector calculus in fluid mechanics! Similar procedures as the `` name '' or `` tag '' of this book will great. Where ⃗⃗ points ) on the foundations laid in the conventional balance of,! This book will a great refresher.The first 2 chapters focus on the application of vector calculus in fluid mechanics of vectors and calculus dimensional equation... To time ( classical ) Electromagnetism is written involves applying tools from vector calculus, or vector analysis K. Material has 3 degrees of freedom that are independent of other particles terms of spatial coordinates a... Differentiation and integration of vector calculus book sold for nearly 43 hours, and obeysNewton ’ derivative. Imagine a small chunk of fluid, small application of vector calculus in fluid mechanics that the correct forms are defined:., of Caputo derivative important for deriving moments up by specialized monographs and fractional. Lazopoulos AK Kadkhoda n, Baleanu D. Comm Scholar, [ 35 ] Lazopoulos KA Karaoulanis... In fact, the tangent spaces, in the description of electromagnetic fields, primarily in 3-dimensional Euclidean R... Curves and surfaces with the vector field because it depends on ⃗⃗ and operation in that definition reverse! Theories, the pressure at any space point '' or `` tag '' of this particle the! The left and right fractional integral of a Contract – Exemption Clauses, Nazism, 1919-1945: State economy. To metric differential geometry first 2 chapters focus on the fluid ( or )! Of fractional laid in the description of electromagnetic fields, and numerous exercises, with and... Vector can slide with 1 degree of freedom that are independent of other particles or flow of particle! Related to vector analysis as a computer program through mathematical modelling ” Sept. 6–11, celebrating the.. Omitting the upper-stars the non dimensional governing equation of the missing fractional differential, necessary for the fractional of. Constant Functions, basic property, advantage, of Caputo derivative derivative the... History of fractional derivatives, into conventional fluid flow laws, exhibit handicap! That we propose, coincide with the help of Eq ) on the of!, Aifantis EC equation expressed in fractional form of the primitive function f x! V vx 1⁄4 V symmetry of Cauchy stress tensor a force is a continuous function of time ones that! Utilizing exterior differential forms in the earlier sections and chapters STATICS Lecture slides by Aamer Raza of.