DOCSLIB.ORG

Useful Identities and Theorems from Vector Calculus

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Useful Identities and Theorems from Vector Calculus Appendix A Useful Identities and Theorems from Vector Calculus A.1 Vector Identities A · (B × C) = C · (A × B) = B · (C × A) A × (B × C) = B(A · C) − C(A · B) (A × B) × C = B(A · C) − A(B · C) ∇×∇f = 0 ∇·(∇×A) = 0 ∇·( f A) = (∇ f ) · A + f (∇·A) ∇×( f A) = (∇ f ) × A + f (∇×A) ∇·(A × B) = B · (∇×A) − A · (∇×B) ∇(A · B) = (B ·∇)A + (A ·∇)B + B × (∇×A) + A × (∇×B) ∇·(AB) = (A ·∇)B + B(∇·A) ∇×(A × B) = (B ·∇)A − (A ·∇)B − B(∇·A) + A(∇·B) ∇×(∇×A) =∇(∇·A) −∇2 A A.2 The Gradient Theorem For two points a, b in a space where a scalar function f with spatial derivatives everywhere well-defined up to first order, b (∇ f ) · d = f (b) − f (a), a independently of the integration path between a and b. P. Charbonneau, Solar and Stellar Dynamos, Saas-Fee Advanced Course 39, 215 DOI: 10.1007/978-3-642-32093-4, © Springer-Verlag Berlin Heidelberg 2013 216 Appendix A: Useful Identities and Theorems from Vector Calculus A.3 The Divergence Theorem For any vector field A with spatial derivatives of all its scalar components everywhere well-defined up to first order, (∇·A)dV = A · nˆ dS , V S where the surface S encloses the volume V . A.4 Stokes’ Theorem For any vector field A with spatial derivatives of all its scalar components everywhere well-defined up to first order, (∇×A) · nˆ dS = A · d , S γ where the contour γ delimits the surface S, and the orientation of the unit nor- mal vector nˆ and direction of contour integration are mutually linked by the right-hand rule. A.5 Green’s Identities For any two scalar functions φ and ψ defined over a volume V bounded by a surface S and with spatial derivatives well-defined up to second order, (φ∇2ψ +∇φ ·∇ψ)dV = φ(∇ψ) · nˆ dS , V S and (φ∇2ψ − ψ∇2φ)dV = (φ∇ψ − ψ∇φ) · nˆ dS . V S These are known respectively as Green’s first and second identities, the latter often simply referred to as Green’s theorem. Appendix B Coordinate Systems and the Fluid Equations This Appendix is adapted in part from Appendix B of the book by Jean-Louis Tassoul (1978) given in the bibliography to this appendix with a number of addi- tions, including the MHD induction equation, expressions for the operators u ·∇ and ∇2 acting on a vector field, and for the divergence of a second rank tensor. Note also, in Sects.B.1.5 and B.2.5, the quantities in square brackets correspond to the components of the deformation tensor D jk = (1/2)(∂ j uk + ∂ku j ). B.1 Cylindrical Coordinates (s, φ, z) Fig. B.1 Geometric defin- ition of cylindrical coordi- nates. The coordinate ranges are s ∈[0, ∞], φ ∈[0, 2π], z ∈[−∞, ∞]. The cylindrical radius s is measured perpen- dicularly from the cartesian z-axis. The zero point of the azimuthal angle φ is on the cartesian x-axis. The local unit vector triad is oriented such that eˆz = eˆs × eˆφ. P. Charbonneau, Solar and Stellar Dynamos, Saas-Fee Advanced Course 39, 217 DOI: 10.1007/978-3-642-32093-4, © Springer-Verlag Berlin Heidelberg 2013 218 Appendix B: Coordinate Systems and the Fluid Equations B.1.1 Conversion to Cartesian Coordinates x = s cos φ, y = s sin φ, s = x2 + y2, φ = atan(y/x), z = z. eˆx = cos φ eˆs − sin φ eˆφ, eˆy = sin φ eˆs + cos φ eˆφ, eˆs = cos φ eˆx + sin φ eˆy, eˆφ =−sin φ eˆx + cos φ eˆy, eˆz = eˆz . B.1.2 Infinitesimals d = dseˆs + sdφ eˆφ + dz eˆz dV = s ds dφ dz B.1.3 Vector Operators D ∂ ∂ uφ ∂ ∂ = + u + + u Dt ∂t s ∂s s ∂φ z ∂z ∂ f 1 ∂ f ∂ f ∇ f = eˆ + eˆφ + eˆ ∂ s ∂φ ∂ z s s z uφ Aφ uφ As (u ·∇)A = u ·∇A − eˆ + u ·∇Aφ + eˆφ + (u ·∇A ) eˆ s s s s z z 1 ∂ 1 ∂ Aφ ∂ A ∇·A = (sA ) + + z ∂ s ∂φ ∂ s s s z 1 ∂ A ∂ Aφ ∇×A = z − eˆ ∂φ ∂ s s z ∂ As ∂ Az 1 ∂(sAφ) ∂ As + − eˆφ + − eˆ ∂z ∂s s ∂s ∂φ z 1 ∂ ∂ 1 ∂2 ∂2 ∇2 = s + + ∂ ∂ 2 ∂φ2 ∂ 2 s s s s z A 2 ∂ Aφ ∇2 A = ∇2 A − s − eˆ s 2 2 ∂φ s s s ∂ 2 Aφ 2 As 2 + ∇ Aφ − + eˆφ + ∇ A eˆ s2 s2 ∂φ z z Appendix B: Coordinate Systems and the Fluid Equations 219 B.1.4 The Divergence of a Second-Order Tensor 1 ∂(sT ) 1 ∂Tφ ∂T Tφφ [∇ · T] = ss + s + zs − s s ∂s s ∂φ ∂z s 1 ∂(sTsφ) 1 ∂Tφφ ∂Tzφ Tφs [∇ · T]φ = + + + s ∂s s ∂φ ∂z s 1 ∂(sT ) 1 ∂Tφ ∂T [∇ · T] = sz + z + zz z s ∂s s ∂φ ∂z B.1.5 Components of the Viscous Stress Tensor ∂us 2 τss = 2μ + ζ − μ ∇·u ∂s 3 1 ∂uφ us 2 τφφ = 2μ + + ζ − μ ∇·u s ∂φ s 3 ∂uz 2 τzz = 2μ + ζ − μ ∇·u ∂z 3 1 1 ∂us ∂ uφ τ φ = τφ = 2μ + s s s 2 s ∂φ ∂s s 1 ∂uφ 1 ∂uz τφ = τ φ = 2μ + z z 2 ∂z s ∂φ 1 ∂uz ∂us τzs = τsz = 2μ + 2 ∂s ∂z B.1.6 Equations of Motion u2 ∂Φ ∂ ∂ ∂ Dus − φ =− − p + Bz Bs − Bz Dt s ∂s ∂s μ0 ∂z ∂s Bφ ∂(sBφ) ∂ Bs 1 ∂ 1 ∂τsφ ∂τsz τφφ − − + (sτss) + + − μ0s ∂s ∂φ s ∂s s ∂φ ∂z s Duφ uφu ∂Φ ∂ ∂(sBφ) ∂ + s =− − 1 p + Bs − Bs Dt s s ∂φ s ∂φ μ0s ∂s ∂φ Bz 1 ∂ Bz ∂ Bφ 1 ∂ 1 ∂τφφ ∂τφz τsφ − − + (sτφs) + + + μ0 s ∂φ ∂z s ∂s s ∂φ ∂z s 220 Appendix B: Coordinate Systems and the Fluid Equations ∂Φ ∂ Bφ ∂ ∂ Bφ Duz =− − p + 1 Bz − Dt ∂z ∂z μ0 s ∂φ ∂z Bs ∂ Bs ∂ Bz 1 ∂ 1 ∂τzφ ∂τzz − − + (sτzs) + + μ0 ∂z ∂s s ∂s s ∂φ ∂z B.1.7 The Energy Equation Ds 1 ∂ ∂T 1 ∂ ∂T ∂ ∂T T = Φν + Φη + χs + χ + χ Dt s ∂s ∂s s2 ∂φ ∂φ ∂z ∂z 2 2 2 2 2 2 2 2 Φν = 2μ(D + Dφφ + D + 2D φ + 2Dφ + 2D ) + (ζ − μ)(∇·u) ss zz s z zs 3 2 2 2 η 1 ∂ Bz ∂ Bφ ∂ Bs ∂ Bz 1 ∂(sBφ) ∂ Bs Φη = − + − + − 2 μ0 s ∂φ ∂z ∂z ∂s s ∂s ∂φ B.1.8 The MHD Induction Equation ∂ Bs 1 ∂ ∂ = u Bφ − uφ B − (u B − u B ) ∂t s ∂φ s s ∂z z s s z ∂η ∂(sBφ) ∂ ∂η ∂ ∂ − 1 − Bs + Bs − Bz s2 ∂φ ∂s ∂φ ∂z ∂z ∂s B 2 ∂ Bφ + η ∇2 B − s − s s2 s2 ∂φ ∂ Bφ ∂ ∂ = uφ B − u Bφ − u Bφ − uφ B ∂t ∂z z z ∂s s s ∂η ∂ ∂ Bφ ∂η ∂(sBφ) ∂ − 1 Bz − + 1 − Bs ∂z s ∂φ ∂z s ∂s ∂s ∂φ ∂ 2 Bφ 2 Bs + η ∇ Bφ − + s2 s2 ∂φ ∂ Bz 1 ∂ 1 ∂ = (su B − su B ) − uφ B − u Bφ ∂t s ∂s z s s z s ∂φ z z ∂η ∂ B ∂ B 1 ∂η 1 ∂ B ∂ Bφ − s − z + z − + η ∇2 B ∂s ∂z ∂s s ∂φ s ∂φ ∂z z Appendix B: Coordinate Systems and the Fluid Equations 221 B.2 Spherical Coordinates (r, θ, φ) Fig. B.2 Geometric defini- tion of polar spherical coordi- nates. The coordinate ranges are r ∈[0, ∞], θ ∈[0, π], φ ∈[0, 2π]. The zero point of the azimuthal angle φ is on the cartesian x-axis and the zero point of the polar angle θ (sometimes called colatitude) is on the cartesian z-axis. Note that in so-called geographical coordinates, longitude ≡ φ, but latitude ≡ π/2 − θ.The local unit vector triad is ori- ented such that eˆr = eˆθ × eˆφ. B.2.1 Conversion to Cartesian Coordinates x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. r = x2 + y2 + z2, θ = atan x2 + y2/z , φ = atan(y/x). eˆx = sin θ cos φ eˆr + cos θ cos φ eˆθ − sin φ eˆφ, eˆy = sin θ sin φ eˆr + cos θ sin φ eˆθ + cos φ eˆφ, eˆz = cos θ eˆr − sin θ eˆθ. eˆr = sin θ cos φ eˆx + sin θ sin φ eˆy + cos θ eˆz, eˆθ = cos θ cos φ eˆx + cos θ sin φ eˆy − sin θ eˆz, eˆφ =−sin φ eˆx + cos φ eˆy. B.2.2 Infinitesimals d = dreˆr + rdθeˆθ + r sin θdφeˆφ dV = r 2 sin θ dr dθ dφ 222 Appendix B: Coordinate Systems and the Fluid Equations B.2.3 Vector Operators D ∂ ∂ uθ ∂ uφ ∂ = + ur + + Dt ∂t ∂r r ∂θ r sin θ ∂φ ∂ f 1 ∂ f 1 ∂ f ∇ f = eˆ + eˆθ + eˆφ ∂ r ∂θ θ ∂φ r r r sin uθ Aθ uφ Aφ (u ·∇)A = u ·∇Ar − − eˆr r r uφ Aφ uθ Ar + u ·∇Aθ − cot θ + eˆθ r r uφ Ar uφ Aθ + u ·∇Aφ + + cot θ eˆφ r r ∂ ∂( θ) ∂ 1 2 1 Aθ sin 1 Aφ ∇·A = (r Ar ) + + r 2 ∂r r sin θ ∂θ r sin θ ∂φ 1 ∂(Aφ sin θ) ∂ Aθ ∇×A = − eˆ θ ∂θ ∂φ r r sin 1 ∂ Ar ∂(Aφr sin θ) 1 ∂(rAθ) ∂ Ar + − eˆθ + − eˆφ r sin θ ∂φ ∂r r ∂r ∂θ 1 ∂ ∂ 1 ∂ ∂ 1 ∂2 ∇2 = r 2 + sin θ + r 2 ∂r ∂r r 2 sin θ ∂θ ∂θ r 2 sin2 θ ∂φ2 2A 2 ∂ Aθ sin θ 2 ∂ Aφ ∇2 A = ∇2 A − r − − eˆ r 2 2 θ ∂θ 2 θ ∂φ r r r sin r sin ∂ θ ∂ 2 2 Ar Aθ 2 cos Aφ + ∇ Aθ + − − eˆθ 2 ∂θ 2 2 θ 2 2 θ ∂φ r r sin r sin ∂ θ ∂ 2 2 Ar 2 cos Aθ Aφ + ∇ Aφ + + − eˆφ r 2 sin θ ∂φ r 2 sin2 θ ∂φ r 2 sin2 θ B.2.4 The Divergence of a Second-Order Tensor 2 1 ∂(r Trr) 1 ∂(Tθr sin θ) 1 ∂Tφr Tθθ + Tφφ [∇ · T]r = + + − r 2 ∂r r sin θ ∂θ r sin θ ∂φ r 2 1 ∂(r Trθ) 1 ∂(Tθθ sin θ) 1 ∂Tφθ Tθr Tφφ cot θ [∇ · T]θ = + + + − r 2 ∂r r sin θ ∂θ r sin θ ∂φ r r 2 1 ∂(r Trφ) 1 ∂(Tθφ sin θ) 1 ∂Tφφ Tφr Tφθ cot θ [∇ · T]φ = + + + + r 2 ∂r r sin θ ∂θ r sin θ ∂φ r r Appendix B: Coordinate Systems and the Fluid Equations 223 B.2.5 Components of the Viscous Stress Tensor ∂u 2 τ = μ r + ζ − μ ∇·u rr 2 ∂ r 3 1 ∂uθ ur 2 τθθ = μ + + ζ − μ ∇·u 2 ∂θ r r 3 1 ∂uφ ur uθ cot θ 2 τφφ = μ + + + ζ − μ ∇·u 2 θ ∂φ r sin r r 3 1 1 ∂ur ∂ uθ τ θ = τθ = μ + r r r 2 ∂θ ∂ 2 r r r ∂ θ θ ∂ uφ τ = τ = μ 1 1 u + sin θφ φθ 2 θ ∂φ ∂θ θ 2 r sin r sin 1 ∂ uφ 1 ∂ur τφ = τ φ = 2μ r + r r 2 ∂r r r sin θ ∂φ B.2.6 Equations of Motion u2 + u2 ∂Φ ∂ Dur − θ φ =− − p Dt r ∂r ∂r Bφ ∂ Br ∂ Bθ ∂(rBθ) ∂ Br + − Bφr θ − − μ θ ∂φ ∂ sin μ ∂ ∂θ 0r sin r 0r r θ ∂ ∂ ∂τ τ + τ 1 sin 2 rφ θθ φφ + (r τrr) + (τ θ sin θ) + − r sin θ r ∂r ∂θ r ∂φ r u2 cot θ ∂Φ ∂ Duθ + ur uθ − φ =− − 1 p Dt r r r ∂θ r ∂θ ∂( θ) ∂ Bφ ∂(Bφ sin θ) ∂ θ + Br rB − Br − − B μ ∂ ∂θ μ θ ∂θ ∂φ 0r r 0r sin θ ∂ ∂ ∂τ τ τ θ 1 sin 2 θφ rθ φφ cot + (r τθ ) + (τθθ sin θ) + + − r sin θ r ∂r r ∂θ ∂φ r r Duφ u uφ uθuφ cot θ ∂Φ ∂ + r + =− − 1 p θ ∂φ θ ∂φ Dt r r r sin rsin θ ∂(Bφ sin θ) ∂ θ ∂ ∂(Bφr sin θ) + B − B − Br Br − μ θ ∂θ ∂φ μ θ ∂φ ∂ 0r sin 0r sin r θ ∂ ∂ ∂τ τ τ θ 1 sin 2 φφ rφ θφ cot + (r τφ ) + (τφθ sin θ) + + + r sin θ r ∂r r ∂θ ∂φ r r 224 Appendix B: Coordinate Systems and the Fluid Equations B.2.7 The Energy Equation ∂ ∂ Ds 1 2 T T = Φν + Φη + χr 2 ∂ ∂ Dt r r r 1 ∂ ∂T 1 ∂ ∂T + χ sin θ + χ r 2 sin θ ∂θ ∂θ r 2 sin2 θ ∂φ ∂φ 2 2 2 2 2 2 2 2 Φν = 2μ(D + Dθθ + Dφφ + 2D θ + 2Dθφ + 2Dφ ) + ζ − μ (∇·u) rr r r 3 2 η ∂(Bφ sin θ) ∂ Bθ Φη = − 2 2 ∂θ ∂φ μ0r sin θ 2 2 ∂ B ∂(Bφr sin θ) ∂(rBθ) ∂ B + r − + sin2 θ − r ∂φ ∂r ∂r ∂θ B.2.8 The MHD Induction Equation ∂ Br 1 ∂ ∂ = ( θ (u Bθ − uθ B )) − uφ B − u Bφ ∂ θ ∂θ sin r r ∂φ r r t r sin ∂η ∂( θ) ∂ ∂η ∂ ∂(Bφr sin θ) − 1 rB − Br + 1 Br − 2 ∂θ ∂ ∂θ 2 2 θ ∂φ ∂φ ∂ r r r sin r ∂( θ) ∂ 2 2Br 2 Bθ sin 2 Bφ + η ∇ Br − − − r 2 r 2 sin θ ∂θ r 2 sin θ ∂φ ∂ Bθ 1 ∂ 1 ∂ = uθ Bφ − uφ Bθ − (ru Bθ − ruθ B ) ∂ θ ∂φ ∂ r r t r sin r r ∂η ∂(Bφ sin θ) ∂ θ ∂η ∂( θ) ∂ − 1 − B + 1 rB − Br 2 2 θ ∂φ ∂θ ∂φ ∂ ∂ ∂θ r sin r r r ∂ θ ∂ 2 2 Br Bθ 2 cos Bφ + η ∇ Bθ + − − r 2 ∂θ r 2 sin2 θ r 2 sin2 θ ∂φ ∂ Bφ 1 ∂ ∂ = ruφ B − ru Bφ − uθ Bφ − uφ Bθ ∂ ∂ r r ∂θ t r r ∂η ∂ ∂(Bφr sin θ) ∂η ∂(Bφ sin θ) ∂ θ − 1 Br − + 1 − B θ ∂ ∂φ ∂ 2 θ ∂θ ∂θ ∂φ r sin r r r sin ∂ θ ∂ 2 2 Br 2 cos Bθ Bφ + η ∇ Bφ + + − r 2 sin θ ∂φ r 2 sin2 θ ∂φ r 2 sin2 θ Appendix B: Coordinate Systems and the Fluid Equations 225 Bibliography For more on all this stuff, see Morse, P.
Load more

Recommended publications
  • General Meteorology
    Dynamic Meteorology 2 Lecture 9 Sahraei Physics Department Razi University http://www.razi.ac.ir/sahraei Stream Function Incompressible fluid uv 0 xy u ; v Definition of Stream Function y x Substituting these in the irrotationality condition, we have vu 0 xy 22 0 xy22 Velocity Potential Irrotational flow vu Since the, V 0 V 0 0 the flow field is irrotational. xy u ; v Definition of Velocity Potential x y Velocity potential is a powerful tool in analysing irrotational flows. Continuity Equation 22 uv 2 0 0 0 xy xy22 As with stream functions we can have lines along which potential is constant. These are called Equipotential Lines of the flow. Thus along a potential line c Flow along a line l Consider a fluid particle moving along a line l . dx For each small displacement dl dy dl idxˆˆ jdy Where iˆ and ˆj are unit vectors in the x and y directions, respectively. Since dl is parallel to V , then the cross product must be zero. V iuˆˆ jv V dl iuˆ ˆjv idx ˆ ˆjdy udy vdx kˆ 0 dx dy uv Stream Function dx dy l Since must be satisfied along a line , such a line is called a uv dl streamline or flow line. A mathematical construct called a stream function can describe flow associated with these lines. The Stream Function xy, is defined as the function which is constant along a streamline, much as a potential function is constant along an equipotential line. Since xy, is constant along a flow line, then for any , d dx dy 0 along the streamline xy Stream Functions d dx dy 0 dx dy xy xy dx dy vdx ud y uv uv , from which we can see that yx So that if one can find the stream function, one can get the discharge by differentiation.
    [Show full text]
  • Potential Flow Theory
    2.016 Hydrodynamics Reading #4 2.016 Hydrodynamics Prof. A.H. Techet Potential Flow Theory “When a flow is both frictionless and irrotational, pleasant things happen.” –F.M. White, Fluid Mechanics 4th ed. We can treat external flows around bodies as invicid (i.e. frictionless) and irrotational (i.e. the fluid particles are not rotating). This is because the viscous effects are limited to a thin layer next to the body called the boundary layer. In graduate classes like 2.25, you’ll learn how to solve for the invicid flow and then correct this within the boundary layer by considering viscosity. For now, let’s just learn how to solve for the invicid flow. We can define a potential function,!(x, z,t) , as a continuous function that satisfies the basic laws of fluid mechanics: conservation of mass and momentum, assuming incompressible, inviscid and irrotational flow. There is a vector identity (prove it for yourself!) that states for any scalar, ", " # "$ = 0 By definition, for irrotational flow, r ! " #V = 0 Therefore ! r V = "# ! where ! = !(x, y, z,t) is the velocity potential function. Such that the components of velocity in Cartesian coordinates, as functions of space and time, are ! "! "! "! u = , v = and w = (4.1) dx dy dz version 1.0 updated 9/22/2005 -1- ©2005 A. Techet 2.016 Hydrodynamics Reading #4 Laplace Equation The velocity must still satisfy the conservation of mass equation. We can substitute in the relationship between potential and velocity and arrive at the Laplace Equation, which we will revisit in our discussion on linear waves.
    [Show full text]
  • Stream Function for Incompressible 2D Fluid
    26/3/2020 Navier–Stokes equations - Wikipedia Continuity equation for incompressible fluid Regardless of the flow assumptions, a statement of the conservation of mass is generally necessary. This is achieved through the mass continuity equation, given in its most general form as: or, using the substantive derivative: For incompressible fluid, density along the line of flow remains constant over time, therefore divergence of velocity is null all the time Stream function for incompressible 2D fluid Taking the curl of the incompressible Navier–Stokes equation results in the elimination of pressure. This is especially easy to see if 2D Cartesian flow is assumed (like in the degenerate 3D case with uz = 0 and no dependence of anything on z), where the equations reduce to: Differentiating the first with respect to y, the second with respect to x and subtracting the resulting equations will eliminate pressure and any conservative force. For incompressible flow, defining the stream function ψ through results in mass continuity being unconditionally satisfied (given the stream function is continuous), and then incompressible Newtonian 2D momentum and mass conservation condense into one equation: 4 μ where ∇ is the 2D biharmonic operator and ν is the kinematic viscosity, ν = ρ. We can also express this compactly using the Jacobian determinant: https://en.wikipedia.org/wiki/Navier–Stokes_equations 1/2 26/3/2020 Navier–Stokes equations - Wikipedia This single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter. Note that the equation for creeping flow results when the left side is assumed zero. In axisymmetric flow another stream function formulation, called the Stokes stream function, can be used to describe the velocity components of an incompressible flow with one scalar function.
    [Show full text]
  • A Dual-Potential Formulation of the Navier-Stokes Equations Steven Gerard Gegg Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1989 A dual-potential formulation of the Navier-Stokes equations Steven Gerard Gegg Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Aerospace Engineering Commons, and the Mechanical Engineering Commons Recommended Citation Gegg, Steven Gerard, "A dual-potential formulation of the Navier-Stokes equations " (1989). Retrospective Theses and Dissertations. 9040. https://lib.dr.iastate.edu/rtd/9040 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS The most advanced technology has been used to photo­ graph and reproduce this manuscript from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are re­ produced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps.
    [Show full text]
  • Div-Curl Problems and H1-Regular Stream Functions in 3D Lipschitz Domains
    Received 21 May 2020; Revised 01 February 2021; Accepted: 00 Month 0000 DOI: xxx/xxxx PREPRINT Div-Curl Problems and H1-regular Stream Functions in 3D Lipschitz Domains Matthias Kirchhart1 | Erick Schulz2 1Applied and Computational Mathematics, RWTH Aachen, Germany Abstract 2Seminar in Applied Mathematics, We consider the problem of recovering the divergence-free velocity field U Ë L2(Ω) ETH Zürich, Switzerland of a given vorticity F = curl U on a bounded Lipschitz domain Ω Ï R3. To that end, Correspondence we solve the “div-curl problem” for a given F Ë H*1(Ω). The solution is expressed in Matthias Kirchhart, Email: 1 [email protected] terms of a vector potential (or stream function) A Ë H (Ω) such that U = curl A. After discussing existence and uniqueness of solutions and associated vector potentials, Present Address Applied and Computational Mathematics we propose a well-posed construction for the stream function. A numerical method RWTH Aachen based on this construction is presented, and experiments confirm that the resulting Schinkelstraße 2 52062 Aachen approximations display higher regularity than those of another common approach. Germany KEYWORDS: div-curl system, stream function, vector potential, regularity, vorticity 1 INTRODUCTION Let Ω Ï R3 be a bounded Lipschitz domain. Given a vorticity field F .x/ Ë R3 defined over Ω, we are interested in solving the problem of velocity recovery: T curl U = F in Ω. (1) div U = 0 This problem naturally arises in fluid mechanics when studying the vorticity formulation of the incompressible Navier–Stokes equations. Vortex methods, for example, are based on the vorticity formulation and require a solution of problem (1) at every time-step.
    [Show full text]
  • The Stream Function MATH1091: ODE Methods for a Reaction Diffusion Equation 2020/Stream/Stream.Pdf
    The Stream Function MATH1091: ODE methods for a reaction diffusion equation http://people.sc.fsu.edu/∼jburkardt/classes/math1091 2020/stream/stream.pdf Stream lines reveal the paths of fluid flow. The Stream Function The flow of an incompressible fluid in a 2D region, which is usually described by a vector field (u,v), can also be represented by stream function (x; y). 1 The continuity equation At a given moment in time, the flow of a fluid in a 2D region can be represented by a velocity field, which we might represent as the vector field ~u(x; y) or as a pair of horizontal and vertical velocity components (u(x; y); v(x; y)). In general, the law of mass conservation must be applied to the mass velocity, that is, to the product of mass and velocity. But if the fluid is incompressible, then the mass conservation law can be applied directly to the velocity field itself, and has the form: @u @v r(u; v) ≡ + = 0 (Continuity equation) @x @y This is generally referred to as the continuity equation since it can be interpreted to say that, at any point, the flow coming in must equal the flow going out. The operatator r is computing the divergence of the vector field, and so the continuity equation is often stated as: \The divergence of an incompressible flow is zero everywhere." 2 Some sample flows We will start by considering some simple examples of velocity flow fields: • channel: parabolic flow through a straight channel in [0; 5] × [−1; +1]; • corner: flow that turns around a corner in [0; 1] × [0; 1]; • shear: layers of flow at different speeds in [0; 1] × [0; 1]; • vortex: flow that rotates around a center in [−1; 1] × [−1; 1]; 1 Each of the flows will be described using data in matrices.
    [Show full text]
  • Streamfunction-Vorticity Formulation
    Streamfunction-Vorticity Formulation A. Salih Department of Aerospace Engineering Indian Institute of Space Science and Technology, Thiruvananthapuram { March 2013 { The streamfunction-vorticity formulation was among the first unsteady, incompressible Navier{ Stokes algorithms. The original finite difference algorithm was developed by Fromm [1] at Los Alamos laboratory. For incompressible two-dimensional flows with constant fluid properties, the Navier{Stokes equations can be simplified by introducing the streamfunction y and vorticity w as dependent variables. The vorticity vector at a point is defined as twice the angular velocity and is w = ∇ ×V (1) which, for two-dimensional flow in x-y plane, is reduced to ¶v ¶u wz = w · kˆ = − (2) ¶x ¶y For two-dimensional, incompressible flows, a scalar function may be defined in such a way that the continuity equation is identically satisfied if the velocity components, expressed in terms of such a function, are substituted in the continuity equation ¶u ¶v + = 0 (3) ¶x ¶y Such a function is known as the streamfunction, and is given by V = ∇ × ykˆ (4) In Cartesian coordinate system, the above relation becomes ¶y ¶y u = v = − (5) ¶y ¶x Lines of constant y are streamlines (lines which are everywhere parallel to the flow), giving this variable its name. Now, a Poisson equation for y can be obtained by substituting the velocity components, in terms of streamfunction, in the equation (2). Thus, we have ∇2y = −w (6) where the subscript z is dropped from wz. This is a kinematic equation connecting the streamfunction and the vorticity. So if we can find an equation for w we will have obtained a formulation that automatically produces divergence-free velocity fields.
    [Show full text]
  • LECTURE – 33 Geometric Interpretation of Stream Function
    04/04/2017 LECTURE – 33 Geometric Interpretation of Stream Function: In the last class, you came to know about the different types of boundary conditions that needs to be applied to solve the governing equations for fluid flow: i.e. the conservation of mass, the conservation of linear momentum, the conservation of energy. You were told that, the most general way of solving a fluid flow problem is to simultaneously solve the above equations on conservation principles & get the values of the unknown or dependent variables (i.e. , p, u, v, w, T). However, we as human beings, it may be difficult for us to solve all three simultaneously even for a simple fluid flow. Moreover, considering engineering applications, there may no need to solve all of them simultaneously. You can reduce the dependent variables & the equations as per the situation. In that light, we introduced you the concept of stream function for horizontal two dimensional flow. i.e. you know that flow varies only in x & y directions & assuming isothermal conditions as well as incompressible flow, the continuity equation is: uv 0 xy 휕푤 (As, = 0 ; no variations in vertical direction) 휕푧 For solving benefit, you were introduced a function (,)xy such that ( ) ( ) 0 x y y x i.e, this means that u & v y x So, the governing equation for motion becomes unknown in only one quantity . How?? Rather than giving in most general form: Recall the velocity gradient term. It consisted of strain rate tensor & vorticity tensor. That is, a fluid flow consist of rate of deformation & rate of rotation.
    [Show full text]
  • ENE 206 – Fluid Mechanics WEEK 9 • Conservation of Mass • the Stream
    Fluid Flow in Differential Formulation ENE 206 – Fluid Mechanics WEEK 9 Conservation of mass The application of the principle of conservation of mass to an infinitesimal fluid element results in an equation which is known as the continuity equation. The principles of conservation of mass requires that the net outflow of mass per unit time from the infinitesimal control volume must be equal to the rate of decrease of mass within the infinitesimal control volume. Then the relevant equations are given as follows: ρ ρu ρv ρw 0 (6.5) t x y z or alternatively: Dρ u v w ρ 0 (6.6) dt x y z Equation 6.5 and 6.6 are known as the differential formulation of the continuity equation. The equation can be also in vector notation as: ρ .ρV 0 (6.7) t and Dρ ρ..V 0 (6.8) dt For a steady flow, .ρV 0 (6.9) For an incompressible fluid flow, .V 0 (6.10) u v w This equation can be alternatively written as 0 . x y z The stream function A stream function is defined to relate the streamlines which are the imaginary lines describing the fluid flow motion to the principle of conservation of mass. It is possible to define a stream function either for a two-dimensional flow of an incompressible fluid or for a two-dimensional and steady flow. When Two-dimensional flow of an incompressible fluid: The continuity equation becomes: u v 0 (6.11) x y Fluid Flow in Differential Formulation Here a continuous function is defined as ψ ψx,y,t , such that ψ ψ u and v (6.12) y x The stream function for a two-dimensional and steady flow The continuity equation for a defined xy-plane of the Cartesian coordinates becomes: ρu ρv 0 (6.13) x y Then it is possible to define a continuous stream function ψ ψx,y such that ψ ψ ρu and ρv (6.14) y x which also satisfies the continuity equation.
    [Show full text]
  • Visualization of Three-Dimensional Incompressible Flows by Quasi- Two-Dimensional Divergence-Free Projections
    Visualization of three-dimensional incompressible flows by quasi- two-dimensional divergence-free projections Alexander Yu. Gelfgat School of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University, Ramat Aviv, Tel-Aviv, Israel, 69978, [email protected] Abstract A visualization of three-dimensional incompressible flows by divergence-free quasi-two- dimensional projections of velocity field on three coordinate planes is proposed. It is argued that such divergence-free projections satisfying all the velocity boundary conditions are unique for a given velocity field. It is shown that the projected fields and their vector potentials can be calculated using divergence-free Galerkin bases. Using natural convection flow in a laterally heated cube as an example, it is shown that the projection proposed allow for a better understanding of similarities and differences of three-dimensional flows and their two-dimensional likenesses. An arbitrary choice of projection planes is further illustrated by a lid-driven flow in a cube, where the lid moves parallel either to a sidewall or a diagonal plane. Keywords: incompressible flow, flow visualization, Galerkin method, natural convection benchmark, lid-driven cavity benchmark 1 1. Introduction In this article we offer a way of visualization of three-dimensional incompressible flows using divergence-free projections of initial velocity vector potential on two-dimensional coordinate planes. Initially, this study was motivated by a need to visualize three-dimensional benchmark flows, which are direct extensions of well-known two-dimensional benchmarks, e.g., lid-driven cavity and convection in laterally heated rectangular cavities. We seek for a visualization that is capable to show clearly both similarities and differences of flows considered in 2D and 3D formulations.
    [Show full text]
  • Stream Function Notes
    Streamlines and Stream Functions Consider the two-dimensional, steady-state flow of an incompressible fluid. The general continuity equation simplifies as follows: =0 ∂ρ ∂ ∂ ∂ ∂Vx ∂Vy =− ()ρVx − ()ρVVyz− ()ρ ⇒=+0 (1.1) N∂t ∂x ∂y ∂ z ∂∂x y Steady-state ρ comes outside of partial derivative =0 conditions and then cancels out of the equation because Two-dimensional fluid is incompressible Thus, any velocity field that describes a physically possible two-dimensional, steady-state flow of an incompressible fluid must satisfy Eq. (1.1) As an example, consider the following velocity field that supposedly describes the two- dimensional, steady-state flow of an incompressible fluid: VUxxo= and VVy yo= Is it physically possible, i.e. does this velocity field satisfy continuity? If not, can it modified so that continuity is satisfied? For the velocity field to satisfy continuity, it must satisfy the continuity equation simplified for the appropriate conditions. In this case, Eq. (1.1). To check this, we substitute the given velocity equations into Eq. (1.1) as follows: ∂V ∂V ∂∂ x +=y ()Ux +() Vy =+≠ U V 0 ∂∂∂xyxoooo ∂ y Since the equation does not equal zero, this velocity field does not satisfy continuity for a steady- state, two-dimensional flow of an incompressible fluid. However, it will satisfy continuity if Vo = –Uo. Thus, the following velocity field will satisfy continuity: VUxxo= and V y=− Uy o One way to visualize this flow is to sketch the velocity field as a forest of little arrows in an x-y grid each showing the direction and the magnitude of the velocity at a point in the flow field.
    [Show full text]
  • Potential Flow Theory
    Potential Flow Theory “When a flow is both frictionless and irrotational, pleasant things happen.” We can treat external flows around bodies as invicid (i.e. frictionless) and irrotational (i.e. the fluid particles are not rotating). This is because the viscous effects are limited to a thin layer next to the body called the boundary layer.. We can define a potential function(x z t) , as a continuous function that satisfies the basic laws of fluid mechanics: conservation of mass and momentum, assuming incompressible, inviscid and irrotational flow. Potential Flow Theory The Stream Function Is a clever device which allows us to wipe out the continuity equation and solve the momentum equation directly for the single variable. Continuity equation Potential Flow Theory The most common application is incompressible flow in the xy plane This equation is satisfied identically if a function ψ(x, y) is defined such that and the above equation becomes The Eq. shows that this new function must be defined such that Potential Flow Theory The vorticity, or curl V, is an interesting function Where One important application is inviscid irrotational flow in the xy plane, where ωZ =0 and the above Eq. is reduced to This is the second-order Laplace equation for which many solutions and analytical techniques are known. Also, boundary conditions like Eq. reduce to Potential Flow Theory Geometric Interpretation of Stream Function The fancy mathematics above would serve by itself to make the stream function immortal and always useful to engineers. Even better, though, stream function has a beautiful geometric interpretation: Lines of constant stream function are streamlines of the flow.
    [Show full text]