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momentum equation

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Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005450
EISBN: 978-1-62708-196-2
... Abstract This article is a comprehensive collection of fluid dynamic equations for properties of fluids, fluid statics, fluid motion, dimensional analysis, and boundary layer flow. It presents equations for analyzing problems in fluid mechanics, continuity equation, momentum equation...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005426
EISBN: 978-1-62708-196-2
... Abstract Computational fluid dynamics (CFD) is a computationally intensive three-dimensional simulation of thermal fluids systems where non-linear momentum transport plays an important role. This article presents the governing equations of fluid dynamics and an introduction to the CFD...
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002444
EISBN: 978-1-62708-194-8
...-conditioning system configuration. Calculations were done using the FIDAP finite-element CFD code ( Ref 73 ). Abstract Abstract Computational fluid dynamics (CFD) is reserved for computationally intensive three-dimensional simulations of thermal fluids systems where nonlinear momentum transport plays...
Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005525
EISBN: 978-1-62708-197-9
.... A reversal of antimony microsegregation at low solid fraction and the tin-controlled flow cause the unusual antimony macrosegregation in (b). Source: Ref 68 Abstract This article presents conservation equations for heat, species, mass, and momentum to predict transport phenomena during...
Series: ASM Handbook
Volume: 6A
Publisher: ASM International
Published: 31 October 2011
DOI: 10.31399/asm.hb.v06a.a0005636
EISBN: 978-1-62708-174-0
... welding. The article presents the mathematical equations of mass, momentum, energy, and species conservation. It reviews the applications of heat transfer and fluid flow models for different welding processes. Finally, the article discusses the approaches to improve reliability of, and reduce uncertainty...
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005237
EISBN: 978-1-62708-187-0
... α ) Equation 3 is applied with α replaced by α ′. Equation 4 accurately describes behaviors in the extremes of no diffusion and complete diffusion in the solid during solidification. The underlying model comprises the transport equations: continuity, momentum, energy...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005437
EISBN: 978-1-62708-196-2
... Abstract This article presents the governing equations for moving a solidification front, based on the balance of mass, momentum, energy, and solute. It reviews how material properties and geometry can be analyzed in the context of the governing equations. The article provides several example...
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005234
EISBN: 978-1-62708-187-0
... MODEL of the solidification of a metal alloy involves coupling of energy, species, mass, and momentum conservation equations ( Ref 1 , 2 , 3 , 4 , 5 , 6 , 7 ). This article examines critical features of four key areas of modeling transport phenomena associated with casting processes: Heat...
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005208
EISBN: 978-1-62708-187-0
... Abstract This article begins with balance equations for mass, momentum, energy, and solute and the necessary boundary conditions for solving problems of interest in casting and solidification. The transport phenomena cover a vast range of length and time scales, from atomic dimensions up...
Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005510
EISBN: 978-1-62708-197-9
... the base control volume is partially occupied and expands in each time step, a special treatment is required for the discretization of the conservation equations over this volume. This involves formulation of the equations for mass, momentum, energy conservation, turbulence quantities, and concentration...
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005233
EISBN: 978-1-62708-187-0
... are also called control volumes. Using FVMs, one can easily construct discrete approximations that have the conservative property; that is, the discrete approximations can mimic the physical laws from which the fluid equations were derived by conserving properties such as computed mass, momentum...
Series: ASM Handbook
Volume: 21
Publisher: ASM International
Published: 01 January 2001
DOI: 10.31399/asm.hb.v21.a0003402
EISBN: 978-1-62708-195-5
... constitutive equations. For thermoset materials, where the viscosity is low, one can usually impregnate the resin within a stationary fiber preform bed and thus use the flow through porous media theory to describe the motion. Hence for such materials, the momentum conservation is not described...
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005238
EISBN: 978-1-62708-187-0
... , ISS , Warrendale, PA , 1996 The modeling of mechanical behavior requires solution of 1) the equilibrium or momentum equations relating force and stress, 2) the constitutive equations relating stress and strain, and 3) compatibility equations relating strain and displacement. This is because...
Series: ASM Handbook
Volume: 6
Publisher: ASM International
Published: 01 January 1993
DOI: 10.31399/asm.hb.v06.a0001480
EISBN: 978-1-62708-173-3
... The velocity in the energy equation is determined from the momentum and continuity equation. The thermal flux from the arc is prescribed on the weld pool and arc interface. With this data, the energy equation determines the position of the liquid-solid interface. With a new temperature distribution...
Series: ASM Handbook
Volume: 6A
Publisher: ASM International
Published: 31 October 2011
DOI: 10.31399/asm.hb.v06a.a0005587
EISBN: 978-1-62708-174-0
... of forces is expressed by the conservation of momentum and the definition of the stress tensor, σ, the traction vector, τ, and the body force, b , in the differential equation: (Eq 17) ∇ ⋅ σ + b = m x ¨ In welding, it is typical to assume that the inertial forces are negligible...
Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005520
EISBN: 978-1-62708-197-9
... of gas-forming elements in multicomponent systems (ranging from aluminum alloys to steels and nickel superalloys) during solidification. Their methodology is based on a volume averaging of the liquid, solid, and gas within each element, allowing the momentum equation to be reformulated for the mushy zone...
Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005540
EISBN: 978-1-62708-197-9
... of dynamic explicit codes has an inertia term similar to the following equation. Different codes may use different formulations, but the concept of the momentum equation is the same: (Eq 1) F i n t ( u ˙ ) + c + m u ¨ = F e x t where F int is the internal...
Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005503
EISBN: 978-1-62708-197-9
..., the continuity and momentum equations for the melt flow in the injection molding cavity are merged into a single Poisson equation in terms of the pressure and fluidity. Computer simulation is usually based on a 2.5-dimensional approach because of the thin wall and axial symmetry. However, the Hele-Shaw model has...
Series: ASM Handbook
Volume: 6A
Publisher: ASM International
Published: 31 October 2011
DOI: 10.31399/asm.hb.v06a.a0005592
EISBN: 978-1-62708-174-0
... is the density, v is the velocity, and ∇ is the spatial gradient operator. Fig. 3 Eulerian versus Lagrangian models in welding The conservation of momentum equation is as follows: (Eq 2) ∇ ⋅ ( ρ v ) v = − ∇ p + ∇ ⋅ σ ′ + ρ g where p is the pressure, σ...
Series: ASM Handbook
Volume: 4C
Publisher: ASM International
Published: 09 June 2014
DOI: 10.31399/asm.hb.v04c.a0005898
EISBN: 978-1-62708-167-2
... ) and F ( e 3 ) acting on the faces of the cube with normals e 1 e 2 , and e 3 ( Fig. 14 ). Table 1 lists dynamic viscosity values for some melts and alloys. Fig. 14 Stress tensor for fluid. The fluid motion equation represents the conservation of momentum...