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| The '''Oldroyd-B model''' is a constitutive model used to describe the flow of [[viscoelastic]] fluids.
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| This model can be regarded as an extension of the [[Upper Convected Maxwell model]] and is equivalent to a fluid filled with elastic bead and spring dumbbells.
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| The model is named after its creator [[James G. Oldroyd]].
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| The model can be written as:
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| :<math> \mathbf{T} + \lambda_1 \stackrel{\nabla}{\mathbf{T}} = 2\eta_0 (\mathbf{D} + \lambda_2 \stackrel{\nabla}{\mathbf{D}}) </math>
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| where:
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| * <math>\mathbf{T}</math> is the [[Stress (physics)|stress]] [[tensor]];
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| * <math>\lambda_1</math> is the relaxation time;
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| * <math>\lambda_2</math> is the retardation time = <math> \frac{\eta_s}{\eta_0}\lambda_1 </math>;
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| * <math> \stackrel{\nabla}{\mathbf{T}} </math> is the [[Upper convected time derivative]] of stress tensor:
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| :<math> \stackrel{\nabla}{\mathbf{T}} = \frac{\partial}{\partial t} \mathbf{T} + \mathbf{v} \cdot \nabla \mathbf{T} -( (\nabla \mathbf{v})^T \cdot \mathbf{T} + \mathbf{T} \cdot (\nabla \mathbf{v})) </math>;
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| *<math>\mathbf{v}</math> is the fluid velocity;
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| *<math>\eta_0</math> is the total [[viscosity]] composed of solvent and polymer components, <math> \eta_0= \eta_s + \eta_p </math>;
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| *<math>\mathbf {D}</math> is the deformation rate tensor or rate of strain tensor, <math>\mathbf{D} = \frac{1}{2}\left[\boldsymbol\nabla \mathbf{v} + (\boldsymbol\nabla \mathbf{v})^T\right]</math>.
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| The model can also be written split into polymeric (viscoelastic) part separately from the solvent part:
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| <math> \mathbf{T} = 2\eta_s \mathbf{D} + \mathbf{\tau} </math>.
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| where
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| :<math> \mathbf{\tau} + \lambda_1 \stackrel{\nabla}{\mathbf{\tau}} = 2\eta_p \mathbf{D} </math>
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| Whilst the model gives good approximations of viscoelastic fluids in shear flow, it has an unphysical singularity in extensional flow, where the dumbbells are infinitely stretched;
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| If the solvent viscosity is zero then the Oldroyd-B becomes the [[Upper Convected Maxwell model]].
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| ==References==
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| * {{cite journal|last=Oldroyd|first=James|title=On the Formulation of Rheological Equations of State|journal=Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences|year=1950|month=Feb|volume=200|issue=1063|pages=523–541}}
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| * {{cite book | author=Owens, R. G.,Phillips, T. N.| title=Computational Rheology.| publisher=Imperial College Press | year=2002 | isbn=978-1-86094-186-3}}
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| [[Category:Non-Newtonian fluids]]
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I'm Roy and I live in Kommel.
I'm interested in Occupational Therapy, American football and Dutch art. I like travelling and reading fantasy.
Here is my homepage bola terpercaya (pop over to this website)