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Advisor(s)
Abstract(s)
Neste trabalho são estudadas as instabilidades estacionárias e não-estacionárias que
ocorrem em escoamento laminar de fluidos não-newtonianos através de condutas curvas, em
que as instabilidades, apesar de origem inercial, são influenciadas pelo carácter viscoelástico
dos fluidos. É assumido escoamento tridimensional, laminar, em desenvolvimento de fluidos
não-newtonianos viscoelásticos, através de canais com curvatura de comprimento angular
igual a 180 e secção transversal rectangular, para diferentes condições de escoamento e
com parâmetros geométricos variáveis. É investigado o efeito da inércia ( 0 Re 2332 ) e a
ausência desta ( Re 0 ), da elasticidade ( 0 Wi 5.00 ), da extensibilidade ( 2 L 500 ) e do
parâmetro de retardamento ( 0 1.0 ), para além do efeito das condições de entrada e da
geometria (1.50 15.10 c R e 0.50 A 5.00 ). A análise é realizada através de simulação
computacional, em que as equações de governo foram resolvidas utilizando um algoritmo
totalmente implícito, tendo como base o método dos volumes finitos, e com a aplicação do
esquema de alta resolução CUBISTA para a discretização dos termos convectivos das equações
da tensão e da quantidade de movimento. As propriedades viscoelásticas dos fluidos foram
definidas pelos modelos constitutivos reológicos FENE-P e FENE-CR.
O escoamento em curvas desenvolve escoamento secundário, perpendicular ao
escoamento principal, cujo padrão mais simples (dito principal) é caracterizado por dois
vórtices simétricos que circulam na direcção da parede exterior da curva ao longo do plano de
simetria. Este escoamento desenvolve-se em escoamentos de fluidos viscoelásticos mesmo na
ausência de inércia. O mecanismo de desenvolvimento do escoamento secundário na presença
de inércia é diferente daquele na ausência de inércia. Enquanto no primeiro a força motriz é
a força centrifuga desenvolvida pela combinação da inércia e da curvatura, no segundo a
força motriz é a elevada tensão norma axial nas paredes.
Em determinadas condições de escoamento e para geometrias com secção transversal
quadrada, o padrão de escoamento pode tornar-se mais complexo, com o desenvolvimento de
dois vórtices junto da parede exterior da curva, também simétricos, mas de tamanho mais
reduzido e com rotação oposta à do par principal de vórtices. Não existe um valor único de
Re crítico de transição, necessário para que ocorra o desenvolvimento do par adicional de
vórtices, pois depende da razão de curvatura, do modelo de fluido, da elasticidade e dos
parâmetros viscoelásticos. Porém, de uma forma geral, os modelos viscoelásticos e as
propriedades fluidificante do modelo, assim como o aumento de Wi e de 2 L , e a diminuição
de , aumentam a intensidade do escoamento secundário, diminuindo o Re crítico de
transição, comparativamente ao escoamento de fluido newtoniano.
Por sua vez, quando considerados os parâmetros geométricos, existe um valor de c R e A
para os quais o Re crítico de transição é mínimo. Os resultados deste trabalho indicam que
estes parâmetros assumem os valores 7.5 c R e A 1.5 , independentemente do modelo reológico. O estudo do efeito da razão de aspecto mostra que para A 1.5 o número de pares
de vórtices adicionais desenvolvidos ao longo da parede exterior da curva aumenta.
No escoamento em desenvolvimento, as condições de entrada são importantes. Aqui é
comparado o desenvolvimento do escoamento quando o perfil de entrada é completamento
desenvolvido e uniforme. Verifica-se que no segundo caso o desenvolvimento do escoamento
secundário é retardado, independentemente das condições de escoamento. Além disso, em
curvas com 3.5 c R , observa-se a existência do efeito da curvatura a montante da curva e,
por esse motivo, a presença ou não do canal de entrada é também avaliada. Os resultados
mostram que, nestas geometrias, a presença do canal de entrada retarda o desenvolvimento
do escoamento, mas apresenta maior estabilidade numérica. Apesar das diferenças na
evolução do escoamento para as diferentes condições de entrada, os escoamentos tendem a
aproximar-se no final da curva sem, no entanto, se igualarem, indicando que o comprimento
angular da curva não é suficiente para que o escoamento atinja o desenvolvimento completo.
Os resultados apresentados mostram ainda que o escoamento secundário persiste até ao
final do canal recto de saída a jusante da curva, apesar do escoamento axial atingir
novamente o desenvolvimento completo.
Para Re 2332 , o escoamento de fluido newtoniano é estacionário, mas para fluido
viscoelásticos o escoamento passa a não-estacionário, embora com natureza não-turbulenta,
mesmo para valores de elasticidade reduzidos (Wi 0.10 ). O escoamento secundário passa a
apresentar oscilações que se concentram junto da parede exterior da curva, em particular na
região do par adicional de vórtices. O padrão de escoamento secundário depende
consideravelmente quer da elasticidade quer do parâmetro de retardamento, mas a variação
deste segue a mesma linha de evolução para estas variáveis.
Finalmente, admitindo escoamento pulsante à entrada da curva com diferentes
parâmetros pulsantes, o escoamento de fluido newtoniano e não-newtoniano é analisado. Os
resultados obtidos mostram que as paredes rectas da secção transversal retardam, ou mesmo
evitam o desenvolvimento do escoamento de Lyne (característico do escoamento pulsante em
curva com secção transversal circular), uma vez que, nas condições de escoamento assumidas
neste trabalho, aquele tipo de escoamento não se desenvolve.
In this work, the stationary and non-stationary instabilities that occur in laminar flows of non-Newtonian fluids through a curved channel, associated with the viscoelastic nature of the fluids, are examined. It is assumed three-dimensional, laminar, developing flow of viscoelastic fluids through curved ducts with angular length 180 and rectangular crosssection, considering different flow conditions and variable geometrical parameters. Both inertial ( 0 Re 2332 ) and creeping ( Re 0 ) flow conditions, and the effects of elasticity ( 0 Wi 5.00), extensibility ( 2 L 500 ) and retardation parameter ( 0 1.0 ) are investigated, in addition to the effects of entry conditions and geometry (1.50 15.10 c R and 0.50 A 5.00 ). The analysis is performed by computer simulation, with the governing equations solved using a fully implicit algorithm based on the finite volume method, and the high resolution scheme CUBISTA applied for the discretization of the convective terms of the equations. The viscoelastic properties of the fluids are defined by the rheological models FENE-P and FENE-CR. The flow in curved ducts develops secondary flow perpendicular to the main flow, in which the main pattern is characterized by two symmetrical vortices with the fluid moving towards the outer wall of the curve along the plane of symmetry. This secondary flow develops for viscoelastic fluids, even in the absence of inertia. The secondary flow development mechanism in the presence of inertia is different from that in the absence of inertia. While in the first case the driving force is the centrifugal force developed by the combination of inertia and curvature, in the second case the driving force is the high axial normal tension at the walls. Under certain flow conditions and considering curved ducts with square cross-section, the flow pattern can become more complex with the development of two vortices near the outer wall of the curve, also symmetric but smaller in size and with opposite rotation compared to the main vortices pair. There is no single critical value of Re for the development of the additional pair of vortices; it depends on the curvature ratio, fluid model, elasticity and viscoelastic parameters. However, in general, the viscoelastic model and the shear-thinning properties of the fluid, as well as the increase of Wi and 2 L , and decrease of , lead to the growth of the intensity of the secondary flow, reducing the critical value of Re , as compared to the corresponding Newtonian fluid flow case. On the other hand, when considering the geometric parameters, there is a value of c R and A for which the critical value of Re is the lowest. In this study, the results indicate that these parameters assume the values of 7.5 c R and A 1.5 , regardless of the rheological model. The analysis of the effect of the aspect ratio also shows that for A 1.5 the number of additional pair of vortices near the outer wall increases. The imposed entry flow conditions are important to the development of the flow. In this work, the flow development is analysed considering uniform and fully developed flows at the curve entrance. It is observed that in the former case the development of the secondary flow is delayed, regardless of flow conditions. In curved channels with 3.5 c R , the flow upstream of the curve is affected by the curvature, therefore the effect of including a straight channel at the entrance of the curve is also evaluated. The results show that, in these geometries, the presence of the inlet channel decelerates the development of the flow, but tends to promote numerical stability. Despite differences in flow development, the flow patterns tend to approach each other (with and without inlet channel) at end of the curve, albeit without becoming exactly equal, indicating that the angular length of the curve is not sufficient for the flow to reach fully development. The results also show that the secondary flow persists downstream of the curve along the straight outlet channel, although in this case full development is achieved. When Re 2332 , Newtonian fluid flow is stationary in a curved channel, but viscoelastic fluid flow becomes non-stationary, with non-turbulent nature, even for low elasticity ( Wi 0.10 ). The secondary flow is then oscillatory, and the oscillations are concentrated near the outer wall, particularly in the region of the additional pair of vortices. The pattern of secondary flow is highly dependent on the elasticity and retardation parameter, but this variation follows the same line of evolution for both variables. Finally, assuming a pulsating flow through the curved duct with different pulsating parameters, the Newtonian and non-Newtonian flow response is also analysed. The results show that the straight walls of the cross-section delay, or even prevent, the development of a Lyne type flow (characteristic of pulsating flow in curved ducts of circular cross-section); apparently, for the assumed conditions, such kind of flow does not tend to develop.
In this work, the stationary and non-stationary instabilities that occur in laminar flows of non-Newtonian fluids through a curved channel, associated with the viscoelastic nature of the fluids, are examined. It is assumed three-dimensional, laminar, developing flow of viscoelastic fluids through curved ducts with angular length 180 and rectangular crosssection, considering different flow conditions and variable geometrical parameters. Both inertial ( 0 Re 2332 ) and creeping ( Re 0 ) flow conditions, and the effects of elasticity ( 0 Wi 5.00), extensibility ( 2 L 500 ) and retardation parameter ( 0 1.0 ) are investigated, in addition to the effects of entry conditions and geometry (1.50 15.10 c R and 0.50 A 5.00 ). The analysis is performed by computer simulation, with the governing equations solved using a fully implicit algorithm based on the finite volume method, and the high resolution scheme CUBISTA applied for the discretization of the convective terms of the equations. The viscoelastic properties of the fluids are defined by the rheological models FENE-P and FENE-CR. The flow in curved ducts develops secondary flow perpendicular to the main flow, in which the main pattern is characterized by two symmetrical vortices with the fluid moving towards the outer wall of the curve along the plane of symmetry. This secondary flow develops for viscoelastic fluids, even in the absence of inertia. The secondary flow development mechanism in the presence of inertia is different from that in the absence of inertia. While in the first case the driving force is the centrifugal force developed by the combination of inertia and curvature, in the second case the driving force is the high axial normal tension at the walls. Under certain flow conditions and considering curved ducts with square cross-section, the flow pattern can become more complex with the development of two vortices near the outer wall of the curve, also symmetric but smaller in size and with opposite rotation compared to the main vortices pair. There is no single critical value of Re for the development of the additional pair of vortices; it depends on the curvature ratio, fluid model, elasticity and viscoelastic parameters. However, in general, the viscoelastic model and the shear-thinning properties of the fluid, as well as the increase of Wi and 2 L , and decrease of , lead to the growth of the intensity of the secondary flow, reducing the critical value of Re , as compared to the corresponding Newtonian fluid flow case. On the other hand, when considering the geometric parameters, there is a value of c R and A for which the critical value of Re is the lowest. In this study, the results indicate that these parameters assume the values of 7.5 c R and A 1.5 , regardless of the rheological model. The analysis of the effect of the aspect ratio also shows that for A 1.5 the number of additional pair of vortices near the outer wall increases. The imposed entry flow conditions are important to the development of the flow. In this work, the flow development is analysed considering uniform and fully developed flows at the curve entrance. It is observed that in the former case the development of the secondary flow is delayed, regardless of flow conditions. In curved channels with 3.5 c R , the flow upstream of the curve is affected by the curvature, therefore the effect of including a straight channel at the entrance of the curve is also evaluated. The results show that, in these geometries, the presence of the inlet channel decelerates the development of the flow, but tends to promote numerical stability. Despite differences in flow development, the flow patterns tend to approach each other (with and without inlet channel) at end of the curve, albeit without becoming exactly equal, indicating that the angular length of the curve is not sufficient for the flow to reach fully development. The results also show that the secondary flow persists downstream of the curve along the straight outlet channel, although in this case full development is achieved. When Re 2332 , Newtonian fluid flow is stationary in a curved channel, but viscoelastic fluid flow becomes non-stationary, with non-turbulent nature, even for low elasticity ( Wi 0.10 ). The secondary flow is then oscillatory, and the oscillations are concentrated near the outer wall, particularly in the region of the additional pair of vortices. The pattern of secondary flow is highly dependent on the elasticity and retardation parameter, but this variation follows the same line of evolution for both variables. Finally, assuming a pulsating flow through the curved duct with different pulsating parameters, the Newtonian and non-Newtonian flow response is also analysed. The results show that the straight walls of the cross-section delay, or even prevent, the development of a Lyne type flow (characteristic of pulsating flow in curved ducts of circular cross-section); apparently, for the assumed conditions, such kind of flow does not tend to develop.
Description
Keywords
Reologia Dinâmica dos fluidos computacional Escoamento laminar - Canais com curvatura Fluidos newtonianos Fluidos não-newtonianos Fluidos viscoelásticos