[pt] A transformação de coordenada introduzida por L. Crocco, para a solução do escoamento de fluidos compressíveis sobre a placa plana isotérmica, é utilizada de uma forma original, com o intuito de se aproveitarem as vantagens inerentes a esta transformação no método de Não-Similaridade Local, idealizado por E. M. Sparrow, na solução de camada limite não-similares incompreensíveis. A transformação de Crocco é aplicada às equações de conservação em convecção forçada que regem os escoamentos laminar, de propriedades físicas constantes e bidimensionais, em torno de sólidos. Dois problemas não-similares devido a forma da velocidade potencial, o cilindro em escoamento transversal e o escoamento desacelerado de Howarth, são resolvidos a fim de ilustrar-se este novo procedimento. Na solução destes casos considera-se inclusive o efeito da função dissipação. Os resultados dos problemas hidrodinâmico e térmico são comparados aos da literatura disponível e uma boa concordância foi observada. / [en] The coordinate trnsformation developed by L. Crocco to obtain the solution of the compressible fluid flows over isotermal flat plates is originally employed in the present work, with the purpose of adding its inherent advantage to the Non-Similarity Method idealized by E. M. Sparrow, in the solution of the incompressible non-similar boundary layers. The Croccos’s transformation is applied to the conservation equation for forced convection, laminar, Constant properties and two-dimensional flows freestream velocity distribution, the cylinder in crossflow and the Howarth’s retarded flow, are solved with a view to illustrating the new procedure. In those solutions the effect of frictional heat is also considered. The results of hydrodynamic and thermal problems are compared with available published information and good agreement was observed.

Esta dissertação apresenta uma modelagem matemática do escoamento laminar em tubos de paredes permeáveis aplicada à micro-filtração de suspensões. A modelagem utilizou-se da formulação integral das equações de conservação e de funções pré- estabelecidas para o representar os campos de velocidade e de concentração ao longo do tubo permeável. As equações integrais da quantidade de movimento e da conservação das espécies químicas forneceram duas equações diferenciais ordinárias de primeira ordem para as variáveis funcionais \"n (z)\" e \"m (z)\" presentes nas funções pré-estabelecidas. Para a solução destas equações optou-se pelo método de Runge-Kutta de quarta ordem devido a sua simplicidade e versatilidade conhecida da literatura. No entanto a equação para a conservação da quantidade de movimento apresentou grande instabilidade ao ser submetida à solução numérica, contornada a partir da imposição de diferentes formas de evolução para o campo de velocidade, através do funcional n(z) cujas formas de variação foram impostas segundo uma dependência linear, exponencial e polinomial. Por outro lado, a solução da equação para conservação das espécies foi numericamente convergente. De posse das funções pré-estabelecidas e ajustadas a partir da equação da conservação das espécies na forma integral, obtém-se neste trabalho os valores correspondentes para o adimensional de Sherwood, quantificando o processo de transferência de massa. Com os valores de Sherwood, os resultados desta modelagem foram comparados com os da literatura, Grober et al. (Apud Zeman & Zydney, 1996) e outros, e apresentaram-se de acordo para estudos de casos particulares, no intervalo de Peclet de 104 - 106 . / This dissertation presents a mathematical modeling of the larninar flow in permeable tubes applied to the micro-filtration of suspensions. The modeling uses of integral formulation of the conservation equations and of functions pre-established for to represent the fields of velocity and concentration along the permeable tube. The integral equations of the momentum and of conservation of the chemical species its supplied two differential ordinary equations if first order for the variables functional \"n(z)\" and \"m(z)\" presents in the pre-established functions. For the solution of these equations was opted for the method of Runge-Kutta of fourth order due to its simplicity and well-known versatility of the literature. However the equation for the conservation of the momentum presented great instability to be submitted to the numeric solution, outlined starting from the imposition forms different from evolution for the field of velocity, through the functional \"n(z)\" with lineal, exponential and polynomial dependence. However, the solution of the equation for conservation of the species was convergent numerical. Through of the pre-established functions and adjusted starting from the equation of the conservation of the species in the integral form, it was obtained in this work the corresponding values for the dimensionless of Sherwood, quantifying the process of mass transfer. With the values of Sherwood, the results of this modeling were compared with the one of the literature, Grober et al. (Apud Zeman & Zydney, 1996) and other, and they came of agreement for particular cases in the interval of Peclet of 104 the 106.

Esta dissertação apresenta uma modelagem matemática do escoamento laminar em tubos de paredes permeáveis aplicada à micro-filtração de suspensões. A modelagem utilizou-se da formulação integral das equações de conservação e de funções pré- estabelecidas para o representar os campos de velocidade e de concentração ao longo do tubo permeável. As equações integrais da quantidade de movimento e da conservação das espécies químicas forneceram duas equações diferenciais ordinárias de primeira ordem para as variáveis funcionais \"n (z)\" e \"m (z)\" presentes nas funções pré-estabelecidas. Para a solução destas equações optou-se pelo método de Runge-Kutta de quarta ordem devido a sua simplicidade e versatilidade conhecida da literatura. No entanto a equação para a conservação da quantidade de movimento apresentou grande instabilidade ao ser submetida à solução numérica, contornada a partir da imposição de diferentes formas de evolução para o campo de velocidade, através do funcional n(z) cujas formas de variação foram impostas segundo uma dependência linear, exponencial e polinomial. Por outro lado, a solução da equação para conservação das espécies foi numericamente convergente. De posse das funções pré-estabelecidas e ajustadas a partir da equação da conservação das espécies na forma integral, obtém-se neste trabalho os valores correspondentes para o adimensional de Sherwood, quantificando o processo de transferência de massa. Com os valores de Sherwood, os resultados desta modelagem foram comparados com os da literatura, Grober et al. (Apud Zeman & Zydney, 1996) e outros, e apresentaram-se de acordo para estudos de casos particulares, no intervalo de Peclet de 104 - 106 . / This dissertation presents a mathematical modeling of the larninar flow in permeable tubes applied to the micro-filtration of suspensions. The modeling uses of integral formulation of the conservation equations and of functions pre-established for to represent the fields of velocity and concentration along the permeable tube. The integral equations of the momentum and of conservation of the chemical species its supplied two differential ordinary equations if first order for the variables functional \"n(z)\" and \"m(z)\" presents in the pre-established functions. For the solution of these equations was opted for the method of Runge-Kutta of fourth order due to its simplicity and well-known versatility of the literature. However the equation for the conservation of the momentum presented great instability to be submitted to the numeric solution, outlined starting from the imposition forms different from evolution for the field of velocity, through the functional \"n(z)\" with lineal, exponential and polynomial dependence. However, the solution of the equation for conservation of the species was convergent numerical. Through of the pre-established functions and adjusted starting from the equation of the conservation of the species in the integral form, it was obtained in this work the corresponding values for the dimensionless of Sherwood, quantifying the process of mass transfer. With the values of Sherwood, the results of this modeling were compared with the one of the literature, Grober et al. (Apud Zeman & Zydney, 1996) and other, and they came of agreement for particular cases in the interval of Peclet of 104 the 106.

Convection set up in a fluid due to the presence of two components of differing diffusivities is known as double diffusive convection. Double diffusive convection is observed in nature, in oceans, in the formation of certain columnar rock structures and in stellar interiors. The major engineering applications of double diffusive convection are in the fields metallurgy and alloy solidification in casting processes. The two components may be any two substances which affect the density of the fluid, heat and salt being the pair found most commonly in nature. Depending upon the initial stratifications of the two components, double diffusive convection can be set up in either the diffusive mode or the finger mode. In this thesis, the linear stability of a double diffusive system prone to finger instability has been studied in the presence of temporally varying non-linear background profiles of temperature and salinity. The motivation for the present study is to bridge the gap between existing theories, which mainly concentrate on linear background profiles independent of time, on the one hand and experiments and numerical simulations, which have time dependent step-like non-linear background profiles, on the other. The general stability characteristics of a double diffusive system with step-like background profiles have been studied using the standard normal mode method. The background temperature and salinity profiles are assumed to follow the hyperbolic tangent function, since it has a step-like character. The sharpness of the step can be altered by changing a suitable parameter in the hyperbolic tangent function. It is found that changing the degree of non-linearity of the background profile of one of the components keeping the background profile of the other component linear affects the growth rate, Wave number and the form of the disturbances. In general, increasing the degree of nonlinearity of background salinity profile makes the system more unstable and results in a reduction in the vertical extent of the disturbances. On the other hand, increasing the degree of non-linearity of the background temperature profile with the salinity profile kept linear results in a reduction in the growth rate and increase in the wave number. The form of the disturbance may change due to enhanced modal competition between the gravest odd and even modes in this case. The method of normal modes inherently assumes that the background profiles of temperature and salinity are independent of time and hence, it cannot be used for studying the stability of systems with time varying background profiles. A pseudo-similarity method has been used to handle such background profiles. Initial steps of temperature and salinity diffuse according to the error function form, and hence, the case of error function background profiles has been studied in detail. Taking into account the time-dependence of background profiles has been shown to significantly change the wave number and the incipient flux ratio. The dependence of the critical wave number (kc) on the thermal Rayleigh number (RaT ) can be determined analytically and is found to change from kc ~ Ra T1/4 for linear background profiles to kc ~ Ra T1/3 for error function profiles. The region of instability in the Rp (density stability ratio) space is found to increase from 1 ≤ R ρ ≤ r−1 for linear background profiles to 1 ≤ Rρ < r−3/2 for error function background profiles, where T denotes the ratio of the diffusivity of the slower diffusing component to that of the faster diffusing one. A parametric study covering a wide range of parameter values has been carried out to determine the effect of the parameters density stability ratio (Rp), diffusivity ratio (ρ ) and Prandtl number (Pr) on the onset time, critical wavenumber and the incipient flux ratio. The wide range of governing parameters covered here is beyond the scope of experimental and numerical studies. Such a wide range can be covered by theoretical approaches alone. It has been shown that the time of onset of convection determines the thicknesses of the temperature and salinity boundary layers, which in turn determine the width of salt fingers. Finally, the theoretical predictions of salt finger widths have been shown to be in agreement with the results of two dimensional numerical simulations of thermohaline system.