In this paper, a model problem of viscous flow and heat transfer in a rectangular converging (diverging) channel has been investigated. The governing equations are presented in Cartesian Coordinates and consequently they are simplified and solved with perturbation and numerical methods. Initially, symmetrical solutions of the boundary value problem are found for the upper half of the channel. Later on, these solutions are extended to the lower half and then to the whole channel. The numerical and perturbation solutions are compared and exactly matched with each other for a small value of the parameters involved in the problem. It is also confirmed that the solutions for the converging/diverging channel are independent of the sign of m (the slope). Moreover, the skin friction coefficient and heat transfer at the upper wall are calculated and graphed against the existing parameters in different figures. It is observed that the heat transfer at walls is decreased (increased) with increasing $${c}_{1}$$ c 1 (thermal controlling parameter) for diverging (converging). It is also decreased against Pr (Prandtle number). For $${c}_{1}=0$$ c 1 = 0 , the temperature profiles may be exactly determined from the governing equations and the rate of heat transfer at the upper wall is $$\theta^{\prime } (1) = \frac{m}{{(1 + m^{2} )\tan^{ - 1} m}}$$ θ ′ ( 1 ) = m ( 1 + m 2 ) tan - 1 m . It is confirmed that the skin friction coefficient behaves linearly against Re* (modified Reynolds number) and it is increased with increasing of Re* (changed from negative to positive). Moreover, it is increased asymptotically against m and converges to a constant value i.e. zero.

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In this paper, effects of deforming walls on peristaltic flow in a two dimensional channel have been investigated. The two dimensional form of the governing equations is simplified by using appropriate transformations and well established approximations, which are used extensively for solution of such models. The transformations are designed so that the complex problem is reduced into an ordinary differential equation (ODE). New and simple non-linear ODE is formed in view of adopted procedures and techniques. Its solutions are exactly matched with the solutions of classical problems. Solutions of the final problem are provided for small values of the surface expansion (contraction) ratio and Reynolds number with the help of the perturbation technique and non-linear shooting method. The velocity field, pressure and shear stress are evaluated analytically and numerically. Meanwhile, effects of all parameters are observed on the velocity field, pressure rise per wavelength and shear stress profiles with the help of tables and different figures. Excellent agreement between solutions is found. Current results are apparently matched with the classical problems of peristaltic flow in deforming and non-deforming walls.

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The well-known classical Jaffery-Hammel flow model of converging and diverging channel in plan Polar Coordinates is reformulated in Cartesian Coordinates with the additional effects of heat and mass transfer. Second, concept of wedge flow is explained in view of Blasius variables in 1931 and it is strictly presented in Cartesian Coordinates. This classical problem of blasius is reformulated with the additional effects of heat and mass transfer in terms of Cartesian Coordinated system, whereas, the classical boundary layer approximations are not taken into account. The well-known two dimensional form of four governing equations are taken in Cartesian Coordinates, whereas, no restrictions have been imposed on the velocity components. The PDE’s are transformed into non-linear coupled ODE’s of variable coefficients. Later on, another set of transformation is found and the non-linear ODE’s with variable coefficients are transformed into ODE’s of constant coefficients and the modeled problem is exactly matched with the classical simulation of converging and diverging flow. Moreover, the problem is solved by different methods and effects of different parameters are seen on field variables.