1.3 Outline of the Thesis 7
(i.e. spectral accuracy). In contrast to high-order (Chebyshev) polynomials, they can work well on uniformly-gridded and scattered data. Integrated RBFNs have several advantages over differentiated RBFNs. The use of integration to construct the approximations helps avoid the problem of reduced convergence rate caused by differentiation as well as provides a powerful means of imple- menting derivative boundary conditions. The use of IRBFNs constructed in one dimension for the solution in two or higher dimensions facilitates the em- ployment of a much larger number of nodes. For most works reported, RBFNs are introduced as trial functions in the point collocation and there are so few RBF results for the simulation of **non**-**Newtonian** **fluid** **flows**.

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In the first part, 1D-IRBFNs are incorporated into the Galerkin formulation to simulate viscous **flows**. The governing equations are taken in the streamfunction- vorticity formulation and in the streamfunction formulation. Boundary condi- tions are effectively imposed with the help of the integration constants. The proposed 1D-IRBFN-based Galerkin methods are validated through the numer- ical simulation of several benchmark test problems including free convection in a square slot and in a concentric annulus.

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This paper presents a numerical collocation technique based on 1D-IRBFNs and Cartesian grids for the simulation of **Newtonian** and **non**-**Newtonian** **flows** in rectan- gular and **non**-rectangular domains. The technique is extremely easy to implement and capable of yielding a high level of accuracy using a relatively coarse grid. The latter is achieved owing to the facts that (i) RBFNs with the multiquadric basis function have spectral approximation power and do not require an underlying mesh (boundary points may not coincide with regular grid nodes), (ii) the problem of reduced convergence rates associated with conventional differentiation approaches is overcome, and (iii) the boundary conditions are presently implemented in an exact manner. The obtained numerical results are very encouraging, showing the great potential of the RBF collocation method in the field of computational **fluid** dynamics.

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Keywords: **non**-**Newtonian** ﬂuid ﬂows; improved L 2 decay
1 Introduction
Mathematical models for ﬂuid dynamics have been attracting more and more attention in theoretical and computational studies. Naiver-Stokes equations [] are generally ac- cepted as proving an accurate model for the incompressible motion of viscous ﬂuids in many practical situations, where the constitutive relation is linear due to the Stokes hy- potheses. However, in industrial application, some ﬂuids which exhibit the nonlinear con- stitutive relation, such as liquid crystals, some polymers, some oils and so on (refer to [,

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A sudden change in the flow rate brings about significant pressure oscillations in the piping system, known as water hammer (**fluid** hammer). Unsteady flow of a **non**-**Newtonian** **fluid** due to instantaneous valve closure is studied. Power law and Cross models are used to simulate **non**- **Newtonian** effects. Firstly, the appropriate governing equations are derived and then, they are solved by a numerical approach. A fourth-order Runge–Kutta scheme is used for the time integration, and the central difference scheme is employed for the spatial derivatives discretization. To verify the proposed mathematical model and numerical solution, a comparison with corresponding experimental results from the literature are made. The results reveal a remarkable deviation in pressure history and velocity profile with respect to the water hammer in **Newtonian** fluids. The significance of the **non**-**Newtonian** **fluid** behaviour is manifested in terms of the drag reduction and line packing effect observed in the pressure history results. A detailed discussion regarding the **fluid** viscosity and its shear-stress diagrams are also included.

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If L(q, O) At + B is the Legendre transform function of a s/reamfunction for a steady, plane, a/igned, incompressible, in6nitely conducting second--grade Buid ttow, then the flow in the [r]

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This paper deals with ψ - ω formulation of finite differences [1], [4] and [6] for assessment of **newtonian** and **non**-**newtonian** incompressible **fluid** flow. For **newtonian** **flows**, the formulation is tested in a driven cavity setup, characterized by small vor- tex formation at low Reynolds number flow, these vortices adding to the numerical difficulties. The problem presented by the behavior of pressure term in the Navier-Stokes equation can be alleviated by applying curl to the momentum conservation equa- tion, thus obtaining a vorticity transport equation. The pressure distribution can later be evaluated explic- itly using an unhindered version of the momentum equation, should the information be necessary. In ad- dition, the velocity field was replaced by streamfunction, thereby achieving better convergence rate by using **non**-integral methods taking into ac- count problems in calculating the boundary values of vorticity. By the implementation of streamfunction, the three equation model is reduced to a two-equa- tion model, while the remaining field properties can be evaluated directly from corresponding conserva- tion equations. A similar mechanism can be used for simpler models of **non**-**newtonian** **flows**, while more sophisticated approaches (e.g. Oldroyd-B) require the solution of a coupled system of differential equations. 0 UVOD

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Figure 13: Residual history of **non**-isothermal blood flow simulations for a single wire problem: **Newtonian** **fluid** flow (left) and **non**-**Newtonian** blood flow (right).
The computational benchmark problem presented in this subsection could be considered as a reliable verification and validation test case for **non**-isothermal Newtonain and **non**-**Newtonian** MHD-FHD **fluid** **flows** when microfluidic applications are concerned, because the numerical solution was converged to a very small threshold value of 10 −7 within a relatively small total number of iterations. Furthermore, the obtained results are well-explainable comparing them to the **non**-isothermal Hartmann flow. It is important to mention again the limitation of the current implementation, because it has been observed relying on numerical experience that when the applied magnetic field strength is higher than 4 T esla, it was difficult to ensure convergence of the numerical solution for both **Newtonian** and **non**-**Newtonian** fluids. In consequence of this, it is recommended to employ higher than second-order treatment of the convective and/or MHD- FHD terms in the system of governing equation in order to overcome the oscillatory numerical behaviour of the stiff MHD and FHD source terms at high dimensionless magnetic numbers.

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observed depends **non**-monotonically on this geometrical parameter. In addition, we 450 were able to show that vortex formation occurs even under creeping ﬂow conditions, 451
despite ﬂow inertia enhancing its appearance. In contrast, the presence of the walls in 452
three-dimensional geometries has a stabilizing effect at low Re, delaying the onset of 453 these recirculations, which are no longer closed recirculations. 454

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(0)
g ′ have been plotted respectively in Figure 4 and Figure 5. The **non**-alignment function g has no effect on heat transfer. For increasing values of the shrinking parameter α , the boundary layer thickness increases and hence the heat transfer rate decreases as depicted in Figure 6.

4.1 Rectilinear and **non**-rectilinear **flows** in a straight duct
MQs are employed to compute ﬂows of a power-law ﬂuid through circular and **non**-circular tubes, while TPSs are used to simulate the ﬂow of a viscoelastic ﬂuid through a square duct. For simplicity, in the MQ-based approximation scheme, the factor β is chosen to be unity. To enhance numerical stability, a stress-splitting formulation is employed. The resulting nonlinear system of equations is solved by a decoupled approach, where a Picard- type iteration scheme is utilized to render nonlinear terms linear. At each iteration, a perturbed **Newtonian** problem and a constitutive model are solved in two sequential steps.

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accepted assumptions and limitations of use. As a rule, a suitable model is chosen with due consideration of available information about the flow structure and the required description accuracy. The methods of solving problems with moving interfaces are reviewed most comprehensively in [1]. Nowadays, the high efficacy and implementation simplicity of the volume-of-**fluid** method (VOF) [2] make it the most popular one among the algorithms of continuous volume markers. It has proven itself in the calculation of **flows** with a free surface [1, 3]. The idea of the method is that a **fluid** and a gas are regarded as a single two- component medium; the spatial distribution of phases within the calculation domain is determined by the special marker function F(x,y,z,t). The volumetric fraction of the liquid phase in the calculation cell is as follows: F(x,y,z,t)=0 for an empty cell, F(x,y,z,t)=1 for a cell entirely filled with liquid, 0
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porous media. Hooper et al. (1993) have studied mixed convection from a vertical plate in porous media with surface injection or suction. Choi (1995) has studied enhancing thermal conductivity of fluids with nanoparticles in developments and applications of **non**-**Newtonian** **flows**. Das et al. (1996) have studied radiation effects on flow past an impulsively started infinite isothermal plate. Yih (1998) has studied Coupled heat and mass transfer in mixed convection over a wedge with variable wall temperature and concentration in porous media: The entire regime. Magyari et al. (1999) have studied heat and mass transfer in the boundary layers on an exponentially stretching continuous surface.

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To the best of our knowledge, this is the first attempt to apply this robust and highly effective analytical technique as well as highly accurate method to study the **non**-**Newtonian** **flows** in this geometry. The results presented in this paper will now be available for experimental verification to give confidence for the well-posedness of this **non**- linear boundary value problem.

Whereas a **Newtonian** **fluid**, under uniform stress, **flows** with uniform shear rate, complex fluids (e.g. foams, par- ticulate suspensions and smectic liquid crystals) can ex- hibit coexistence of macroscopic regions with distinct shear rates [8–15]. This phenomenon, known as shear banding, has been explained in terms of various models with **non**-trivial stress fields [16–19]. The definition of shear flow requires at least two spatial dimensions. In the 1D XY model, if we represent the unit vectors s j ly- ing in a plane perpendicular to the one spatial dimension of their locations j, then we can interpret their rotational velocity differences as shear rates. Surprisingly, we find the 1D XY model exhibits shear-banding and other **non**- **Newtonian** phenomena routinely seen in soft matter.

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Introduction
Comments At constant Re recirculation zone larger and reattachment length longer for small El. ‘Overshoot’ seen in **non**- **Newtonian** case. Good agreement with experimental results observed. Corner vortex decreases with increasing flow rate. No evidence of lip vortex. Experimental results are quantitatively reproduced f the **fluid** is well characterised. Shear-thinning reduces the intensity of the singularity near the re-entrant corner. Square/square contraction shows correlation between vortex activities and extentional properties. Planar expansion does not show the same correlation. As De increases the lip vortex also increases and the corner vortex decreases. At De = 5 the vortices merge and the lip vortex is dominant. Vortex behaviour more pronounced in size and strength in axisymmetric contractions, **fluid** **flows** through central ‘funnel’ that elongates as vortex grows. No upper limit toDe found for exponential PTT **fluid**. Approximate limit of De~200 for linear PTT. Previous results had achievedDe~9 for linear PTT andDe~35 for exponential PTT. C – contraction, E – expansion, C-E – contraction-expansion
**Fluid** Viscoelastic Oldroyd-B Shear-thinning, UCM, Boger and PTT Viscoelastic, PTT and UCM UCM **fluid** Oldroyd-B **fluid** Oldroyd-B, PTT
Ratio 1:2 4:1 4:1 4:1 4:1 4:1 4:1

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users particularly on roads where connection between motorized and **non**-motorized traffic is high, such as residential roads, school and community zones. Even though speed limit signs are located, much is left to the ethics of drivers whether they should abide by them. Hence, controlling vehicular speed is an important outcome in traffic management. One way of controlling speed is to use speed breaker which produces distress while driver experiences while crossing over it. It plays a decisive role in implementing speed limits, thereby preventing over speeding of vehicles. It considerably assists to the overall road safety objective through the prevention of accidents. **Newtonian** **fluid** speed breaker reduces the speed of any over speeding vehicles travelling on a roadway. It is formed by at least one hollow band of ductile material. Each vessel is filled with a dilatants shear-thickening **fluid**. If the vehicle travels at a low speed the **fluid** has a low viscosity and the strip is easily deformed, whereas if the speed of the vehicle is high the viscosity of the **fluid** is high and as a result has great resistance to deformation, thus forming a rigid obstacle to the movement of the vehicle. Drivers must slow down when driving over the conventional speed breaker to prevent damage to their vehicle. However, the **Non** **Newtonian** **fluid** speed breaker is sensitive to the speed of the vehicle. The vehicle needs to slow down only if it is over speeding.

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there is an additional diculty in the present case because of the nonlinear dependence of the viscosity on shear rate. The presence of the nonlinearity in expression (3) would make the derivation of the desired ordinary dierential equations governing the coecients extremely dicult (if at all possible) if the projection were attempted. In order to make the problem more tractable, we carry out the binomial expansion in (3), and take E small enough to guarantee convergence. The present formulation and subsequent numerical results are thus limited to ows at moderately small shear rate or relaxation time. This assumption is not unreasonable since, although large strains may be present during thermal convection, only small shear rates are usually involved. Thus, keeping terms to O ( E 2 ), the binomial expansion of Eq. 3, and in a manner similar to the case of a **Newtonian**

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Examples of shear thickening fluids include quicksand and cornstarch. When the index is less than one, the **fluid** is considered shear thinning and the viscosity decreases with shear rate. Examples of common shear thinning fluids include shaving cream, ketchup and nail polish. Since this problem deals with three-dimensional flow, it is necessary to analyze the shear stress and shear rate as a tensor. Thus, state of stress should be taken across a cube of the material, resulting in a square matrix with dimension three. For the purpose of this analysis all fluids were assumed to be incompressible. The normal stresses can be negated, and only the shear stresses remain.

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– The normal direction n points from the inside to the outside and the traction is the force per unit area exerted on the **fluid** inside by the **fluid** on the outside.
Normal stresses are negative in compression for us – the usual fluids convention

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