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1. Two questions occurred to me on first seeing this diagram which incidentally is taken from: Fluid Mechanics Fundamentals and Applications Y. A. Cengel and J. M. Cimbala. McGraw Hill If the Navier-Stokes equations really are the corner stone of Fluid Mechanics shouldn't they be inserted as part of the Fluid Mechanics foundation course. Now come I had manages to teach fluid mechanics all these years without needing to make use of these strange equations. A talk by T. Swann
2. z2 z1 F2 = P δaR = gz2aR z2 z Area of surface elementδaS Mathematics From Fluids -1 Dr Anthony B Swann Dept of CEE University of Auckland, Auckland, New Zealand Abstract Integration over the Region Integration over the Surface Mathematics From Fluids - A review of Gauss' Theorem The successful use Computational Fluid Dynamic software such as Flexpde5 to model complex flow problems requires an appreciation of three dimensional field theory and in particular of Gauss' theorem. Traditionally, Gauss' theorem has been derived mathematically from a consideration of a general three dimensional field. The aim of this seminar is to explore an alternative "Fluid Mechanics" approach to teaching Gauss' Theorem in which a consideration of the well known principle due to Archimedes provides the context. Once established, the approach can be readily adapted to the application of the conservation laws of mass and momentum to fluid flows as well as to a derivation of the well known Navier–Stokes equations. Diagram 4 shows the hydrostatic pressure force acting on a small element of the surface of the body resolved in the vertical direction. Integrating this over the whole of the surface gives the buoyancy force. Also since and the total Diagram 3 shows how integration over the region is used to find the volume of the body. Diagram 1 shows the force of the water on the upper surface of the discretised body pressure change can be found by integrating the vertical gradient of the pressure from the bottom to the top of the column it follows that z1 Taking the water surface as datum δF = -(P δaS)n k n F1 = - PδaR = -gz1aR γ z1 z2 and hence that Column area =δaR N.B. -ve because a positive buoyancy force requires a negative gradient or as a volume integral and hence This establishes an equivalence relation between the region and the surface integral, namely: which reduces to Region Ron a horizontal plane which is the shadow of the object Conclusions Vertical height of the elementary volume =z2 - z1 Therefore volume of the element =δ = (z2 - z1)δaR Total Force is obtained by integrating over the region Introduction Integration over the Volume Though rarely used, it is the region integration that provides the link between the surface and volume integrations and hence provides the theoretical basis for Gauss' theorem. The situation is conveniently illustrated by the diagram: The total volume of the body Archimedes principle deals with the upthrust or buoyancy force which acts on a body immersed in a fluid. To establish a link with Gauss' theorem we will reinterpret it as the resultant force acting on a body immersed in a fluid when linearly varying pressure field (the hydrostatic field) exists in the fluid. Archimedes principle states that: Diagram 2 shows the force of the water on the lower surface of the body Diagram 5 is similar to diagram 3.It shows a small element of the elementary column. By combining an inner integration w.r.t z with an outer regional integrationwe can integrate over the entire volume of the body. Hence So far so good but instead of stopping here we consider calculation the buoyancy force by integrating over the surface of the body. To do this we first introduce two unit vectors k in the vertical direction and n in the direction normal to the surface on the fluid side. (called the outwardunit normal). Bonaventura Cavalieri When a body is placed in a fluid it experiences an upward force which is equal the weight of the fluid displaced This result is at the very heart of our use of control volumesin the study fluid dynamics of flows. This is because it shows that, in analysing the dynamics of a flow, we can, in many cases, consider only the flow conditions that exist at the surface of the control volume and can ignore those that exist within. born 1598, Milan [Italy]died Nov. 30, 1647, Bologna, Papal States Cavalieri, in 1629, developed his method of indivisibles, a means of determining the size of geometric figures similar to the methods of integral calculus. Resultant buoyancy force This leads to the double integral expression Further Work  ABS October 2007
3. Region Ry Region Rx Control Surface Control Volume Region Rz Region Mathematics From Fluids - 2 Dr Anthony B Swann Dept of CEE University of Auckland, Auckland, New Zealand Abstract Integration over the Region Integration over the Surface Application to the mass conservation law of Fluid Mechanics The application of conservation laws to fluid flows requires the use of a control volume to define the extent of the body of fluid under consideration. Such a control volume can be conceptualised by first considering Archimedes's submerged body and then removing its substance. This leaves the essence of the body in much the same way that the removal of the "Cheshire cat" leaves only the "Grin". Being only essence, the flow is not affected by the presence of the C.V. We can however use it to determine whether the C.V. has a net inflow or a net outflow or whether the flow is balanced. Diagram 9 shows a patch of the surface above a single tread. Here n is the unit normal to the patch and V is the local velocity vector. The expressions for the other directions are. To get the total flow crossing the surface we must include the flows across the two sets of risers. Diagram 8 shows all three integration regions. Diagram 6 is based on diagram 3 except that body is now an insubstantial shape (the C.V.). The moving fluid, now passes through it unhindered. The moving fluid constitutes a 3-D flow field described by lines called streamlines which are tangential to the velocity vector at every point along their length. n V Surface Patch Rate of flow through this surface element Now we can add the three expressions to get the net outflow from the U.C.V. The result is: and hence This establishes an equivalence relation between the region and the surface integrals, namely: We may note that the expression Introduction is called the divergence of the vector field div(V) and hence that The continuity law of incompressible fluid flow effectively says that in the absence of any sources or sinks, any net inflow to the C.V. through one part of its surface must be just balanced by a net outflow through another part of its surface. So if we first divide the surface into small elements then multiply the area of each element by the normal component of the flow velocity where the flow crosses the element and finally sum the results for all the elements then the result should be zero. This concept forms the basis of an important theorem called the "Divergence Theorem". It is in effect Gauss' Theorem applied to the continuity problem. Integration over the Volume The result for the three regions is: Diagram 7 shows flow through a single tread element of area aron the upper control surface. Diagram 10 is similar to diagram 6 except that it shows a small element of the elementary column. Now since integrating the gradient of Vz over the height of the column gives the total change in Vz across the column we can use it to substitute for the expression under the regional integral in the previous result. This leads to a double integral with the inner integration being performed over the entire height of the column. V Conclusions Vz Vy Again we see that the regional integration provides the link between the surface and volume integrations and hence it is another example of Gauss' theorem at work. The situation in this case is: Area = δaR Next we calculate the flow by integrating over the surface of the C.V. Vx Gauss z The flow through this element Qu is given by: z2 born 1777, Braunschweig, [Germany]died 1855, Göttingen, Hanover, {Germany] Carl Friedrich Gauss, in 1813, developed the divergence theorem (also called Gauss' Theorem) which states that "The net flux of any vector field through any closed surface is equal to the divergence of the field integrated throughout the volume enclosed by the surface". z1 The net outflow flow through all the treads is obtained by integrating over the entire region for both upper and lower control surfaces and subtracting thus: The divergence of the velocity field is a measure of the overall rate at which fluid is either approaching or retreating from a point in the flow. The resulting expression for the z-direction velocity component is:  ABS January 2008
4. Mathematics From Fluids - 3 Dr Anthony B Swann Dept of CEE University of Auckland, Auckland, New Zealand Abstract Volume Integration Gravity Force Integration over the Surface Viscous Force Grad(Vz2)z2 Application to the linear momentum conservation law of Fluid Mechanics The linear momentum conservation law is simply an extension of the law of continuity with the linear momentum vector replacing the velocity vector. The U.C.V. is used as before to control the calculation. In all other respects the derivation is the same According to Newton, Viscous forces in a fluid can be characterised as "The resistance which arises from the lack of slipperiness originating in a fluid which, other things being equal, is proportional to the velocity by which the parts of the fluid are being separated from each other. "In other words, viscous stress is proportional to the rate of strain of the fluid. The rate at which fluid particles are separating from each other at the tread surface is simply Vz/z or in other words grad(Vz)k.The viscous force on the step is obtained by multiplying by the tread area aR and the coefficient of proportionality .. Of course in this case other forces will be acting on the risers. Similarly for the x and y direction components To handle the forces active on the control surface we turn to our poster 1 template. Further, since gravity is a body force it is logical to consider the volume integral from our template which is: When integrated over the entire surface, these expressions give the change of momentum of the component in each direction. Further, by the momentum principle this change will be equal to the sum of all external forces resolved in that direction. thus: z2 and z Grad(Vz1)z1 However, for bodies on or near the earths surface, the gradient of the gravity field will be sensibly constant, directed in the vertical and equal to g times the mass of the body . Thus we can substitute ρg for -grad(P) to get: Since these are all z-direction components of the viscous force the total component will be Introduction Poster 1 introduced Gauss, theorem as it applies to forces acting at the surface of a body (real or imagined) while poster 2 applied Gauss' theorem a quantity (namely volume) which the field is convecting through an imaginary body (referred to as the Control Volume). In this poster we will use Gauss' theorem to apply the momentum principle to the C.V. using the previous applications as templates. As momentum is convected by the flow we will use the surface integral from poster 2 diagram 9 as our template and simply replace volume by momentum. Thus: z1 as the gravity force acting on the fluid inside the C.V., the other components being zero. A more general result is obtained by letting X, Yand Z be the component of the gravity force per unit mass in the x, y and z directions respectively. This gives three components of the gravity force namely: Region Area element daR Integration over the Volume Which is simply: However, our template also tells us that : or: Similar results for the other velocity component scalar fields leads to the final result : Which implies that for instance for the x-direction momentum : Diagram 11 is similar to diagram 10 except that the velocity vectors must be replaced by the component gradient vectors since these give the magnitude and direction of the viscous stress components. This lets us assess the total for the C.V. For instance, for the z-direction component, of the gradient we have : Pressure Force Becomes: Expanding this expression we get: The pressure force comes directly from the poster 1 template except that we need to consider each coordinate direction separately. For instance, for the x-direction we can write: Conclusions Where  Mx is the x-direction component of the momentum convected through the surface patch. In this poster we have obtained the ingredients required to apply the momentum principle to the C.V. The result for say the z-direction in volume integral form is: Similarly: Again the double integration can be replaced by an integration over the volume of the C.V. giving: But conservation of mass requires that the divergence term be zero leaving only the 2nd term whence: Newton Born 1642, at Woolsthorpe in Lincolnshire (UK)died 1727, Kensington, London In Book II of his Principia, Newton laid the foundations of a theory of fluid mechanics which included the effects of viscosity on a flow.  ABS January 2008
5. Mathematics From Fluids - 4 Dr Anthony B Swann Dept of CEE University of Auckland, Auckland, New Zealand Abstract RHS Terms The Constitutive equations Part 2. The constitutive equations relate the forces acting on a fluid particle to the state of stress generated inside the fluid. To derive the constitutive equations first consider the net forces acting on the surface of an oblong fluid particle at point P: In part 1, the formula embodying the application of the momentum principle to the Universal Control Volume is presented in different forms . Part 2 investigates the implications of the formula for the gravity, pressure and viscous forces which must act on a fluid particle having the same shape as the Control Volume used previously. The result is a set of equations known collectively as The Constitutive Equations Now because, for an incompressible flow, μ div(V) is zero we may add it to the LHS of the equation The result is The first term on the RHS which is the gravity force term becomes: Thus: Introduction The second or pressure term becomes: Associating the terms in brackets, the equation reduces to Part 1. Versions of the basic momentum principle formula: P From the diagram Conclusions Combining terms having a common denominator we get : We might for instance write the formula for a control volume of infinitesimal volume d we can then omit the volume integrals thus: While the third or viscosity term becomes: : The Navier-Stokes equations are derived from the volume integral form of the momentum principle equations applied to an elementary oblong control volume. This is in contrast with the conventional method which is to use the surface integral form of the equations and integrate over a piecewise continuous control surface chosen to make the integration process as straight forward as possible. The practical difference is that the Navier-Stokes solution will give information about velocities, pressures etc everywhere in the flow field. In other words to “model” the flow. Conventional momentum principle solutions yield results only bulk results which apply to the whole control volume. The constitutive equations allow us the use velocity gradient information from the Navies-Stokes solution to find the distribution of normal and shear forces throughout the field. If we now equate the expressions for say Fx we get: The above can now be expanded in partial differential form. In this case the LHS becomes: Then integrating both sides of each equation w.r.t. its denominator gives the constitutive equations: Navier, Claude-Louis Back substitution then gives the well known Navier-Stokes equation (see background): Borne 1785 in Dijon France died 1836 in Paris Although he made great contributions relating to elasticity and stress in solids, His major contribution however remains the Navier-Stokesequations(1822), central to Fluid Mechanics. the other six constitutive equations can be easily generated by the cyclic interchange of suffices technique.  ABS January 2008