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FLUID MECHANICS Fluid mechanics is the study of the behaviour of fluids (liquids and gases). These materials will look at the various aspects of fluid behaviour. PRESSURE DUE TO DEPTH IN A LIQUID If we consider a point below the surface of a fluid, then we can see that there must be sufficient pressure in the fluid to support the column of fluid above that point. Surface of liquids h Column of liquid Area of column The weight (W) of the column of fluid is given by the equation W = mg Where m is the mass of fluid and g is the acceleration due to gravity. Also, m = V x Where V = Volume and = Density Therefore W = V x x g The volume of the column of fluid is given by V=Axh Therefore: W = A x h x x g This weight must be supported by the pressure of the liquid, providing the force F F=PxA Where P = Pressure of fluid Therefore if W = F Axhx xg= xA Fluid/docs/Science/DC/SL Therefore h x x g = P (cancelling A) P = gh Or We can now check that the units of the equation balance N m2 = kg m3 x m s2 x m Substitute 1N = 1 Kg m s2 Kgm S2m2 = Kgm2 S2m2 cancelling Kg ms2 = Kg ms2 We can see that it is important that we use values of , g and h in base units if we are to calculate pressure in N/m2 Fluid/docs/Science/DC/SL CONTINUITY EQUATION 1 2 Flo w . If fluid flows along a pipe then the mass flow rate (m) at any point (1) is equal to the mass flow rate at any other point (2), assuming there are no leakages. . . Therefore m, = m2 If the fluid has constant density then as . m=Qx Where Q = Volumetric flow rate (m3/s) = density of fluid (Kg/m3) Q1 = Q 2 And A1 x C1 = A2 x C2 Where A = cross sectioned area And C = velocity of fluid flow . If the fluid does not have a constant density (gas) then we have m = m 2 Q1 x 1 = Q 2 x 2 And as Q = A x C A1 x C1 x 1 = A2 x C2 x 2 Fluid/docs/Science/DC/SL Bernoulli’s Equation Bernoulli’s equation states that the total energy at one point (1) of flow in a pipe is equal to the total energy at another point (2) in the pipe providing that: The flow is steady The points lie on a streamline There is no friction 1 2 Pressure Energy1 + Kinetic Energy1 + Potential Energy1 = Pressure Energy2 + Potential Energy2 P1 V1 + mC12 + mgh1 = P2 V2 +mC22 + mgh2 2 2 Where V = Volumetric flow rate Dividing through by mass gives P1 V1 + C12 + gh1 = P2 V2 + C22 +gh2 m 2 m 2 As m= V P1 + C12 + gh1 = P2 + C22 + gh2 2 2 2 1 Multiply by P1 + C12 + 1 gh1 = P2 + C22 + gh2 2 2 Or we can state P + C2 + gh = Constant 2 Fluid/docs/Science/DC/SL If we consider a constant density fluid flowing horizontally (i.e. neglecting potential energy) then we have: P1 + C12 = P2 + C22 1 2 2 Also from continuity, we have A1 C1 = A2 C2 2 From equation 2 we can see that as the area of flow decreases, the velocity increases. A1 C1 = A2 C2 Also, from equation 1 we can see that the pressure decreases. P1 + C12 = P2 + C22 2 2 Therefore, when fluid passes over a stationary body forcing the streamlines to get closer together, the velocity must increase (as the area decreases), and therefore the pressure decreases. This is how an airfoil works, the flow over the top surface is faster than the flow over the bottom surface, therefore there is low pressure on the top and an upward force is produced on the airfoil (lift) The Bernoulli equation allows us to determine how energy transfers take place between elevation, velocity head and pressure head. We can examine piping systems and determine the variation of fluid properties and the balance of energy. Sometimes the changes in the energy levels are confusing and seem to contradict common sense; students can often find the outcomes of the Bernoulli equation confusing. For example, if an incompressible fluid is flowing along a horizontal pipe which expands in diameter, then, as the area increases the velocity must decrease (continuity, Q = CA). As there is no change in potential head (horizontal pipe), the loss in velocity head must result in a gain in pressure head (conservation of energy or head). Therefore, as the velocity decreases the pressure increases. This may seem counterintuitive (surely fluid cannot flow in the direction of increasing pressure), however, we must remember that the fluid has inertia and the increase in pressure does have the effect of decelerating the fluid (F=ma), in the same way that a car moves forward when a brake (force in the opposite direction to motion) is applied. There are many fluid problems to which Bernoulli can be applied, it can be applied to problems in which more than one flow may enter or leave the system at the same time, or to series and parallel piping systems. Fluid/docs/Science/DC/SL