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FLUID MECHANICS FORMULA SHEET FLUID PROPERTIES ρwater = 1000 kg/m3 ρair = 1.23 kg/m3 ρmercury = 13600 kg/m3 = ρg g = 9.81 m/s2 PHYSICAL PROPERTIES Newton’s Law of Viscosity Capillary Rise in Circular Tubes du dy ( / ) h 2 cos r Bulk Modulus of Elasticity Ev Compressibility dp dp d / 0 d / K d 1 d 1 dp dp HYDROSTATICS Pressure Distribution Hydrostatic Force on a Plane Surface Point of Application of FR Buoyancy Force P = h FR = PcA xR=xc+Ixyc/ycA yR=yc+Ixc/ycA FB = f sub 4R/(3) C x y a b C C 4R/(3) 4R/(3) d a C C R R x y (b+d)/3 b x y 4 3 r 3 1 r 2 h 3 sphere cone Rectangle Triangle Circle Semicircle A Ixc ba (ba3)/12 Iyc (ab3)/12 Ixyc 0 R2 R4/ 4 R4/ 4 0 ab/2 ba3/36 ba2(b- R2/2 0.1098R4 Quarter circle R2/4 0.05488R4 0.3927R4 0.05488R4 0 -0.01647R4 a/3 KINEMATICS Position Vector r xi yj zk Velocity V ui vj wk r t Equation of Pathline Equation of Streamline dx dy dz dt u v w dx dy dz ds dt u v w V a Acceleration Vorticity DV V ( V )V Dt t ( xV ) Del Operator i j k x y z Change Equation Select an equation to solve for a different unknown Solve for flow rate Solve for flow area Solve for flow velocity Where Q = flow rate A = flow area v = flow velocity Volumetric Dilatation Rate 1 d( ) V dt Note Bernoulli Equation Assumes: 1. 2. 3. 4. flow is streamline steady state flow inviscid fluid incompressible fluid Select an equation to solve for a different unknown Solve for head loss Solve for static head or elevation at point 1 Solve for pressure at point 1 Solve for velocity at point 1 Where h Z P V p g Q = = = = = = = head loss static head or elevation Pressure fluid velocity fluid density acceleration of gravity flow rate Quantity Symbol Object Units pressure p scalar N/m2 velocity v vector m/s density scalar kg/m3 viscosity scalar kg/m-s body force b vector N/kg time t scalar s barotropic compressible density, incompressible inviscid A barotropic fluid is one whose pressure and density are related by an equation of state that does not contain the temperature as a dependent variable. Mathematically, the equation of state can be expressed as p = p() or = (p). A fluid flow is compressible if its density changes appreciably (typically by a few percent) within the domain of interest. Typically, this will occur when the fluid velocity exceeds Mach 0.3. Hence, low velocity flows (both gas and liquids) behave incompressibly. The mass of fluid per unit volume. For a compressible fluid flow, the density can vary from place to place. An incompressible fluid is one whose density is constant everywhere. All fluids behave incompressibly (to within 5%) when their maximum velocities are below Mach 0.3. Not viscous. irrotational An irrotational fluid flow is one whose streamlines never loop back on themselves. Typically, only inviscid fluids can be irrotational. Of course, a uniform viscid fluid flow without boundaries is also irrotational, but this is a special (and boring!) case. laminar (nonturbulent) An organized flow field that can be described with streamlines. In order for laminar flow to be permissible, the viscous stresses must dominate over the fluid inertia stresses. Mach Mach number is the relative velocity of a fluid compared to its sonic velocity. Mach numbers less than 1 correspond to sub-sonic velocities, and Mach numbers > 1 correspond to super-sonic velocities. Newtonian A Newtonian fluid is a viscous fluid whose shear stresses are a linear function of the fluid strain rate. Mathematically, this can be expressed as: ij = Kijqp*Dpq, where ij is the shear stress component, and Dpq are fluid strain rate components. perfect A perfect fluid is defined as a fluid with zero viscosity (i.e. inviscid). rotational A rotational fluid flow can contain streamlines that loop back on themselves. Hence, fluid particles following such streamlines will travel along closed paths. Bounded (and hence nonuniform) viscous fluids exhibit rotational flow, typically within their boundary layers. Since all real fluids are viscous to some amount, all real fluids exhibit a level of rotational flow somewhere in their domain. Regions of rotational flow correspond to the regions of viscous losses in a fluid. Inviscid fluid flows can also be rotational, but these are special nonphysical cases. For an inviscid fluid flow to be rotational, it must be set up that way by initial conditions. The amount of rotation (called the velocity circulation) in an inviscid fluid flow is conserved, provided that the fluid is also barotropic and subject only to conservative body forces. This conservation is known as Kelvin's Theorem of constant circulation. Stokesian A Stokesian (or non-Newtonian) fluid is a viscous fluid whose shear stresses are a non-linear function of the fluid strain rate. streamline turbulent viscosity, A path in a steady flow field along which a given fluid particle travels. A flow field that cannot be described with streamlines in the absolute sense. However, time-averaged streamlines can be defined to describe the average behavior of the flow. In turbulent flow, the inertia stresses dominate over the viscous stresses, leading to small-scale chaotic behavior in the fluid motion. A fluid property that relates the magnitude of fluid shear stresses to the fluid strain rate, or more simply, to the spatial rate of change in the fluid velocity field. Mathematically, this is expressed as: = *(dV/dy), where is the shear stress in the same direction as the fluid velocity V, and y is a direction perpendicular to the fluid velocity direction.