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The first viscous terms in
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Fluid Mechanics: Fundamentals and Applications
- the differential equation for conservation of mass, the continuity equation. In cylindrical coordinates, and for steady flow, 1/ r ∂(rur) /∂r + 1/ r ∂u? /∂? + ∂uz /∂z = 0 Write the primary dimensions of each additive term in the equation, and verify that the equation is dimensionally homogeneous. Show all your work.arrow_forwardQ1:: Explain all the terms of the Continuity Equation and their physical meanings with the help of examples.arrow_forwardSimplify the Navier–Stokes equation as much as possible for the case of incompressible hydrostatics, with gravity acting in the negative z-direction. Begin with the incompressible vector form of the Navier–Stokes equation, explain how and why some terms can be simplified, and give your final result as a vector equation.arrow_forward
- volumetric strain rate as the rate of increase of volume of a fluid element per unit volume. In Cartesian coordinates we write the volumetric strain rate as 1/ V DV/Dt = ∂u/∂x + ∂?/∂y + ∂w/∂zWrite the primary dimensions of each additive term, and verify that the equation is dimensionally homogeneous. Show all your work.arrow_forwardHow could the fluid flow variable be introduced into the following simplified Navier-Stokes equation? If you consider: -The fluid is incompressible P dv at µAv + VP = 0 (Ctrl) -arrow_forwardA proposed three-dimensional incompressible fl ow fi eldhas the following vector form:V = Kxi + Kyj - 2Kzk( a ) Determine if this fi eld is a valid solution to continuityand Navier-Stokes. ( b ) If g = - g k, fi nd the pressure fi eldp ( x , y , z ). ( c ) Is the fl ow irrotational?arrow_forward
- An incompressible velocity field is given by u=a(x°y²-y), v unknown, w=bxyz where a and b are constants. (a)What is the form of the velocity component for that the flow conserves mass? (b) Write Navier- Stokes's equation in 2-dimensional space with x-y coordinate system.arrow_forwardfluid mechanics experts please solve with details and give reasons for steps if needed. parts a b and c are solved . in this question i want the last two questions. please use the P that isobtained from g part in h part with P0 as example. σxx= -P + τxx = 0.125 ρ - P0 + 2μ = 0.75 Paarrow_forwardThe Stokes-Oseen formula for drag force on a sphere at low speed is given asD = 3dV +916V 2d2, where D is drag, V is velocity, is density, d is the sphere diameter, and is the viscosity coe¢ cient.(a) Using the formula given, Önd the dimensions of the viscosity coe¢ cient. (Donít simply look upthe dimensions; use the formula to show them.) Be sure to show your work. Find the primaryunits of viscosity in SI and British units.(b) Verify that the Stokes-Oseen formula is dimensionally homogeneous.arrow_forward
- 4s-1, Given the velocity field V = Axî – Ayĵ, where A %3D (a) Sketch the velocity field. (you can do this by hand or use software of your choice)arrow_forwardhe velocity at apoint in aflued for one-dimensional Plow wmay be aiven in The Eutkerian coordinater by U=Ax+ Bt, Show That X Coordinates Canbe obtained from The Eulerian system. The intial position by Xo and The intial time to zo man be assumeal · 1. x = foxo, yo) in The Lagrange of The fluid parficle is designatedarrow_forwardEngine oil at 60°C rotates as a rigid body about the z-axis in a spinning cylindrical container. There are no viscous stresses since the water moves as a solid body; thus the Euler equation is appropriate. (We neglect viscous stresses caused by air acting on the water surface.) Integrate the Euler equation to generate an expression for pressure as a function of r and z everywhere in the water. Write an equation for the shape of the free surface (zsurface as a function of r). (Hint: P = Patm everywhere on the free surface. The flow is rotationally symmetric about the z-axis.)arrow_forward
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