Computational Fluid Dynamics Multiple Choice Questions on “Partial Differential Equation”.
1. Which of these models of fluid flow give complete partial differential equations directly?
a) Finite control volume moving along with the flow
b) Finite control volume fixed in space
c) Infinitesimally small fluid element fixed in space
d) Infinitesimally small fluid moving along with the flow
Answer: c
Clarification: Infinitesimally small fluid element gives partial differential equations. When they are fixed in space, the equations are directly in partial differential form and there will be no need for changing a substantial derivative into partial differentials.
2. Where do we encounter partial differential equations in CFD?
a) Physical models
b) Assumptions
c) Governing equations
d) Discretized equations
Answer: c
Clarification: The governing equations of CFD are in partial differential form. This is because the flow variables depend upon four independent variables (three spatial coordinates and one time point). When a flow variable is differentiated with respect to one of the independent variables, the others are kept constant.
3. What is the method used in CFD to solve partial differential equations?
a) Variable separation
b) Method of characteristics
c) Change of variables
d) Discretization
Answer: d
Clarification: In CFD, partial differential equations are discretized using Finite difference or Finite volume methods. These discretized equations are coupled and they are solved simultaneously to get the flow variables.
4. After discretizing the partial differential equations take which if these forms?
a) Exponential equations
b) Trigonometric equations
c) Logarithmic equations
d) Algebraic equations
Answer: d
Clarification: After discretization, the partial differential equations become algebraic equations with the flow variables as the unknowns which are then solved using some iterative method.
5. These are essential for solving partial differential equations.
a) Boundary conditions
b) Physical principle
c) Mathematical model
d) Algebraic equations
Answer: a
Clarification: The analytical solutions of partial differential equations depend upon boundary conditions and this is employed in CFD to some extent. Though the same PDE is solved, the solutions may differ based on the boundary conditions.
6. Find the order of the continuity equation for steady two-dimensional flow.
a) 1
b) 0
c) 2
d) 3
Answer: a
Clarification: Continuity equation for steady two-dimensional flow is given by (frac{partialrho u}{partial x}+frac{partial rho v}{partial y}). This is a first order partial differential equation.
7. The y-momentum equation falls into which of these types of PDEs?
a) 1-D first order equation
b) 2-D second order equation
c) 2-D first order equation
d) 1-D first order equation
Answer: b
Clarification: The y-momentum equation is
(frac{partial (rho u)}{partial t}+frac{partial (rho u^2 )}{partial x}+frac{partial (rho uv)}{partial y}=-frac{partial p}{partial x}+frac{partial}{partial x}[(nabla.vec{V})+2mufrac{partial u}{partial x}]+frac{partial}{partial y}[mu(frac{partial u}{partial y}+frac{partial v}{partial x})]+rho f_x)
When expanded it gets second derivatives and hence it is a second order equation of x and y dimensions.
8. Which of these does not come under partial differential equations?
a) Laplace’s equation
b) Equations of motion
c) 1-D wave equation
d) Heat equation
Answer: b
Clarification: Equations of motion comes under ordinary differential equations. Laplace’s equation, wave equation and heat equations are all partial differential equations.
9. Which of these is not an analytical method to solve partial differential equations?
a) Change of variables
b) Superposition principle
c) Finite Element method
d) Integral transform
Answer: c
Clarification: Change of variables, Superposition principle, and Integral transform are all analytical methods. It is difficult to solve partial differential equations using analytical methods. Finite Element method is a numerical method to solve partial differential equations.
10. Linear partial differential equations are reduced to ordinary differential equations in which of these methods?
a) Change of variables
b) Fundamental equations
c) Superposition principle
d) Separation of variables
Answer: d
Clarification: In the separation of variables method, linear partial differential equations are reduced to ordinary differential equations and then these ODEs are solved.
11. The governing equations of CFD are ____________ partial differential equations.
a) Linear
b) Quasi-linear
c) Non-linear
d) Non-homogeneous
Answer: b
Clarification: The governing equations of CFD are quasi-linear partial differential equations. They have their highest order terms linearly and the coefficients are functions of the dependent variables itself.
12. Which of these is a quasi-linear partial differential equation?
a) (frac{partial^2 u}{partial x^2}+frac{partial^2 u}{partial y^2}=0)
b) (frac{partial^2 u}{partial x^2}+a(x,y)frac{partial^2 u}{partial y^2}=0)
c) (frac{partial u}{partial x}frac{partial ^2 u}{partial x^2}+frac{partial u}{partial y}frac{partial^2 u}{partial y^2}=0)
d) ((frac{partial ^2 u}{partial x^2})^2+frac{partial^2 u}{partial y^2}=0)
Answer: c
Clarification: (frac{partial u}{partial x}frac{partial ^2 u}{partial x^2}+frac{partial u}{partial y}frac{partial^2 u}{partial y^2}=0) is a quasi-linear partial differential equation. (frac{partial^2 u}{partial x^2}+frac{partial^2 u}{partial y^2}=0) and (frac{partial^2 u}{partial x^2}+a(x,y)frac{partial^2 u}{partial y^2}=0) are linear partial differential equations. ((frac{partial ^2 u}{partial x^2})^2+frac{partial^2 u}{partial y^2}=0) is non-linear.