For solid mechanics and fluid mechanics you need to start with your three equations of evolution (mass, momentum and energy) combined with the laws of thermodynamics (energy conservation and entropy) then obtain solutions that satisfies these for your specific problem. Similarly, for heat transfer you have laws of conduction, convection, and radiation combined with the laws of thermodynamics. Therefore, you start by satisfying these equations to obtain a realistic solution for your specific problem (which is the hard part). For electricity and magnetism, you need to satisfy Maxwell's field equations. Therefore, every physical phenomenon has a group of laws that needs to be satisfied; and for a physically realistic solutions that satisfy the physics of the problem, these laws are where you start from. Every time.

I agree with Thomas Lynn Warren and will add that single variable governing differential equations will not be enough to model the systems you have described. If you are trying to build a model from first principles, the complexity can be overwhelming. Modelling the interactions of several physical systems requires multi-variable partial differential equations. Simplifying assumptions may need to be introduced, depending on the complexity of the physical variables and geometry. For example is heat or magnetic attraction coming from a single point source, or is gravity constant, or is the fluid flow laminar? Etc., etc. As an alternative, there may be commercially available software that takes many of these issues off the table.

Depending on the problem you are going to end up having to solve a non-linear coupled set of PDE's. You will most likely need a 3d PDE SOLVER (I.E. FEM, Peridynamics, finite volume, etc. You first formulate the PDE's required, then you have to sovle them (the hard part). And yes in your formulation look for any problem try and simplify it as much as possible, but yes commercial software is your best bet and that is what it is designed for, or you can go through the misery and write your own like I often do. And yes that is what Arbitrary Lagrangian Eulerian codes do, They solve multi-physics problems and have contact interfaces, Like if you want both heat transfer and solid deformation you run a heat transfer code together with a solid dynamics code and they share the energy equation, and you usually run them staggered, Plastic deformation generates heat through the Taylor-Quiney parameter, Then the heat transfer FEM does the conduction, convection, and radiation. And no these are 3d with time.

It will be a system of PDE's with 3 spatial components and a time component. A good example is just for solids and thermal effects by Camacho G and Ortiz M, Adaptive Lagrangian modeling of ballistic penetration of metallic targets. Compt Meth Appl Mech Eng 1997; 142:169-301. In this paper they derive the conservation equations and thermal equations then solves them with a solid mechanics FEM and a heat transfer FEM run simultaneously. For solids and liquids people use arbitrary Lagrangian Eulerian codes which you can buy commercially from all the major code developers. where generally the Lagrangian is for the solid and the Eulerian is for the fluid, and then there is coupling which can get quite complicated.

For each specific problem (i.e solid mechanics, heat transfer, fluid mechanics) apply the principle of virtual work to the relevant balance law for for each system being considered. From this you will get the governing equlibrium equations for the system along with the corresponding boundary conditions required to solve the problem.